Topological Quantum Information and the Jones Polynomial

TOPOLOGICAL QUANTUM INFORMATION AND THE JONES POLYNOMIAL

Louis H. Kauffman Math, UIC 851 South Morgan St. Chicago, IL 60607-7045

Arxiv: 1001.0354 Background Ideas and References Topological Quantum Information Theory (with Sam Lomonaco)

quant-ph/0603131 and quant-ph/0606114 Spin Networks and Anyonic TQC arXiv:0805.0339 Quantum Knots and Mosaics arXiv:0910.5891 arXiv:0804.4304 The Fibonacci Model and the Temperley-Lieb Algebra arXiv:0706.0020 A 3-Stranded Quantum Algorithm for the Jones Polynomial

arXiv:0909.1080 NMR Quantum Calculations of the Jones Polynomial Authors: Raimund Marx, Amr Fahmy, Louis Kauffman, Samuel Lomonaco, Andreas Spörl, Nikolas Pomplun, John Myers, Steffen J. Glaser arXiv: 0909.1672 Anyonic topological quantum computation and the virtual braid group. H. Dye and LK. Untying Knots by NMR: first experimental implementation of a quantum algorithm for approximating the Jones polynomial Raimund Marx1, Andreas Spörl1, Amr F. Fahmy2, John M. Myers3, Louis H. Kauffman4, Samuel J. Lomonaco, Jr.5, Thomas-Schulte-Herbrüggen1, and Steffen J. Glaser1

3 1Department of Chemistry, Technical University Munich, Gordon McKay Laboratory, Harvard University, 29 Oxford Street, Cambridge, MA 02138, U.S.A. 4 Lichtenbergstr. 4, 85747 Garching, Germany University of Illinois at Chicago, 851 S. Morgan Street, Chicago, IL 60607-7045, U.S.A. 5 2Harvard Medical School, 25 Shattuck Street, Boston, MA 02115, U.S.A. University of Maryland Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250, U.S.A.

A knot is defined as a closed, non-self-intersecting curve roadmap of the example #1 example #2 example #3 that is embedded in three dimensions. quantum algorithm Trefoil Figure-Eight Borromean rings example: “construction” of the Trefoil knot:

knot or link make a fuse the make it “knot” free ends “look nice” start with a rope end up with a Trefoil

J. W. Alexander proved, that any knot can be represented s1 s - as a closed braid (polynomial time algorithm) s1 1 s 1 “trace- -1 2 generators of the 3 strand braid group: s2 s1 s -1 -1 closed” 1 -1 1 s s s s s s2 1 1 2 2 braid 1 s1 -1 s1 s2 -1 s2

3 2 3 It is well known in knot theory, how to obtain the unitary matrix representation -1 -1 UTrefoil = (U1 ) U Figure-Eight = (U 2 ×U1 ) U Borrom.R. = (U 2 ×U1 ) of all generators of a given braid goup (see “Temperley-Lieb algebra” and “path

model representation”). The unitary matrices U1 and U2, corresponding to the æ ö æ iq sin(6q ) -iq ö -iq iq sin(6q )sin(2q ) generators s1 and s2 of the 3 strand braid group are shown on the left, where the unitary ç e 0 ÷ ç - e + e - e ÷ -iq ç ÷ ç sin(4q ) sin(4q ) ÷ variable “q” is related to the variable “A” of the Jones polynomial by: A = e rad. matrix U = U = The unitary matrix representations of s-1 and s-1 are given by U-1 and U-1. 1 ç ÷ 2 ç ÷ 1 2 1 2 iq sin(4q ) -iq sin(6q )sin(2q ) iq sin(2q ) -iq ç 0 - e + e ÷ ç - eiq - e + e ÷ ç sin(2q ) ÷ ç sin(4q ) sin(4q ) ÷ The knot or link that was expressed as a product of braid group generators can è ø è ø therefore also be expressed as a product of the corresponding unitary matrices.

Step #1: Step #2: Step #3: Instead of applying the unitary matrix U, we apply it’s controlled variant cU. from the 2x2 matrix U application of cU on the measurement This matrix is especially suited for NMR quantum computers [4] and other thermal state expectation value quantum computers: you only have to apply to the 4x4 matrix cU: NMR product operator I1x : of I 1x and I 1y : cU to the NMR product operator I1x and measure I1x and I1y in order to obtain controlled † † 0 U the trace of the original matrix U. . † 1 0 1 0 1 1 0 tr I 1 = 1 Â (tr{U}) unitary 1 0 cU I 1x cU = 2 1x 2 (U 0 ) 2 cU = ( 0 U ) ( 1 0 ) ( 0 U ) { } Independent of the dimension of matrix U you only need ONE extra qubit for the matrix ( 0 U ) † 1 0 U 0 U † implementation of cU as compared to the implementation of U itself. = tr I 1 = 1 Á(tr{U}) 2 (U 0 ) { 1y 2 (U 0 )} 2 The measurement of I1x and I1y can be accomplished in one single-scan experiment.

-b b -b b All knots and links can be expressed as a product of braid group generators (see I I I I above). Hence the corresponding NMR pulse sequence can also be expressed as z z z z a sequence of NMR pulse sequence blocks, where each block corresponds to the I -a I -1 p -p a I g -a -g I -1 g+p -p a -g

NMR cU cU cU cU . S 1 S 1 S 2 S 2 controlled unitary matrix cU of one braid group generator. S S S S pulse means z means y y z means y z y means y y z y a a a a sequence pJ pJ pJ pJ This modular approach allows for an easy optimization of the NMR pulse I cU cU cU I cU cU -1 cU cU -1 I cU cU -1 cU cU -1 cU cU -1 sequences: only a small and limited number of pulse sequence blocks have to S 1 1 1 S 1 2 1 2 S 1 2 1 2 1 2 be optimized. .

Comparison of experimental results, theoretical predictions, and simulated ex-

periments, where realisitic inperfections like relaxation, B1 field inhomogeneity,

and finite length of the pulses are included. . NMR experiment For each data point, four single-scan NMR experiments have been performed:

measurement of I1x , measurement of I1y , reference for I1x , and reference for I1y . If necessary each data point can also be obtained in one single-scan experiment

by measuring amplitude and phase in a referenced setting. .

Jones Polynomial Jones Polynomial Jones Polynomial The Jones Polynomials can be reconstructed out of the NMR experiments by: “Trefoil": “Figure-Eight": “Borromean rings": 3 -w(L) I(L) 2 -2 2 V (A)=(-A ) (tr{U}+A [(-A -A ) -2]) Jones L ( -4 + -12 - -16) + 8 - 4 - 3 + 2 - 1 polynomial A A A A A A 3A 2A 2 -2 -8 -4 -3 -2 -1 where: w(L) is the writhe of the knot or link L (- A - A ) + A - A - A + 3A - 2A tr{U} is determined by the NMR experiments 0 0 I(L) is the sum of exponents in the braid word + A + 4A corresponding to the knot or link L

References: 1) 1) L. Kauffman, AMS Contemp. Math. Series, 305, edited by S. J. Lomonaco, (2002), 101-137 (math.QA/0105255) 2) R. Marx, A. Spörl, A. F. Fahmy, J. M. Myers, L. H. Kauffman, S. J. Lomonaco, Jr., T. Schulte-Herbrüggen, and S. J. Glaser: paper in preparation 3) Vaughan F. R. Jones, Bull. Amer. Math. Soc., (1985), no. 1, 103-111 4) J. M. Myers, A. F. Fahmy, S. J. Glaser, R. Marx, Phys. Rev. A, (2001), 63, 032302 (quant-ph/0007043) 5) D. Aharonov, V. Jones, Z. Landau, Proceedings of the STOC 2006, (2006), 427-436 (quant-ph/0511096) 6) M. H. Freedman, A. Kitaev, Z. Wang, Commun. Math. Phys., (2002), 227, 587-622

Quantum Inf Process S. J. Lomonaco, L. H. Kauffman Keywords Quantum knots Knots Knot theory Quantum computation DOI 10.1007/s11128-008-0076-7Quantum algorithms Quantum· vortices· · · · Mathematics Subject Classiﬁcation (2000) Primary 81P68 57M25 81P15 57M27 Secondary 20C35 · · · ·

1 Introduction

The objective of this paper is to set the foundation for a research program on quantum knots.1 For simplicity of exposition, we will throughout this paper frequently use the term “knot” to mean either a knot or a link.2 In part 1 of this paper, we create a formal system (K, A) consisting of (1) A graded set K of symbol strings, called knot mosaics, and (2) A graded subgroup A, called the knot mosaic ambient group, of the group of all Quantum knotspermutations of the set of knotandmosaics K. mosaics We conjecture that the formal system (K, A) fully captures the entire structure of tame knot theory. Three examples of knot mosaics are given below: with SamuelSam J. Lomonaco Louis H. Kauffman Lomonaco · Each of these knot mosaics is a string made up of the following 11 symbols

called mosaic tiles. An example of an element in the mosaic ambient group A is the mosaic Reidemeister 1 move illustrated below: Each mosaic is a tensor product of elementary tiles. 1 A PowerPoint presentation of this paper can be found at http://www.csee.umbc.edu/~lomonaco/Lectures. html. 2 For references on knot theory, see for example [4,10,13,20].

123

Abstract In this paper, we give a precise and workable definition of a quantum knot system, the states of which are called quantum knots. This definition can be viewed as a blueprint for the construction of an actual physical quantum system. Moreover, this definition of a quantum knot system is intended to represent the “quantum embodi- ment” of a closed knotted physical piece of rope. A quantum knot, as a state of this system, represents the state of such a knotted closed piece of rope, i.e., the particular spatial conﬁguration of the knot tied in the rope. Associated with a quantum knot system is a group of unitary transformations, called the ambient group, which represents all possible ways of moving the rope around (without cutting the rope, and without letting the rope pass through itself.) Of course, unlike a classical closed piece of rope, a quantum knot can exhibit non-classical behavior, such as quantum superposition and quantum entanglement. This raises some interesting and puzzling questions about the relation between topological and quantum entanglement. The knot type of a quantum knot is simply the orbit of the quantum knot under the action of the ambient group. We investigate quantum observables which are invariants of quantum knot type. We also study the Hamiltonians associated with the generators of the ambient group, and brieﬂy look at the quantum tunneling of overcrossings into undercrossings. A basic building block in this paper is a mosaic system which is a formal (rewriting) system of symbol strings. We conjecture that this formal system fully captures in an axiomatic way all of the properties of tame knot theory.

S. J. Lomonaco (B) University of Maryland Baltimore County (UMBC), Baltimore, MD 21250, USA e-mail: [email protected] URL: http://www.csee.umbc.edu/ lomonaco ∼ L. H. Kauffman University of Illinois at Chicago (UIC), Chicago, IL 60607-7045, USA e-mail: [email protected] URL: http://www.math.uic.edu/ kauffman ∼ 123 Quantum knots and mosaics

Here is yet another way of ﬁnding quantum knot invariants: (n) Theorem 3 Let Q , A(n) be a quantum knot system, and let ! be an observable K on the Hilbert space (n). Let St (!) be the stabilizer subgroup for !, i.e., ! K " 1 St (!) U A(n) U!U − ! . = ∈ : = # $ Then the observable

1 U!U − U (n)/St(!) ∈A% is a quantum knot n-invariant, where U!U 1 denotes a sum over a U A(n)/St(!) − complete set of coset representatives for the∈stabilizer subgroup St (!) of the ambient & group A(n). 1 Proof The observable g (n) g!g− is obviously an quantum knot n-invariant, since ∈A 1 1 1 g# g (n) g!g− g&#− g (n) g!g− for all g# A(n). If we let St (!) ∈A = ∈A ∈ | | denote' the order of (St (!) , and if we let c1, c2, . . . , cp denote a complete set of & | | & p 1 1 coset representatives of the stabilizer subgroup St (!), then j 1 cj !cj− St(!) 1 = = | | g (n) g!g− is also a quantum knot invariant. ∈A & $% &We end this section with an example of a quantum knot invariant: Example 2 The following observable ! is an example of a quantum knot 4-invariant:

Remark 6 For yet another approach to quantum knot measurement, we refer the reader toThisthe brief obserdiscussionvableon quantum isknot a tomographquantumy found knotin item ( 11)ininvariantthe conclusion of this paper. for 4x4 tile space. Knots have characteristic 4 Conclusion: Openinvariantsquestions and infutur nxne directions tile space. There are many possible open questions and future directions for research. We mention only a few. (1) WSuperpositionshat is the exact structure of tofhe ambient combinatorialgroup A(n) and its directknotlimit conﬁgurations are seen as quantum A lim A(n). = states.−→ Can one write down an explicit presentation for A(n)? for A? The fact that the ambient group A(n) is generated by involutions suggests that it may be a Coxeter group. Is it a Coxeter group?

123 The Grand Generalization Universe as a Quantum Knot: Self-Excited Circuit Producing its Own Context The Wheeler Eye Papers on Quantum Computing, Knots, Khovanov Homology and Virtual Knot Theory arXiv:1001.0354 Title: Topological Quantum Information, Khovanov Homology and the Jones Polynomial Authors: Louis H. Kauffman arXiv:0907.3178 Title: Remarks on Khovanov Homology and the Potts Model Authors: Louis H. Kauffman arXiv:0173783 Title: Introduction to virtual knot theory Author: Louis H. Kauffman Quantum Mechanics in a Nutshell

0. A state of a physical system corresponds to a unit vector |S> in a complex vector space. 1. (measurement free) Physical processes are modeled by unitary transformations U applied to the state vector: |S> -----> U|S> 2. If |S> = z |1> + z |2> + ... + z |n> 1 2 n in a measurement basis {|1>,|2>,...,|n>}, then measurement of |S> yields |i> with probability |zi |^2. |0> |1> |0> |1> -|1>

|0> |0> |1> |0>

Mach-Zender Interferometer

|0> |1> |1> |0>

|0> -|1>

1 1 0 1 H = /Sqrt(2) M = [ 1 -1 ] [ 1 0 ] 1 0 HMH = [ 0 -1 ] Hadamard Test

|0> H H Measure frequency of |phi> U |0>

|0> occurs with probability 1/2 + Re[]/2. May 16, 2010 0:40 WSPC/INSTRUCTION FILE QITTJMP

May 16, 2010 0:40 WSPC/INSTRUCTION FILE QITTJMP

Instructions for Typing Manuscripts (Paper’s Title) 37

Instructions for Typing Manuscripts (Paper’s Title) 37 It follows from this calculation that the question of computing the bracket poly- nomialItffoorllotwhes fcrolomsuthreis ocaflctuhleattiohnrethe-astttrhaenqduebsrtaioind obf cisommpauttihnegmthaetibcraalclkyeteqpouliyv-alent to thenpormoibalefmor othfeccolmospuruetionfgthtehethtrreae-csetroanf dthberauidnibtaisrymamthaetmriaxtiΦca(lbly).equivalent to the problem of computing the trace of the unitary matrix Φ(b). The Hadamard Test The Hadamard Test In Ionrdoerdretroto(q(quuaannttuumm) ccoommppuutetethtehteratcreacoef aofunaituarnyitmaraytrimx aUt,rioxneUc,anonuesecan use thetHheaHdaamdaamradrdtetsetstttoo oobbtaiinntthheeddiaigaognoanl amlamtraixtreilxemeelenmts enψtUs ψψ Uof Uψ. TohfeUtr.aTcehe trace ! | |! "| | " is thisetnhetnhethseusmumoof ftthheessee maattrrixixeleelmemenetns tass aψs ψrunsruonvesroavnerorathnonoorrtmhoanl boarmsisaflobr asis for | "| " thetvhecvteocrtosrpsapcaec.e.WWeefifirrsst oobbttaainin 11 1 1 ++ ReRψe UψψU ψ 22 2 2 ! |! || "| " as an expectation by applyingApplthe Hy aHadamardamard gda tGatee H as an expectation by applying the Hadamard gate H 1 H 0 = 1( 0 + 1 ) H| "0 =√2 | (" 0 |+" 1 ) | " √2 | " | " 1 H 1 = ( 0 1 ) | " √21| " − | " H 1 = ( 0 1 ) to the first qubit of | " √2 | " − | " to first1 qubit of to the first qubit oCfU (H 1) 0 ψ = ( 0 ψ + 1 U ψ . ◦ ⊗ | "| " √2 | " ⊗ | " | " ⊗ | " 1 Here C denoteCsUcon(tHrolled1U),0acψting=as U (w0hen thψe c+ont1rol bUit iψs .1 and the U ◦ ⊗ | "| " √ | " ⊗ | " | " ⊗ | "| " identity mapping when the control bit is 02. We measure the expectation for the | " HerfiersCt qubditen0oteosf tchoenrtersoultleindg Ust,ataecting as U when the control bit is 1 and the U | " | " identity1mapping when the control b1it is 0 . We measure the expectation for the (H 0 ψ + H 1 U ψ ) = (( 0 +| "1 ) ψ + ( 0 1 ) U ψ ) first qub2it 0| "o⊗f |th"e res|ul"t⊗ing |st"ate 2 | " | " ⊗ | " | " − | " ⊗ | " | " 1 1 1 (H 0 ψ += H(10 (Uψψ+)U=ψ )(+( 01 + (1ψ) Uψψ +)).( 0 1 ) U ψ ) 2 | " ⊗ | " 2 || "" ⊗ | |" " | "2 || "" ⊗ || "" −⊗ | | "" | " − | " ⊗ | " This expectation is 1 1= ( 0 ( ψ + U ψ ) + 1 1( ψ U ψ )). ( ψ + ψ U †)( ψ + U ψ ) = + Re ψ U ψ . 2 !2| | "! ⊗| | |" " | | "" |2" ⊗2 | !" −| | |" " ThiTsheexipmeacgtiantairoynpiasrt is obtained by applying the same procedure to 1 1 1 1 ( ψ + ψ U †()0( ψ ψ+ U iψ1 ) =U ψ+ Re ψ U ψ . 2 ! | ! √| 2 | "| ⊗"| " −| | "" ⊗ 2| " 2 ! | | " TheThimisaisgtinhearmyetphaordtuissedobinta[1in],eadndbtyheaprepaldyeirnmgatyhweishamtoecopnrtoecmepdluatreeitsoefficiency in the context of this simple model. Note that the Hadamard test enables this 1 quantum computation to estima(te0the trψace ofi a1ny unUitaψry matrix U by repeated trials that estimate individua√l m2 a|tr"ix⊗en|tri"es− ψ| U" ψ⊗. W| e "shall return to quantum ! | | " Thiaslgisortihthemms efothr othdeuJsoendesinp[o1ly],naomndiatl haendreoatdheer mknaoyt wpoilsyhnotmo icaolsntinemapsluabtseeqitusenetfficiency paper. in the context of this simple model. Note that the Hadamard test enables this quantum computation to estimate the trace of any unitary matrix U by repeated trials that estimate individual matrix entries ψ U ψ . We shall return to quantum ! | | " algorithms for the Jones polynomial and other knot polynomials in a subsequent paper. May 16, 2010 0:40 WSPC/INSTRUCTION FILE QITTJMP May 16, 2010 0:40 WSPC/INSTRUCTION FILE QITTJMP

Instructions for Typing Manuscripts (Paper’s Title) 37 Instructions for Typing Manuscripts (Paper’s Title) 37 It follows from this calculation that the question of computing the bracket poly- MayI1t6,fo2l0l1o0ws0:f4r0omWtShPiCs/cIaNlScTuRlaUtCioTnIOtNhaFtILtEhe quQeITstTiJoMnPof computing the bracket polynomial for the closure of the three-strand braid b is mathematically equivalent to nomial for the closure of the three-strand braid b is mathematically equivalent to the problem of computing the trace of the unitary matrix Φ(b). the problem of computing the trace of the unitary matrix Φ(b). The Hadamard Test The Hadamard Test In order to (quantum) compute the trace of a unitary matrix U, one can use In order to (quantum) computeIntshtreucttiornascfoer oTyfpiang uMnanitusacrriyptsm(Paatperri’xs TUitle,) on37e can use thteheHHaaddaammaarrddtteessttttoo oobbttaaiinn tthe ddiiaaggoonnaallmmaattrrixixeelelemmenentsts ψψUUψψofoUf .UT. hTehteratcreace It follows from this calculation that the question of computin!g!th|e|b|ra|c"k"et poly- isisththenenththeessuummoofftthheessee mmaattrrix elleemmeennttssaass ψψ rruunnssoovvererananorotrhtohnoonromrmalablabsaissifsorfor nomial for the closure of the three-strand br|a|id""b is mathematically equivalent to thtehevevcetcotorrtshspepapacrcoebe.l.eWmWeoef ficfiorrmsstptuootibbnttgatihne trace of the unitary matrix Φ(b). The Hadamard Test 11 11 ++ RRee ψψUUψψ In order to (quantum) com22pute 2t2he tr!!ace||of||a "u"nitary matrix U, one can use the Hadamard test to obtain the diagonal matrix elements ψ U ψ of U. The trace ! | | " asasananexexppeiecsctttahatetnioiotnhnebsbuyymaaoppf ppthllyeysiiennmggatrhixeeeHHlemaadednaatmsmaasarrψddgrgauanttseeoHvHer an orthonormal basis for | " the vector space. We first obtain 11 H 00 1== 1 ((00 ++ 11)) || "" + √√R2e ψ||U""ψ || "" 2 2 2! | | " as an expectation by applying the Hadamard gate H 11 H 1 = 1 ( 0 1 ) H H1 0== ((00+ 1 )1 ) || ""| " √√√222 |||"""−−| "|| "" totoththeefifirsrtstqquubbititooff 1 H 1 = ( 0 1 ) | " √12 | " − | " C (H 1) 0 ψ = 1 ( 0 ψ + 1 U ψ . to the firCstUqUubi(tHof 1) 0 ψ = ( 0 ψ + 1 U ψ . ◦◦ ⊗⊗ | "|| "" √√22 || ""⊗⊗|| "" | |""⊗⊗ | |" " 1 Here C denotes contCroU lle(dH U,1)a0ctψin=g as (U0 whψen+ t1he Ucoψnt. rol bit is 1 and the Here C U denotes controll◦ed ⊗U, |ac"|ti"ng √a2s |U" ⊗w| h"en| t"h⊗e |co"ntrol bit is 1 and the U | "| " identity mHaerpepCingdwenhoteens ctohnetroclloendtUro, lacbtiintgisas 0U .wWhene tmheecaosnutrroel tbhiteisex1peacntdatthioe n for the identity mappinUg when the control bit is | 0" . We measure the |e"xpectation for the first qubitide0ntitoyfmtahpepirnegswuhlteinntghestcaontterol bit is 0| ."We measure the expectation for the first qubit 0| "of the resulting state | " first qubit 0 of the resTheultin grsesultingtate state is 1 | " | " 1 1 (H 0 1 ψ + H 1 U ψ ) = 11(( 0 + 1 ) ψ + ( 0 1 ) U ψ ) 2(H 0| " ⊗(ψ|H "0+ Hψ |1+"H⊗1U|ψU"ψ) =) =2 (((|00"++1| )1" )⊗ψ |+ψ("0 +|(1"0) −U| ψ"1)⊗) |U "ψ ) 2 | " ⊗2| "| " ⊗ | |" " ⊗| " ⊗| "| " 22 || "" | "| ⊗" | ⊗" | |"" − | |" "⊗−|| "" ⊗ | " 1 1 = 1 ( =0 ( 0( ψ( ψ++UUψψ )) + 11 ( ψ( ψ U ψ U)).ψ )). = 2( |0"2⊗| ("|⊗ψ"| +" U| |ψ"") +||"1"⊗⊗| "(|−ψ" −| " U| ψ" )). This expectatio2n i|s " ⊗ | " | " | " ⊗ | " − | " This expectation is 1 The expectation for1 |0>1 is This expectation is ( ψ + ψ U †)( ψ + U ψ ) = + Re ψ U ψ . 1 2 ! | ! | | " | " 2 1 2 1! | | " ( ψ + ψ U †)( ψ + U ψ ) = + Re ψ U ψ . The imag1i2nary part is obtained by applying the sa2m1e pr2o1cedure to ( !ψ |+ !ψ|U †)(|ψ" + U| ψ" ) = + R!e ψ| U| ψ" . 1 The imaginary p2art! is| obt!ai|ned b|y( 0a"pplψyi|ngi"1theUs2aψme2proc!ed|ur|e t"o The imaginary par√2 t| is" ⊗ obtained| " − | " ⊗by |appl" ying the The imaginary part is obtain1esamed by prapocedurplyinget tohe same procedure to This is the method used in [1(], 0and theψreaderim1ay wiUsh tψo contemplate its efficiency in the context of this s√im1ple model. Note that the Hadamard test enables this 2(|0" ⊗ | ψ" − i| 1" ⊗ U| ψ" quantum computation t√o e2sti|ma"te⊗th|e t"ra−ce o|f a"n⊗y unit|ary" matrix U by repeated This is thetrimalsetthhaotdesutismeadteinind[i1v]i,duaanldmtahtreixreenatrdieesr mψ Uayψw. iWshe sthoalcl orenttuernmtpo lqautaentiutms efficiency ! | | " Tihnisthisethcoenamltgeoexrtithhomodfs futohrsietshdesiiJnmon[p1esl]e,paomlnynododmtehila.el aNrneodatdoetehrtehrmaknatoytthwpeoilsyHhnoatmdoiaclmsoinantraedmsutpbessleatqtueennitatsbleeffis ctiheinscy inqutahnetucmonptcaeopxmetrp. ouftatthioisn stiomepstleimmatoedtehl.e Ntroatcee tohf aatnythueniHtaardyammaatrrdixtUestbyenraepbeleastetdhis qutrainaltsutmhactoemstpimutaatteioinndtivoideustailmmaatetritxheenttrraiecse oψf Uanψy u. nWiteasrhyamll arettruixrnUtobqyuraenpteuamted ! | | " traialglosrtihthamt sesftoirmtahtee JinodnievsidpuoalylnmomatirailxaenndtroietsherψkUnoψt p.oWlyenosmhaialllsreintuarnsutobsqeuquanentut m ! | | " alpgaopreitrh. ms for the Jones polynomial and other knot polynomials in a subsequent paper.

Figure 1 - A knot diagram.

III

Figure 2 - The Reidemeister Moves. That is, two knots are regarded as equivalent if one embedding can be obtained from the othReidemeisterer through a contin uMoous favmesily of embeddings of circles

reformulate knot4 theory in terms of graph combinatorics. [16], and Bar-Natan’s emphasis on tangle cobordisms [2]. We use similar considera- tions in our paper [10]. [16], and Bar-Natan’s emphasis on tangle cobordisms [2]. We use similar considera- tions in our paper [10]. Two key motivating ideas are involved in finding the Khovanov invariant. First of all, one woTwulodkelyikmeottiovactinagteigdeoarsiafyre ainvloinlvkedpion lfiynndoinmg ithael Ksuhocvhanaosv inKvaria.nTt. hFeirset are many meanings toof athll,eotneermwoucladtleikgeotroifcya,tebgourtifhy earleintkhpeolqyunoemstiailssutcoh fiasndKa.wT!aheyre"toareexmpanreyss the link meanings to the term categorify, but here the quest is to find a w!ay "to express the link polynomiaploalysnoamgiarlaads eadgrEadueldeEruclehracrhaarcatcetreriissttiicc KK = =χ χHq(KH) (fKor )somfeohrosmoomlogeyhomology !! " " q! ! " " theory assothceioartyeadsswociitahtedKwith. K . ! " ! " The bracket polynomial [7] model for the Jones polynomial [4, 5, 6, 17] is usually The bradceskceritbepdoblyytnhoemexipaalns[i7on] model for the Jones polynomial [4, 5, 6, 17] is usually described by the expansion Bracket Polynomial Model for the Jones Polynomial1 = A + A− (4) ! " ! " ! " 1 and we have = A + A− (4) ! " ! " ! " 2 2 K = ( A A− ) K (5) and we have ! #" − − ! "

= ( 2A3) 2 (6) K ! = "( A− !A−" ) K (5) ! #" − − ! " 3 = ( A− ) (7) ! " − ! " = ( A3) (6) Letting c(K) denote the n!umbe"r of cro−ssings i!n the "diagram K, if we replace K c(K) 1 ! " by A− K , and then replace A by q− , the bracket will be rewritten in the fol- ! " − lowing form: 3 = ( A− ) (7) = q (8) ! ! " " !− " − !! " " 1 with = (q +q− ). It is useful to use this form of the bracket state sum for the sake Lettingofc(thKe!#g)r"addeinngointethtehKehnouvamnobvehromofolcorgoys(tsoinbegsdeisncritbheed bdeilaogwr).aWmeKsha,llifcownteinrueeplace K c(K) 11 ! " by A− toKrefe,r atondthethsmenoortheipnglasclaebeAledbqy(or qA−− ,inththeeborraigcinkaeltbrwaciklletbfoermreuwlatriiotnt)enas in the following formB!-:sm"oothings. We should further note−that we use the well-known convention of enhanced states where an enhanced state has a label of 1 or X on each of its component = q1 (8) loops. We then regard the v!alue o"f the !loop q"+−q− !as the"sum of the value of a circle labeled with 1a 1 (the value is q) added to the value of a circle labeled with an X (the 1 with =val(uqe i+s qq−−). )W.eIctoiuslduhsaevfeuclhtooseunstheetmhiosrefnoerumtraol lfabtheles obfr+a1ckanedt st1atseo tshuatm for the sake !#" − of the grading in the Khovanov hom+o1logy (to be described below). We shall continue q +11 1 to refer to the smoothings labeled q (o⇐r⇒A− ⇐i⇒n the original bracket formulation) as B-smoothianngds. We should further note that we use the well-known convention of en- 1 q− 1 X, hanced states where an enhanced state⇐h⇒as−a l⇐ab⇒el of 1 or X on each of its component 1 loops. We bthute, nsinrceegaanrdalgtehberavianvluoleviongf 1thaendloXopnaqtur+allyq−appeaasrstlhateers, uwme taokfe tthies fvoarmluoef of a circle labeled witlhabealin1g(ftrohme tvhaelbueegininsinqg). added to the value of a circle labeled with an X (the 1 value is q− ). We could have chosen the more neutral labels of +1 and 1 so that − q+1 +1 1 ⇐⇒ ⇐⇒ 5 and 1 q− 1 X, ⇐⇒ − ⇐⇒ but, since an algebra involving 1 and X naturally appears later, we take this form of labeling from the beginning.

5 26

Figure 9 26 26 26

26 26

The Khovanov Complex

Figure 9Figure 9 Figure 9 Figure 9

Figure 9 9 Figure 9 Figure Figure 9 [16], and Bar-Natan’s emphasis on tangle cobordisms [2]. We use similar considera- tions in our paper [10].

[16], and Bar-Natan’s emphasTiswoon ktaenyglme ocotibvoartdinisgmsid[e2a].s Wareeuisnevsoilmvieldaricnonfisniddeirnag- the Khovanov invariant. First tions in our paper [10]. of all, one would like to categorify a link polynomial such as K . There are many ! " Two key motivating imdeeaasnairnegisnvtooltvheed tienrmfincdainteggtohreifKy,hobvuatnhoevreintvhaeriaqnute. sFt iirsstto find a way to express the link polynomial as a graded Euler characteristic K = χ H(K) for some homology of all, one would like to categorify a link polynomial such as K . There a!re "many q! " meanings to the term catetghoeroifryy, baustsohecrieattehde wquiethst isKto.find a w!ay "to express the link ! " polynomial as a graded Euler characteristic K = χ H(K) for some homology ! " q! " theory associated with K . The bracket polynomial [7] model for the Jones polynomial [4, 5, 6, 17] is usually ! " described by the expansion The bracket polynomial [7] model for the Jones polynomial [4, 5, 6, 17] is usually described by the expansion 1 = A + A− (4) ! " ! " ! " 1 = A + A− (4) ! and" we h!ave" ! " and we have 2 2 K = ( A A− ) K (5) 2 2 ! #" − − ! " K = ( A A− ) K (5) ! #" − − ! " = ( A3) (6) = ( A3) ! " − ! (6") ! " − ! " Reformulating the Bracket 3 3 = ( A− ) (7) = ( A− ) ! " − ! (7") Let !c(K) =" number− of crossings! "on link K. -c(K) -2 Letting c(K) denote the FnormLume tAbt ie n r g o fcc(rK oands)s irdneplaceegns oint Aet h t eh bedy i-qnaug rm a. mbeKr o, fifcwroesrseipnlgacseinKthe diagram K, if we replace K c(K) 1 ! " c(K) Then the skein relation1 for will ! " by A− K , and then breyplAac−e A byKq,−an, dthethbernacrkeeptlwacilel bAe breywritqte−n i,nththeebforal-cket will be rewritten in the fol- be! replaced" by: − lowing form! : " lowing form:− = q = q (8) (8) ! " ! " − ! " ! " ! " − ! " 1 with = (q +q− ). It is useful to use this form1 of the bracket state sum for the sake !#" with = (q +q− ). It is useful to use this form of the bracket state sum for the sake of the grading in the Khovoafntohve!h#gorm"adolionggyi(ntothbee Kdehsocrviabnedovbehloowm)o. lWogeysh(atoll bcoendtiensuceribed below). We shall continue 1 to refer to the smoothings labeled q (or A− in the original bracket formu1lation) as to refer to the smoothings labeled q (or A− in the original bracket formulation) as B-smoothings. We should further note that we use the well-known convention of en- B-smoothings. We should further note that we use the well-known convention of enhanced states where an enhanced state has a label of 1 or X on each of its component loops. We then regard thehvaanluceeodfstthaetelosowp hqe+reqan1 eans hthaenscuemd sotfattheehvaasluaeloafbaelciorcfl1e or X on each of its component − 1 labeled with a 1 (the valuleoiospqs). aWddeedthteonthreegvaarlduethoef avacliurceleolfatbheeleldoowpithq a+n qX−(thaes the sum of the value of a circle 1 value is q− ). We could hlaavbeeclheodsewnithheam1or(ethneeuvtaralluleabiselqs )ofa+dd1eadndto t1hesovtahlaute of a circle labeled with an X (the 1 − value is q− ). We could have chosen the more neutral labels of +1 and 1 so that q+1 +1 1 − ⇐⇒ ⇐⇒ q+1 +1 1 and ⇐⇒ ⇐⇒ 1 q− 1 X, and ⇐⇒ − ⇐⇒ 1 but, since an algebra involving 1 and X naturally appears latqe−r, we take t1his formXo,f labeling from the beginning. ⇐⇒ − ⇐⇒ but, since an algebra involving 1 and X naturally appears later, we take this form of labeling from the beginning.

5 Use enhanced states by labeling each loop with +1 or -1.

+1 -1

= +

-1 q + q 1 with a 1 (the value is q) added to the value of a circle labeled with an X (the value is q− ). We could have chosen the more neutral labels of +1 and 1 so that − q+1 +1 1 ⇐⇒Enhanced⇐ ⇒States and q 1 1 X, − 1 with a 1 (the value is q) added to the value of a circle⇐⇒ −labeled⇐⇒with an X (the value is q− ). We could have chosenbut, sincethe manorealgebraneutralinlabelsvolvingof1+1andandX naturally1 so that appears later, we take this form of labeling from the beginning. − q+1 +1 1 To see how the Khovanov grading arises,⇐consider⇒ ⇐the⇒ form of the expansion of this version of the andbracket polynonmial in enhanced states. We have the formula as a sum over enhanced states s : For reasons1 that will soon become apparent, we q− 1 X, let K-1 be=⇐ denoted⇒ −( b1)y⇐ Xn⇒B and(s) q+1j( sbe) denoted by 1. but, since an algebra involving 1 and X(The$naturally module% s V appearswill− be generatedlater, weby 1tak ande X.)this form of labeling from the beginning. ! where nB(s) is the number of B-type smoothings in s, λ(s) is the number of loops in s labeled 1 minus theTnumbero see hoowf loopsthe KlabeledhovanovXgrading, and j(sarises,) = nBc(onsiders) + λ(sthe). Tformhis cofan thebe reewrittenxpansionin othef thisfollovwingersionform:of the bracket polynonmial in enhanced states. We have the formula as a sum over enhanced states s : K = ( 1)iqjdim( ij) $ % − n (s) j(Cs) K = !i ,j ( 1) B q $ % s − where we define ij to be the linear span (ove!r k = Z/2Z as we will work with mod 2 coefficients) of C wherethe setnofB(senhanced) is the numberstates withof Bn-typeB(s) smoothings= i and j(s)in=s,jλ. (Thens) is thethe numbernumberooffsluchoopsstatesin s labeledis the dimension1 minus thedimnumber( ij). of loops labeled X, and j(s) = nB(s) + λ(s). This can be rewritten in the following form: C i j ij We would like to have a bigraded compleK x=composed( 1)ofq dimthe (ij with) a differential $ % i ,j − C C ! ∂ : ij i+1 j. where we define ij to be the linear span (overCk =−→Z/C2Z as we will work with mod 2 coefficients) of C theThesetdifofferentialenhancedshouldstatesincreasewith nBthe(s)homolo= i andgicalj(sg)rading= j. Theni by 1theandnumberpreservoef thesuchquantumstates isgrtheadingdimensionj. Then dimwe (couldij). write C j i ij j j K = q ( 1) dim( ) = q χ( • ), We would like to have a bigraded$ % complex composed− of theC ij with a difCferential !j !i C !j j ij i+1 j j where χ( • ) is the Euler characteristic of∂the: subcomplex .• for a fixed value of j. C C −→ C C TheThisdifformulaferentialwshouldould constituteincrease tahecatehomologorificationgical groadingf the bracki by 1etandpolynomial.preserve theBeloquantumw, we shallgradingsee jh.oThenw the weoriginalcould wKhoritevanov differential ∂ is uniquely determined by the restriction that j(∂s) = j(s) for each j i ij j j j enhanced state s. Since j isKpreserv= edq by the( d1)ifferential,dim( )these= subcompleq χ( • )x,es have their own Euler $ % − C C C• characteristics and homology. We h!ajve !i !j j j where χ( • ) is the Euler characteristic of the subcomplej x • j for a fixed value of j. C χ(H( • )) = χ(C • ) C C This formula jwould constitute a categorification of the brackj et polynomial. Below, we shall see how the where H( • ) denotes the homology of the complex • . We can write original KhoC vanov differential ∂ is uniquely determinedC by the restriction that j(∂s) = j(s) for each j j enhanced state s. Since j is preserved byKthe=differential,q χ(H(these• ))s.ubcomplexes j have their own Euler $ % C C• characteristics and homology. We have !j The last formula expresses the bracket polynomialj as a gradedj Euler characteristic of a homology theory χ(H( • )) = χ( • ) associated with the enhanced states of the brackC et state sCummation. This is the categorification of the wherebracketHpolynomial.( j) denotesKhothevanohomologyv proveofs thatthetchisomplehomologyx j. Wtheorye caniswritean invariant of knots and links (via the C• C• Reidemeister moves of Figure 1), creating a new andj strongerjinvariant than the original Jones polynomial. K = q χ(H( • )). $ % j C ! 6 The last formula expresses the bracket polynomial as a graded Euler characteristic of a homology theory associated with the enhanced states of the bracket state summation. This is the categorification of the bracket polynomial. Khovanov proves that this homology theory is an invariant of knots and links (via the Reidemeister moves of Figure 1), creating a new and stronger invariant than the original Jones polynomial.

6 26

1 1 An enhanced state that contributes

[(q)(q)(1/q)] [(-q) (-q) (-q)] 1 1 -1 B B B

to the revised -1 bracket state sum.

Figure 9 and j(∂ s ) = j( s ) | ! | ! for each enhanced state s. In the next section, we shall explain how the boundary operator is constructed.

2. Lemma. By deﬁning U : (K) (K) as in the previous section, via C −→ C U s = ( 1)i(s)qj(s) s , | ! − | ! we have the following basic relationship between U and the boundary operator ∂ :

U∂ + ∂U = 0.

Proof. This follows at once from the deﬁnition of U and the fact that ∂ preserves j and increases i to i + 1. //

3. From this Lemma we conclude that the operator U acts on the homology of (K). We can regard H( (K)) = Ker(∂)/Image(∂) as a new Hilbert space on which theC unitary operator U acts.C In this way, the Khovanov homology and its relationship with the Jones polynomial has a natural quantum context.

4. For a ﬁxed value of j, ,j i,j • = C ⊕iC is a subcomplex of (K) with the boundary operator ∂. Consequently, we can speak of ,jC ij the homology H( • ). Note that the dimension of is equal to the number of enhanced C C states s with i = Enhancedi(s) and Statej Sum= Formj(ulas) f.orConsequently the Bracket , we have | ! K = qj(s)( 1)i(s) = qj ( 1)idim( ij) % ! s − − C ! !j !i

j ,j j ,j = q χ( • ) = q χ(H( • )). C C !j !j Here we use the deﬁnition of the Euler characteristic of a chain complex

,j i ij χ( • ) = ( 1) dim( ) C − C !i and the fact that the Euler characteristic of the complex is equal to the Euler characteristic of its homology. The quantum amplitude associated with the operator U is given in terms of the Euler characteristics of the Khovanov homology of the link K.

j ,j K = ψ U ψ = q χ(H( • (K))). % ! % | | ! C !j

7 One advantage of the expression of the bracket polynomial via enhanced states is that it is now a sum of monomials. We shall make use of this property throughout the rest of the paper.

One advantage of the expression of the bracket polynomial via enhanced states is that it is 3 noQw auantumsum of monomials.StatisticsWe shall makande usetheof thisJonespropertyPthroughoutolynomialthe rest of the paper.

We now use the enhanced state summation for the bracket polynomial with variable q to give a quantum3 Qformulationuantum oStatisticsf the state sum.andLettheq beJoneson thePunitolynomialcircle in the complex plane. (This is equivalent to letting the original bracket variable A be on the unit circle and equivalent to letting the JWonese nowpolynmialuse the enhancedvariablestatet besummationon the unitfor ctheircle.)brackLetet polynomial(K) denotewith tvheariablecompleq toxgivvectore a space C withquantumorthonormalformulationbasisof sthe statewheresum.s runsLet qobeveonr thetheenhancedunit circlestatesin the cofomplethe dxiagramplane. (ThisK. Theis vector spaceequiv(alentK) istotheletting(finitethe{|originaldimensional)! } bracket vHilbertariable Aspacebe onforthe unitour circlequantumand eformulationquivalent to lettingof the Jones polynomial.theCJones polynmialWhile it isvariablecustomaryt be onfortheaunitHilbertcircle.)spaceLet to(Kbe) denotewrittenthewithcomplethex lettervectorHspace, we do not with orthonormal basis s where s runs over the enhancedC states of the diagram K. The vector follow that convention h{|ere,! }due to the fact that we shall soon regard (K) as a chain complex space (K) is the (finite dimensional) Hilbert space for our quantum formulationC of the Jones andpolynomial.take iCts homologyWhile it. isOcustomaryne can hardlyfor a aHvilbertoid usingspace to beforwrittenhomologywith.the letter H, we do not A Quantum Statistical ModelH for the Bracket follow that convention here, due to the fact thatPolynonmial.we shall soon regard (K) as a chain complex With q on the unit circle, we define a unitary transformation C and take its homology. One can hardlyLeta C(K)void denoteusing a Hilberfort spacehomology. with basis |s> where s Hruns over the With q on the unit circle, we defineenhanceda unitaryU states: of(transformation Ka knot) or link diagram(K) K. We defineC a unitary− transf→ormation.C U : (K) (K) by the formula C −→ C U s = ( 1)i(s)qj(s) s by the formula | ! − | ! for each enhanced state s. Here i(s)Uqand iss chosen=j(s( on) 1)arethei (unitsa)qs jcir(dscle)efineds in the in the previous section of this paper. | ! complex− plane. | ! for each enhanced state s. Here i(s) and j(s) are as defined in the previous section of this paper. Let Let ψ = s . ψ| =! ss .| ! | ! s !| ! The state vector ψ is the sum over the basis s!tates of our Hilbert space (K). For convenience, The state vector| !ψ is the sum over the basis states of our Hilbert space (K).CFor convenience, we dowenotdo notnormalizenormalize| ! ψψ totolengthlength oneoneininthetheHilbertHilbertspacespace(K).(KWe).thenWCehathenve thehave the | | !! C C Lemma.Lemma.TheTheeveavluationaluationoofftthehe brackbracketetpolynomialpolynomialis giisveginvbeynthebyfollothewingfolloformulawing formula

KK==ψ Uψ ψU .ψ . $$ ! ! $ $| || !| ! Proof. Proof. i(s) j(s) i(s) j(s) ψ U ψ = s! ( 1) i(qs) j(ss) = ( 1) q i(s)s!js(s) ψ$ U| ψ| != s $ s|! −( 1) q | ! s = s − ( 1) $ q| ! s! s $ | | ! !s! ! $ | − | !!s! ! − $ | ! s! s s! s ! ! = ( 1)i(s)qj(s) = K !, ! i(s) j(s) s − $ ! =! ( 1) q = K , s − $ ! since ! since s! s = δ(s, s!) $ | ! s! s = δ(s, s!) where δ(s, s!) is the Kronecker delta, equal$ to| 1!when s = s! and equal to 0 otherwise. // where δ(s, s!) is the Kronecker delta, equal to 1 when s = s! and equal to 0 otherwise. // 4 4 One aOnedvantageadvantageof theof tehexpressionexpressionofofthethebrackbracketet polynomialpolynomialviaviaenhancedenhancedstatesstatesis thatisitthatis it is now a nosumw aosumf monomials.of monomials.WeWsehallshallmakmakeeuuseseofof thisthis propertypropertythroughoutthroughoutthe therest roestf theofpaperthe .paper.

3 Quantum Statistics and the Jones Polynomial 3 Quantum Statistics and the Jones Polynomial We now use the enhanced state summation for the bracket polynomial with variable q to give a We noquantumw use theformulationenhancedofstatethe statesummationsum. Let forq betheon thebrackunitetcirclepolynomialin the complewithxvplane.ariable(Thisq toisgive a quantumequiformulationvalent to lettingofttheheoriginalstate sum.brackLetet vaqriablebe onA betheonunitthe circleunit circlein theandceomplequivalentx plane.to letting(This is equivalentthe Jonesto lettingpolynmialthe originalvariable tbrackbe onetthevaunitriablecircle.)A beLeton (theK)unitdenotecirclethe compleand equix vectorvalentspaceto letting with orthonormal basis s where s runs over the enhancedC states of the diagram K. The vector the Jones polynmial variable{| t! }be on the unit circle.) Let (K) denote the complex vector space space (K) is the (finite dimensional) Hilbert space for ourC quantum formulation of the Jones with orthonormalpolynomial.C Wbasishile it sis customarywhere sforrunsa Hoilbertver thespaceenhancedto be writtenstateswithof thethe letterdiagramH, weK.doThenotvector space (K) is the (finite{| dimensional)! } Hilbert space for our quantum formulation of the Jones folloC w that convention here, due to the fact that we shall soon regard (K) as a chain complex polynomial.and takeWitshilehomologyit is customary. One can hardlyfor a aHvilbertoid usingspacefortohomologybe written. Cwith the letter H, we do not follow that convention here, due to the fact that weHshall soon regard (K) as a chain complex and takWeithitsqhomologyon the unit.cOircle,ne canwe definehardlya unitaryavoid usingtransformationfor homology. C H U : (K) (K)

With q on the unit circle, we define aOneunitaryadvantage of theCexpressiontransformationof −the →bracket polynomialC via enhanced states is that it is now a sum of monomials. We shall make use of this property throughout the rest of the paper. by the formula 3 QuantumUU :s Statistics=(K( )and1)thei(sJones)qj(P(so)Klynomials) We now use the enhanced state summation for the bracket polynomial with variable q to give a quantum formulation| !Cof the state sum.− Let−q→be on theCunit circle| in!the complex plane. (This is equivalent to letting the original bracket variable A be on the unit circle and equivalent to letting for each enhanced state s. Here ithe(sJones) polynmialand vjariable(st)be areon the unitacsircle.)dLetefined(K) denote tihencomplethex vectorprespacevious section of this paper. by the formula with orthonormal basis s where s runs over the enhancedC states of the diagram K. The vector space (K) is the (finite{| dimensional)! } Hilbert space for our quantum formulation of the Jones polynomial.C While it is customary for a Hilberti(spaces) tojbe(writtens) with the letter H, we do not followUthat consvention=here, due( to the1)fact that we shallq soon regards (K) as a chain complex Let and take its homology. One can hardly avoid using for homology. C | ! − H | ! With q on the unit circle,ψwe define=a unitary transformations . for each enhanced state s. Here i(s) and j(s) areU : a(Ks) d(efinedK) in the previous section of this paper. | ! Cs −|→ C! by the formula ! U s = ( 1)i(s)qj(s) s The state vector ψ is the sum over the basis states| ! − of |our! Hilbert space (K). For convenience, Let | ! for each enhanced state s. Here i(s) and j(s) are as defined in the previous section of this paper. C we do not normalize ψ to lengthLet one in the Hilbert space (K). We then have the ψ = s . | ! | ! s | ! C ψ = ! s . The state vector ψ is the sum over the basis states of our Hilbert space (K). For convenience, we do not normalize| !|ψ to!length one in the Hilbert| space! (K). We thenC have the Lemma. The evaluation of the bracket polynomial| ! s is giCven by the following formula Lemma. The evaluation of the bracket polynomial! is given by the following formula

K = ψ U ψ . The state vector ψ is the sum over the basis states$ ! $ |of| ! our Hilbert space (K). For convenience, Proof. K = ψ U ψ . | ! i(s) j(s) i(s) j(s) C ψ U ψ = s! ( 1) q s = ( 1) q s! s we do not normalize ψ to length one in$ |the| ! s Hs $ ilbert| − | ! spaces s − $ (| K! ). We then have the $ !!! ! $ | | !!!! | ! = ( 1)i(s)qj(s) = K , C This gives a new quantums − algorithm$ ! for the ! Proof. since Jones polynomial (via Hadamard Test). s! s = δ(s, s!) Lemma. The evaluation of the bracket polynomiali(s) j($s|)! is given by thei(sfollo) j(s)wing formula ψ U ψ = where δs(s,!s!)(is the K1)ronecker delta,q equal to 1swhen s== s! and equal to 0 otherwise.( 1)// q s! s $ | | ! s $ | − | ! s − $ | ! !s! ! 4 !s! ! K =i(s)ψj(Us) ψ . = $ ( ! 1) $ q| |=! K , s − $ ! ! Proof.since i(s) j(s) i(s) j(s) ψ U ψ = s! ( 1)s! s q= δ(ss, s=!) ( 1) q s! s $ | | ! $ | − $ | ! | ! − $ | ! s! s s! s where δ(s, s!) is the Kroneck! !er delta, equal to 1 when s =!s! and! equal to 0 otherwise. // = ( 1)i(s)qj(s) = K , s − $ ! ! 4 since s! s = δ(s, s!) $ | ! where δ(s, s!) is the Kronecker delta, equal to 1 when s = s! and equal to 0 otherwise. //

4 Generalization to Virtual Knot Theory

planar isotopy vRI

vRII RI

RII vRIII B

RIII mixed RIII A C

Figure 1: Moves

Figure 2: Detour Move

Another way to understand virtual diagrams is to regard them as representatives for oriented Gauss codes [8], [21, 22] (Gauss diagrams). Such codes do not always have planar realizations. An attempt to embed such a code in the plane leads to the production of the virtual crossings. The detour move makes the particular choice of virtual crossings irrelevant. Virtual isotopy is the same as the equivalence relation generated on the collection of oriented Gauss codes by abstract Reidemeister moves on these codes.

Figure 3 illustrates the two forbidden moves. Neither of these follows from Reidmeister moves plus detour move, and indeed it is not hard to construct examples of virtual knots that are non-trivial, but will become unknotted on the application of one or both of the forbidden moves. The forbidden moves change the structure of the Gauss code and, if desired, must be considered separately from the virtual knot theory proper.

4 F F 1 2

Figure 3: Forbidden Moves

Figure 4: Surfaces and Virtuals

3 Interpretation of Virtuals Links as Stable Classes of Links in Thickened Surfaces

There is a useful topological interpretation [21, 23] for this virtual theory in terms of embeddings of links in thickened surfaces. Regard each virtual crossing as a shorthand for a detour of one of the arcs in the crossing through a 1-handle that has been attached to the 2-sphere of the original diagram. By interpreting each virtual crossing in this way, we obtain an embedding of a collection of circles into a thickened surface Sg ×R where g is the number of virtual crossings in the original diagram L, Sg is a compact oriented surface of genus g and R denotes the real line. We say that two such surface embeddings are stably equivalent if one can be obtained from another by isotopy in the thickened surfaces, homeomorphisms of the surfaces and the addition or subtraction of empty handles (i.e. the knot does not go through the handle). We have the Theorem 1 [21, 28, 23, 3]. Two virtual link diagrams are isotopic if and only if their corresponding surface embeddings are stably equivalent.

In Figure 4 we illustrate some points about this association of virtual diagrams and knot and link diagrams on surfaces. Note the projection of the knot diagram on the torus to a diagram in the plane (in the center of the ﬁgure) has a virtual crossing in the planar diagram where two arcs that do not form a crossing in the thickened surface project to the same point in the plane. In this way, virtual crossings can be regarded as artifacts of projection. The same ﬁgure shows a

5 A Quantum Algorithm for the Virtual Jones Polynomial

The bracket polynomial for virtual knots extends naturally by assigning the same values to extended states for loops that cross themselves virtually. Thus the set of enhanced states of a virtual knot or link gives a basis for a Hilbert space, and therefore we have a quantum algorithm for the Jones polynomial extended to virtuals. AT THE PRESENT TIME THIS IS THE ONLY FORMULATION OF A QUANTUM ALGORITHM FOR THIS INVARIANT. Khovanov Homology - Jones Polynomial as an [16], and Bar-Natan’s empEulerhasis on Characteristictangle cobordisms [2]. We use similar considera- tions in our paper [10].

Two key motivating ideas are involved in ﬁnding the Khovanov invariant. First of all, one would like to categorify a link polynomial such as K . There are many meanings to the term categorify, but here the quest is to ﬁnd a w!ay "to express the link polynomial as a graded Euler characteristic K = χq H(K) for some homology theory associated with K . ! " ! " ! " The bracket polynWomeia lwill[7] m oformdel forulatethe Jo nKhoes polvanoynomiavl [4, 5, 6, 17] is usually described by the expansion Homology 1 in the context= A of +ourA− quantum (4) ! " ! " ! " statistical model for the bracket and we have polynomial. 2 2 K = ( A A− ) K (5) ! #" − − ! "

= ( A3) (6) ! " − ! "

3 = ( A− ) (7) ! " − ! " Letting c(K) denote the number of crossings in the diagram K, if we replace K c(K) 1 ! " by A− K , and then replace A by q− , the bracket will be rewritten in the following form! : " − = q (8) ! " ! " − ! " 1 with = (q +q− ). It is useful to use this form of the bracket state sum for the sake of the!#gr"ading in the Khovanov homology (to be described below). We shall continue 1 to refer to the smoothings labeled q (or A− in the original bracket formulation) as B-smoothings. We should further note that we use the well-known convention of enhanced states where an enhanced state has a label of 1 or X on each of its component 1 loops. We then regard the value of the loop q + q− as the sum of the value of a circle labeled with a 1 (the value is q) added to the value of a circle labeled with an X (the 1 value is q− ). We could have chosen the more neutral labels of +1 and 1 so that − q+1 +1 1 ⇐⇒ ⇐⇒ and 1 q− 1 X, ⇐⇒ − ⇐⇒ but, since an algebra involving 1 and X naturally appears later, we take this form of labeling from the beginning.

5 CATEGORIFICATION View the next slide of states of the bracket expansion as a CATEGORY. The cubical shape of this category suggests making a homology theory. In order to make a non-trivial homology theory we need a functor from this category of states to a module category. Each state loop will map to a module V. Unions of loops will map to tenor products of this module. We will describe how this comes about after looking at the bracket polynomial in more detail. 26 26 26

26 26

The Khovanov Complex

Figure 9Figure 9 Figure 9 Figure 9

Figure 9 9 Figure 9 Figure Figure 9 We will construct the differential in this complex first for mod-2 coefficients. The differential is based on regarding two states as adjacent if one differs from the other by a single smoothing at some site. Thus if (s, τ) denotes a pair consisting in an enhanced state s and site τ of that state with τ of type A, then we consider all enhanced states s! obtained from s by smoothing at τ and relabeling only those loops that are affected by the resmoothing. Call this set of enhanced states S![s, τ]. Then we shall define the partial differential ∂τ (s) as a sum over certain elements in S![s, τ], and the differential by the formula

∂(s) = ∂τ (s) τ The boundary is a sum! of partial differentials corresponding to resmoothings on the states. with the sum over all type A sites τ in s. It then remains to see what are the possibilities for ∂τ (s) so that j(s) is preserved. Each state loop is a module. Note that if s S [s, τ], then n (s ) = n (s) + 1. Thus ! ∈ ! B ! B A collection of state loops corresponds to j(s!) = nB(s!) + λ(s!) = 1 + an tensorB(s) +productλ(s! )of. these modules. From this we conclude that j(s) = j(s ) if and only if λ(s ) = λ(s) 1. Recall that ! ! − λ(s) = [s : +] [s : ] − − where [s : +] is the number of loops in s labeled +1, [s : ] is the number of loops labeled 1 (same as − − labeled with X) and j(s) = nB(s) + λ(s).

Proposition. The partial differentials ∂τ (s) are uniquely determined by the condition that j(s!) = j(s) for all s! involved in the action of the partial differential on the enhanced state s. This unique form of the partial differential can be described by the following structures of multiplication and comultiplication on the algebra A = k[X]/(X2) where k = Z/2Z for mod-2 coefﬁcients, or k = Z for integral coefﬁcients.

1. The element 1 is a multiplicative unit and X2 = 0.

2. ∆(1) = 1 X + X 1 and ∆(X) = X X. ⊗ ⊗ ⊗ These rules describe the local relabeling process for loops in a state. Multiplication corresponds to the case where two loops merge to a single loop, while comultiplication corresponds to the case where one loop bifurcates into two loops.

(The proof is omitted.) Partial differentials are deﬁned on each enhanced state s and a site τ of type A in that state. We consider states obtained from the given state by smoothing the given site τ. The result of smoothing τ is to produce a new state s! with one more site of type B than s. Forming s! from s we either amalgamate two loops to a single loop at τ, or we divide a loop at τ into two distinct loops. In the case of amalgamation, the new state s acquires the label on the amalgamated circle that is the product of the labels on the two circles that are its ancestors in s. This case of the partial differential is described by the multiplication in the algebra. If one circle becomes two circles, then we apply the coproduct. Thus if the circle is labeled X, then the resultant two circles are each labeled X corresponding to ∆(X) = X X. If the orginal circle is labeled 1 ⊗ then we take the partial boundary to be a sum of two enhanced states with labels 1 and X in one case, and labels X and 1 in the other case, on the respective circles. This corresponds to ∆(1) = 1 X + X 1. ⊗ ⊗

7 The Functor from the cubical category to the module category demands multiplication and comultiplication in the module.

Module V

m to produce reliable computation. Nevertheless, one has the freedom to create spaces and operators on the mathematical level and to conceptualize these in a quantum mechanical framework. The resulting structures may be realized in nature and in present or future technology. In the case of our Hilbert space associated with the bracket state sum and its corresponding unitary transformation U, there is rich extra structure related to Khovanov homology that we discuss in the next section. One hopes that in a (future) realization of these spaces and operators, the Khovanov homology will play a key role in quantum information related to the knot or link K.

There are a number of conclusions that we can draw from the formula K = ψ U ψ . First of all, this formulation constitutes a quantum algorithm for the computation! "of the! brack| | et" polynomial (and hence the Jones polynomial) at any specialization where the variable is on the unit circle. We have defined a unitary transformation U and then shown that the bracket is an evaluation in the form ψ U ψ . This evaluation can be computed via the Hadamard test [24] and this gives the desired! | quantum| " algorithm. Once the unitary transformation is given as a physical construction, the algorithm will be as efficient as any application of the Hadamard test. The present algorithm requires an exponentially increasing complexity of construction for the associated unitary transformation, since the dimension of the Hilbert space is equal to the 2c(K) where c(K) is the number of crossings in the diagram K. Nevertheless, it is significant that the Jones polynomial can be formulated in such a direct way in terms of a quantum algorithm. By the same token, we can take the basic result of Khovanov homology that says that the bracket is a graded Euler characteristic of the Khovanov homology as telling us that we are taking a step in the direction of a quantum algorithm for the Khovanov homology itself. This will be discussed below.

26 26

4 Khovanov Homology26 and a Quantum Model for the Jones

26 Polynomial 26

In this section we outline how the Khovanov homology is related with our quantum model. This can be done essentially axiomatically, without giving the details of the Khovanov construction. We give these details in the next section. The outline is as follows: 2 1. There is a boundary operator ∂ deﬁned on the Hilbert space of enhanced states of a link 1 diagram K 1

∂2 : (K) (K) C −→ C such that ∂∂ = 0 and so that if i,j = i,j(K) denotes the subspace of (K) spanned by 3 Figure 9 enhanced states s with i = i(s)CandFigurej C= 9 j(s), then C | " Figure 9 ∂ : ij i+1,j. CFigure 9− → CFigure 9 That is, we have the formulas i(∂ s ) = i( s ) + 1 | " | " 6 to produce reliable computation. Nevertheless, one has the freedom to create spaces and operators on the mathematical level and to conceptualize these in a quantum mechanical framework. The resulting structures may be realized in nature and in present or future technology. In the case of our Hilbert space associated with the bracket state sum and its corresponding unitary transformation U, there is rich extra structure related to Khovanov homology that we discuss in the next section. One hopes that in a (future) realization of these spaces and operators, the Khovanov homology will play a key role in quantum information related to the knot or link K.

4 Khovanov Homology and a Quantum Model for the Jones Polynomial

1. There is a boundary operator ∂ deﬁned on the Hilbert space of enhanced states of a link diagram K

2 ∂ 2: (K) (K) 1 C 1 2’ −→ C such that ∂∂ = 0 and so that if i,j = i,j(K) denotes the subspace of (K) spanned by 1 enhanced states s with i = i(s)Cand j C= j(1s), then C | " 2 1’ 1’ 2 2’ ∂ : ij i+1,j. C −→ C For d^2 =0, want partial boundaries to commute. That is, we have the formulas i(∂ s ) = i( s ) + 1 | " | " 6 A d A-1

A ! A-1

m -1 A A

Figure 2: SaddlePoints and State Smoothings

the relationships between Frobenius algebras and the surface cobordism category. The proof of invariance of Khovanov homology with respect to the Reidemeister moves (respecting grading changes) will not be given here. See [12, 1, 2]. It is remarkable that this version of Khovanov homology is uniquely speciﬁed by natural ideas about adjacency of states in the bracket polynomial.

Remark on Integral Differentials. Choose an ordering for the crossings in the link diagram K Theand d commenote thutationem by 1 ,of2, then .parLettials b eboundariesany enhance leadsd state toof Ka and let · · · ∂i(s) denote the structurchain obtaei nofed Frfroobeniusm s by a palgebraplying a pforart itheal bo algebraundary a t the i-th site 1 of s. If the i-th site is a smassociatedoothing of ty tope aA −state, th ecirn ∂clei(s.) = 0. If the i-th site is

m !

F G H

Figure 3: Surface Cobordisms

9 that the existence of a bigraded complex of this type allows us to further write: !K" = qj (−1)idim(Cij ) = qj χ(C• j ), X X X j i j where χ(C• j ) is the Euler characteristic of the subcomplex C• j for a ﬁxed value of j. Since j is preserved by the diﬀerential, these subcomplexes have their own Euler characteristics and homology. We can write

!K" = qj χ(H(C• j )), X j where H(C• j ) denotes the homology of this complex. Thus our last formula expresses the bracket polynomial as a graded Euler characteristic of a homology theory associated with the enhanced states of the bracket state summation. This is the categorification of the bracket polynomial. Kho- vanov proves that this homology theory is an invariant of knots and links, creating a new and stronger invariant than the original Jones polynomial. It turns out that one can take the algebra We explain the differential in thgeneratedis complex fboyr m1 oandd-2 Xco ewithfficients and leave it to the reader to see the references Xfo 2r =th e0 r e andst. T he differential is defined via the algebra A = k[X]/(x2) so that X2 = 0 with coproduct ∆ : A −→ A ⊗ A defined by ∆(X) = X ⊗ X and ∆ (1) = 1 ⊗ X + X ⊗ 1. Partial differentials (which are defined on an enhanced state with a chosen site, whereas the differentialTheis a chainsum o complexf these m aisp pthenings) generatedare defined boyn each enhanced state s and aenhancedsite κ of tstatesype A within th aloopt sta tlabelse. We 1c oandnsid X.er states obtained from the given state by smoothing the given site κ. The result of smoothing κ is to produce a new state s! with one more site of type B than s. Forming s! from s we either amalgamate two loops to a single loop at κ, or we divide a loop at κ into two distinct loops. In the case of amalgamation, the new state s acquires the label on the amalgamated circle that is the product of the labels on the two circles that are its ancestors in s. That is, m(1⊗X) = X and m(X ⊗X) = 0. Thus this case of the partial differential is described by the multiplication in the algebra. If one circle becomes two circles, then we apply the coproduct. Thus if the circle is labelled X, then the resultant two circles are each labelled X corresponding to ∆(X) = X ⊗ X. If the orginal circle is labelled 1 then we take the partial boundary to be a sum of two enhanced states with labels 1 and X in one case, and labels X and 1 in the other case on the respective circles. This corresponds to ∆(1) = 1 ⊗ X + X ⊗ 1. Modulo two, the differential of an enhanced state is the sum, over all sites of type A in the state, of the partial differential at these sites. It is not hard to verify directly that the square of the differential mapping is zero and that it behaves as advertised, keeping j(s) constant. There is more to say about the nature of this construction with respect to Frobenius algebras and tangle cobordisms. See [Kh, BN, BN2]

Here we consider bigraded complexes Cij with height (homological grading) i and quantum grading j. In the unnormalized Khovanov complex [[K]] the index i is the number of B-smoothings of the bracket, and for every enhanced state, the index j is equal to the number of components

8 to produce reliable computation. Nevertheless, one has the freedom to create spaces and operators on the mathematical level and to conceptualize these in a quantum mechanical framework. The resulting structures may be realized in nature and in present or future technology. In the case of our Hilbert space associated with the bracket state sum and its corresponding unitary transformation U, there is rich extra structure related to Khovanov homology that we discuss in the next section. One hopes that in a (future) realization of these spaces and operators, the Khovanov homology will play a key role in quantum information related to the knot or link K.

4 Khovanov Homology and a Quantum Model for the Jones Polynomial

1 X 1. There is a boundary operatorX ∂ deﬁned on the Hilbert space of enhanced states of a link 2 diagram K

∂1: (K) (1 K) C −X→ C X X Xi,j 1 i,j such that ∂∂ = 0 and so that if = (2K) denotes the subspace of (K) spanned by enhanced states s with i = i(s)Cand j C= j(s), then C | " An example of the commutation of partials. ∂ : ij i+1,j. C −→ C That is, we have the formulas i(∂ s ) = i( s ) + 1 | " | " 6 to produce reliable computation. Nevertheless, one has the freedom to create spaces and operators on the mathematical level and to conceptualize these in a quantum mechanical framework. The resulting structures may be realized in nature and in present or future technology. In the case of our Hilbert space associated with the bracket state sum and its corresponding unitary transformation U, there is rich extra structure related to Khovanov homology that we discuss in the next section. One hopes that in a (future) realization of these spaces and operators, the Khovanov homology will play a key role in quantum information related to the knot or link K.

4 Khovanov Homology and a Quantum Model for the Jones Polynomial

1. There is a boundary operatorVirtual Problem∂ deﬁnedof Single Circle Morphismson the Hilbert space of enhanced states of a link diagram K ∂ : (K) (K) C −→ C such that ∂∂ = 0 and so that if i,j = i,j(K) denotes the subspace of (K) spanned by enhanced states s with i = i(s)Cand j C= j(s), then C | " ∂ : ij i+1,j. C −→ C That is, we have the formulas i(∂ s ) = i( s ) + 1 | " | " 6

Standard Khovanov Complex works modulo two for virtual knots and links by setting the self -morphism to zero.

A non-trivial modiﬁcation of the Khovanov Complex due to Vassily Manturov solves the problem for integral homology.

We will not discuss the use of Manturov’s idea in this talk,but it enables the ideas that follow to work for virtual knots and links. and j(∂ s ) = j( s ) | ! | ! for each enhanced state s. In the next section, we shall explain how the boundary operator is constructed.

2. Lemma. By deﬁning U : (K) (K) as in the previous section, via C −→ C U s = ( 1)i(s)qj(s) s , | ! − | ! we have the following basic relationship between U and the boundary operator ∂ :

U∂ + ∂U = 0.

Proof. This follows at once from the definition of U and the fact that ∂ preserves j and with a 1 (the value is q) added to the value of a circle labeled with an X (the value is q 1). We could have increases i to i + 1. // − chosen the more neutral labels of +1 and 1 so that − 3. From this Lemma weq+1conclude+1that the1 operator U acts on the homology of (K). We can regard H( (K)) = K⇐⇒er(∂)/I⇐mag⇒ e(∂) as a new Hilbert space on which theC unitary and operator U acts.C In this way, the Khovanov homology and its relationship with the Jones with a 1 (the value is q) added to the value o1f a circle labeled with an X (the value is q 1). We could have polynomial has a naturalq− quantum1 conteXx,t. − chosen the more neutral labels of +1 and 1 so⇐⇒that− ⇐⇒ but, since an algebra involving 1 and X naturally− appears later, we take this form of labeling from the 4. For a fixed value of j,q+1 +1 1 beginning. ⇐⇒ ⇐⇒ ,j i,j • = and C ⊕iC 1 To see how the Khovanov grading arises,q− consider1 theXform, of the expansion of this version of the is a subcomplex of (K⇐) ⇒with− the⇐⇒boundary operator ∂. Consequently, we can speak of bracket polynonmial in enhanced states. W,jCe have the formula as a sum overijenhanced states s : but, since an algebrathe homologyinvolving 1 andH( X• naturally). Note thatappearsthelaterdimension, we takeothisf formis equalof labelingto thefromnumberthe of enhanced beginning. C C states s with i = Ki(s)=and j(=1)j(nsB).(sConsequently)qj(s) , we have | ! Enhanced$ % State Sum− Formula for the Bracket To see how the Khovanov grading arisTeos,seeconshiodwer tthehe fKorhomvoafnothevegradingxpansionarises,of thisconsiders the form of the expansion of this version of the version of the bracket polynonmial in enhanced states. We have the formula as a sum! j(s) i(s) j i ij bracket polynonmial in enhanced states. WKe ha=ve the qformula( 1)as a sum= overqenhanced( 1)statesdims(: ) over enhanced states s : where nB(s) is the number of B-type smoothings% ! sin s, λ(−s) is the numberj iof −loops in sClabeled 1 minus nB (s) j(s) ! ! ! theK number= !( o1f) loopsq labeled X, and j(s) = nB(s) + λ(nsB)(.s)Tjhis(s) can be rewritten in the following form: ! " − K = ( 1) q s $ % s − j ,j j ,j i(s) = n (s) = number of iB-smoothingsj ij in the B != q χ( • ) = q χ(H( • )). where n (s) is the number of B-type smoothings in s, λ(s) is the number oKf loo=ps in (state1) s.q dimC ( ) C B where nB(s) is the number of B-type$ smoothings% −in js, λ(s) is theC numberj of loops in s labeled 1 minus s labeled 1 minus the number of loops labeled X, and j(s) = n (s) + λ(s).=T nhumberis cani of,j +1 !loops minus number of! -1 loops. the number of loops labeledB X, and j(s) = nB!(s) + λ(s). This can be rewritten in the following form: (X) be rewritten in the following form: ij Here we use the definition of the Euler characteristic of a chain complex where we define to be the linear spanK(o=ver k( =1)iZqj/dim2Z (asijwe) will work with mod 2 coefficients) of i j C ij $ % − C theK s=et of(enhanced1) q dim(states) with n (s) = i and ji ,(js) = j. Then,j the numberi of suchij states is the dimension ! B ij ! χ( • ) = ( 1) dim( ) ! " ij − C C = module generated by enhanced states dim( i ,j). ij C − C where we define to be the linear span (owithver ik =n= andZ/ 2j Zas aboas weve.!iwill work with mod 2 coefficients) of C C B ij the set of enhanced states with n (s) = i and j(s) = j. Then the number of such states is the dimension where we define to be theWlienewarouldspan lik(oveertok h=avZe/and2aZbaigradedtshewefactwillcthatwompleBorkthewixthEcomposedmulerod characteristicof the ijofwiththe acompledifferentialx is equal to the Euler characteristic 2 coefficients) ofCthe set of enhanceddimstat(esijw)i.th n (s) = i and j(s) = j. Then the C B C number of such states is the dimension dim( ij). of its homology. The quantumij amplitudei+1 j associated with the operator U is given in terms C ∂ : ij. We would like to ofhavthee a bEigradeduler characteristicscomplex composedC of−the→ofCtheKhovawithnovahomologydifferentialof the link K. ij C We would like to have a bigraded complex composed of the with a differential ij i+1 j The differential should increaseC the homolog∂ical: grading i by. 1 and preserve the quantum grading j. Then C −→ C j ,j we couldij write i+1 j K = ψ U ψ = q χ(H( • (K))). ∂ : The differential. should increase the homological% g!rading% i|by|1 and! preserve the quantumC grading j. Then C −→ C j i ij !j j j we could write K = q ( 1) dim( ) = q χ( • ), The differential should increase the homological grading i by 1$and%preserjve thej qiuan−- i Cij j j jC tum grading j. Then we could write K =! q! ( 1) dim( ) = !q χ( • ), j $ % j i − C jj7 C where χ( • ) is the Euler characteristic! of !the subcomplex •!for a fixed value of j. j Ci ijj j j Cj K = ! q !( where1) dimχ(( • ))=is!the qEulerχ( •characteristic), of the subcomplex • for a fixed value of j. ! " − C C C j Thisi formula would constitutej a categorification of the bracket polynomial. Below, we shall see how the This formula would constitute a categorification of the bracket polynomial. Below, we shall see how the j original Khovanov differentialj ∂ is uniquely determined by the restriction that j(∂s) = j(s) for each where χ( • ) is the Euler characterisoriginaltic of theKhosubcvanoompvledxiffer• entialfor a fi∂xeisd vuniquelyalue of j.determined by the restriction that j(∂s) = j(s) for each C C j enhancedenhancedstate states. Sinces. Sincej isjpreservis preservededbybythetheddififferential,ferential, thesethesesubcomplesubcomplexes xesj ha•ve hatheirveotheirwn Eulerown Euler C• C This formula would constitucharacteristicste a catecharacteristicsgorificatiaondn ofhomologyatndhe homologybracke.t pWo.leyWnheoamhvaieavle. Below, we shall see how the original Khovanov differential ∂ is uniquely determined by the j j j restriction that j(∂s) = j(s) for each enhanced state s. Since j is preservedχbχ(y(HtHh(e( •• ))))==χχ((• •) ) j CC CC differential, these subcomplexes • have their own Euler characteristics and homol- C where Hj ( j) denotes the homology of the complex jj. We can write ogy. We have where H( • ) denotesC• the homology of the complexC•• . We can write jC j C χ(H( • )) = χ( • ) j j K = q jχ(H( • ))j. C C K = q χ(H( • )). j j $ % j C where H( • ) denotes the homology of the complex • . We can write $ % C C C !j The last formula expresses the bracket polynomial!as a graded Euler characteristic of a homology theory j j K = ! q χ(H( • )). The! "lastassociatedformulawithCexpressesthe enhancedthe brackstateset ofpolynomialthe bracketasstatea grsummation.aded EulerThischarisacteristicthe categorificationof a homologyof the theory associatedbrackj withet polynomial.the enhancedKhovanostatesv provofes thethat tbrackhis homologyet statetheorysummation.is an invariantThisoisf knotsthe candatelinksgorification(via the of the The last formula expresses thbracke bracketeReidemeistertpolynomial.polynomial amos KhoavgersaovdfeanoFdigureEvulepror1),chvacreatingersacthatteristtaihiscneowfhomologyand strongertheoryinvariantis anthanintvheariantoriginalof knotsJones polynomial.and links (via the a homology theory associatedReidemeisterwith the enhancmoed svtaetsesooffFthiguree brac1),ket screatingtate summaatneionw. and stronger invariant than the original Jones polynomial. This is the categorification of the bracket polynomial. Khovanov proves that this ho- 6 mology theory is an invariant of knots and links (via the Reidemeister moves of Figure 1), creating a new and stronger invariant than the original Jones polynomial. 6

6 To see how the Khovanov grading arises, consider the form of the expansion of this version of the bracket polynonmial in enhanced states. We have the 1formula as a sum with a 1 (the value is q) added to the value of a circle labeled with an X (the value is q− ). We could have ovchosener enhtheanmceored sneutraltateslabelss : of +1 and 1 so that − nB (s) j(s) K =+1!( 1) q ! " q +1− 1 ⇐s⇒ ⇐⇒ and 1 where nB(s) is the number of B-tqy−pe smo1othingXs, in s, λ(s) is the number of loops in ⇐⇒ − ⇐⇒ s lbaubt,esinceled 1anmalgebrainus tinhevonlvingumb1eandr ofXlonaturallyops labappearseled Xlater, a,nwed jtak(se)this= nformB(sof)labeling+ λ(s)from. Ththeis can bebereginning.written in the following form:

To see how the Khovanov grading arises, consider ithejform of theij expansion of this version of the bracket polynonmial in enhanced states.K W=e ha!ve(the 1formula) q daisma(sum o)ver enhanced states s : ! " − C i ,j K = ( 1)nB (s)qj(s) $ % s − where we define ij to be the linear sp!an (over k = Z/2Z as we will work with mod where nB(s) is the Cnumber of B-type smoothings in s, λ(s) is the number of loops in s labeled 1 minus 2 ctheoenumberfficienotfsl)oopsof labeledthe seXt o, andf enj(hsa)n=cnedB(s)ta+teλs(sw). iTthhisncBan(bes)re=writteni anindthej(follos) wing= j.form:Then the ij number of such states is the dimKens=ion d(im1)i(qjdim).( ij) $ % − C C !i ,j ij To see how the KhovWanewhereovwgroaduwinledgdalefineriiksees,tcooijnhstoiadvebereththeae fbolinearirKhogmroavanofdtshepanevd econstructsxc(pooavmnesripoknl ediffo=xf ertZhcentialio/s2mZ pactingasoswee ind willtheof ftormwhorke withwmodith 2a coefdiffficients)erentiaolf version of the bracket polynonmial in enhancCed states. We have the formula as a sum C the set of enhanced states with nB(s) = i and j(s) = j. Then the number of such states is the dimension over enhanced states s : ij ij i+1 j dim( ). nB (s) j(s) ∂ : . KC = !( 1) q ! " − C −→ C We would liks e to have a bigraded complex composed of the ij with a differential For j to be constant as i increasesC by 1, we need where nB(s) is the numTbehreofdBif-tfyepreesnmtoiaotlhsinhgos uinlsd, λin(sc)riesathseentuhmebehroomf loolposginical grading i by 1 and preserve the quan- ∂ : ij i+1 j. s labeled 1 minus the nutmubmergofralodopins lgabje.leTd Xhe, nandwje(sc)o=unldB(ws)r+itλe(s)C.toT hdecr−is→ceaseaCn by 1. be rewritten in the following form: The differential should increase the homolog[gicalo backgrading two slides]i by 1 and preserve the quantum grading j. Then we could write i j ij j i ij j j K = !( 1) q dKim( =) ! q !j ( 1)i dim(ij ) = j! qj χ( • ), ! " − ! "C K = q (−1) dim( C) = q χ( • ), C i ,j $ %j i − C j C !j !i !j ij where we define to be twherehe lineχar( spajn) is(ovtheer kEuler= Zcharacteristic/2Z as we wilofl wtheorkswubcompleith mod x j for a fixed value of j. C C• j C• j 2 coefficients) of the sewt ohf eenrheanχc(ed s•tat)esiwsitthhenBE(su)le=r icahnadrja(cst)e=risjt.iTchoenf tthhee subcomplex • for a fixed value of j. number of such states is theThisdimeformulansioCn diwmould( ij)c.onstitute a categorification of the bracket polynomial.CBelow, we shall see how the C original Khovanov differential ∂ is uniquely determined by the restriction that j(∂s) = j(s) for each ij We would like to have aTbihgenhancedirsadefdocrommstateuplleax cswo. moSincepuolsdedjcooisf nthpreservsetitutweedithabyacdtheaiftfedrgeifnoferential,triailficatiothesen osfubcomplethe braxcesketjphaovlye ntheiromoiwaln. EulerBelow, C C• wecharacteristicsshall see handowhomologythe or.igWienhaalveKhovanov differential ∂ is uniquely determined by the ∂ : ij i+1 j. restrictionC th−a→t jC(∂s) = j(s) for each ejnhancedj state s. Since j is preserved by the χ(H( • )) = χ( • ) The differential should increase the homological grading i by 1 and preserjveCthe quan- C differential,jthese subcomplexes • have thjeir own Euler characteristics and homol- tum grading j. Then we cowhereuld wriHte ( • ) denotes the homology of theC complex • . We can write ogy. We hCave C j j j i ij j j K = jq χ(H( • )). j K = ! q !( 1) dim( ) = ! q χ( • )χ, (H( • )) = χ( • ) ! " − C C $ % j C j i j C! C The last formulaj expresses the bracket polynomial as a graded Euler charj acteristic of a homology theory j where H( • ) denotes the hjomology of the complex • . We can write where χ( • ) is the Euler characteristic of the subcomplex • for a fixed value of j. C associatedCwith the enhanced statesC of the bracket state summation.C This is the categorification of the bracket polynomial. Khovanov proves that this homology theory is an invariant of knots and links (via the This formula would constitute a categorification of the bracket polynomial. Belojw, j Reidemeister moves of Figure 1), creatingK a=new!andqstrongerχ(H(invariant• )).than the original Jones polynomial. we shall see how the original Khovanov differential ∂ is unique!ly de"termined by the C j restriction that j(∂s) = j(s) for each enhanced state s. Since j is preserved by the j differential, these subcomplexes • have their own Euler characteristics and ho6mol- ogy. We have The lasCt formula expresses the bracket polynomial as a graded Euler characteristic of j j χ(H( • )) = χ( • ) a homologCy theoryCassociated with the enhanced states of the bracket state summation. j j where H( • ) denotes tThehhisomioslotghyeofctahteecgoomrpilfiexca•tio. Wneocfanthweritebracket polynomial. Khovanov proves that this ho- C C mology theoryj is an ijnvariant of knots and links (via the Reidemeister moves of Figure K = ! q χ(H( • )). ! " C 1), creatingj a new and stronger invariant than the original Jones polynomial.

The last formula expresses the bracket polynomial as a graded Euler characteristic of a homology theory associated with the enhanced states of the bracket state summation. This is the categoriﬁcation of the bracket polynomial. Khovanov proves that this homology theory is an invariant of knots and links (via the Reidemeister moves of Figure 1), creating a new and stronger invariant than the original Jones polynomial. 6

6 1 with a 1 (the value is q) added to the value of a circle labeled with an X (the value is q− ). We could have chosen the more neutral labels of +1 and 1 so that − q+1 +1 1 ⇐⇒ ⇐⇒ and 1 q− 1 X, 1 with a 1 (the value is q) added to the value of ⇐a circle⇒ −labeled⇐⇒with an X (the value is q− ). We could have but, sincechosenanthealgebramore neutralinvolvinglabels1ofand+1Xandnaturally1 so thatappears later, we take this form of labeling from the − beginning. q+1 +1 1 ⇐⇒ ⇐⇒ Toandsee how the Khovanov grading arises, consider the form of the expansion of this version of the 1 q− 1 X, bracket polynonmial in enhanced states. We hav⇐e⇒the−formula⇐⇒ as a sum over enhanced states s : but, since an algebra involving 1 and X naturally appears later, we take this form of labeling from the K = ( 1)nB (s)qj(s) beginning. $ % s − 1 with a 1 (the value is q) added to the value of !a circle labeled with an X (the value is q− ). We could have To see how the Khovanov grading arises, consider the form of the expansion of this version of the where nB(chosens) is thethe numbermore neutralof labelsB-typeof +1smoothingsand 1 so thatin s, λ(s) is the number of loops in s labeled 1 minus bracket polynonmial in enhanced states. We−have the formula as a sum over enhanced states s : the number of loops labeled X, and j(s) = nBq+1(s) + λ+1(s). T1his can be rewritten in the following form: ⇐⇒ n⇐B (⇒s) j(s) K = ( 1)i j q ij and K$ =% s( −1) q dim( ) $ % 1 !− C q−i ,j 1 X, where nB(s) is the number of B-type smoothings!⇐⇒in−s, ⇐λ(⇒s) is the number of loops in s labeled 1 minus but, since an algebra involving 1 and X naturally appears later, we take this form of labeling from the where thewenumberdefine ofijloopsto belabeledthe linearX, andsjpan(s) =(ovneBr(sk) +=λZ(s/)2. ZThisascwean bewillrewrittenwork inwiththe follomodwing2 coefform:ficients) of beginning.C the set of enhanced states with nB(s) = iKand= j(s() =1)ijq.jThendim( theij) number of such states is the dimension ij To see how the Khovanov grading$ arises,% consider− the formC of the expansion of this version of the dim( ). i ,j C bracket polynonmial in enhanced states. We ha!ve the formula as a sum over enhanced states s : ij where we define to be the linear span (over k = Z/2Z asijwe will work with mod 2 coefficients) of We would like to haveCa bigraded complex composedK = ( of1)ntheB (s)qj(s) with a differential the set of enhanced states with nB(s) = i$and% j(ss) =− j. ThenCthe number of such states is the dimension ij ! dim( ). ∂ : ij i+1 j. whereC nB(s) is the number of B-type smoothingsC −in→s,Cλ(s) is the number of loops in s labeled 1 minus the number of loops labeled X, and j(s) = n (s) + λ(s). Thisij can be rewritten in the following form: The difWferentiale would likshoulde to haincreaseve a bigradedthe homolocomplexgicalcomposedBgradingof thei by 1withandapdreservifferentiale the quantum grading j. Then i j C ij K = ij ( 1) qi+1dimj ( ) we could write $ %∂ : − . C j C!i ,ji −→ C ij j j K = q ( 1) dim( ) = q χ( • ), The dwhereifferentialwe defineshouldijincrease$to be% thethelinearhomolospang(ical−overgkrading= Z/Ci2Zbyas1 weandwillpreservworkCe withthe quantummod 2 coefgradingficients)j.oThenf C j i j we couldthe setwofriteenhanced states with!nB(s) != i and j(s) = j. Then the!number of such states is the dimension j j i ij j j j where χ( •dim) (isijthe). Euler characteristicK = ofq the(subcomple1) dim( x) =• forq aχ(fix•ed),value of j. C C $ % j i − C C j C ! ! ij ! This formulaWe wwouldouldj likceonstituteto have a baigradedcategorificationcomplex composedof theofbrackthe etj withpolynomial.a differentialBelow, we shall see how the where χ( • ) is the Euler characteristicThe diffoferentialthe s ubcomple increases thex Chomological• for a fixed value of j. C ij i+1Cj original Khovanov differential ∂gradingis uniquely i by 1 and∂ determinedlea: ves fixed theb quantumy. the r gradingestriction j. that j(∂s) = j(s) for each This formula would constitute a categorificationCof−the→ brackC et polynomial. Below, wej shall see how the enhanced state s. Since j is preserved by the differential, these subcomplexes • have their own Euler originalThe dKhoifferentialvanov shoulddifferentialincrease∂ istheuniquelyhomologicaldeterminedThengrading ibbyy 1theandrepstrictionreserve thethatquantumj(C∂s)gr=adingj(sj). forTheneach characteristicswe couldandwhomologyrite . We have j enhanced state s. Since j is preserved byj the differential,i ijthese subcomplej j xes • have their own Euler K = q ( 1) dim( ) = q χ( • ), C characteristics and homology. $We%have − j C j C !j χ(H!i ( • )) = χ( • !)j j C j C jj wherej χ( • ) is the Euler characteristicχof(Hthe( subcomple• )) = χj x( •• )for a fixed value of j. where H( • ) denotesC the homology of the compleC x • . CWC e can write CThis formulaj would constitute a categorification of theC brackj et polynomial. Below, we shall see how the where H( • ) denotes the homology of the complexj • . We cjan write originalC Khovanov differential ∂ is uniquelyK = determinedq χC(Hb(y •the))re.striction that j(∂s) = j(s) for each $ % j C j enhanced state s. Since j is preservedKby the= djifferential,q χ(H(these• ))s.ubcomplexes j have their own Euler $ % ! C C• characteristics and homology. We have j The last formula expresses the bracket polynomial!as a graded Euler characteristic of a homology theory The last formula expresses the bracket polynomial asj a gradedj Euler characteristic of a homology theory χ(H( • )) = χ( • ) associatedassociatedwith withthe ethenhancedenhancedstatesstatesofofthethebrackbrackCetet statestate ssummation.Cummation.ThisThisis theis thecategorificationcategorificationof theof the bracket polynomial.where H( Khoj) denotesvanovtheprohomologyves thatoftthehischomologyomplex j. theoryWe can writeis an invariant of knots and links (via the bracket polynomial.C• Khovanov proves that this homologyC• theory is an invariant of knots and links (via the Reidemeister moves of Figure 1), creating a new and strongerj jinvariant than the original Jones polynomial. Reidemeister moves of Figure 1), creating aKne=w andqstrongerχ(H( •in))variant. than the original Jones polynomial. $ % C !j The last formula expresses the bracket polynomial as66 a graded Euler characteristic of a homology theory associated with the enhanced states of the bracket state summation. This is the categorification of the bracket polynomial. Khovanov proves that this homology theory is an invariant of knots and links (via the Reidemeister moves of Figure 1), creating a new and stronger invariant than the original Jones polynomial.

6 1 with a 1 (the value is q) added to the value of a circle labeled with an X (the value is q− ). We could have chosen the more neutral labels of +1 and 1 so that − q+1 +1 1 ⇐⇒ ⇐⇒ and 1 q− 1 X, ⇐⇒ − ⇐⇒ but, since an algebra involving 1 and X naturally appears later, we take this form of labeling from the beginning.

To see how the Khovanov grading arises, consider the form of the expansion of this version of the bracket polynonmial in enhanced states. We have the formula as a sum over enhanced states s :

K = ( 1)nB (s)qj(s) $ % s − ! where nB(s) is the number of B-type smoothings in s, λ(s) is the number of loops in s labeled 1 minus the number of loops labeled X, and j(s) = nB(s) + λ(s). This can be rewritten in the following form: K = ( 1)iqjdim( ij) $ % − C !i ,j where we define ij to be the linear span (over k = Z/2Z as we will work with mod 2 coefficients) of C the set of enhanced states with nB(s) = i and j(s) = j. Then the number of such states is the dimension dim( ij). C We would like to have a bigraded complex composed of the ij with a differential C ∂ : ij i+1 j. C −→ C The differential should increase the homological grading i by 1 and preserve the quantum grading j. Then we could write j i ij j j K = q ( 1) dim( ) = q χ( • ), $ % − C C !j !i !j where χ( j) is the Euler characteristic of the subcomplex j for a fixed value of j. C• C• This formula would constitute a categorification of the bracket polynomial. Below, we shall see how the original Khovanov differential ∂ is uniquely determined by the restriction that j(∂s) = j(s) for each enhanced state s. Since j is preserved by the differential, these subcomplexes j have their own Euler C• characteristics and homology. We have j j χ(H( • )) = χ( • ) C C where H( j) denotes the homology of the complex j. We can write C• C• j j K = q χ(H( • )). $ % C !j The last formula expresses the bracket polynomial as a graded Euler characteristic of a homology theory associated with the enhanced states of the bracket state summation. This is the categorification of the bracket polynomial. Khovanov proves that this homology theory is an invariant of knots and links (via the Reidemeister moves of Figure 1), creating a new and stronger invariant than the original Jones polynomial.

6 One advantage of the expression of the bracket polynomial via enhanced states is that it is now a sum of monomials. We shall make use of this property throughout the rest of the paper.

We now use the enhanced state summation for the bracket polynomial with variable q to give a quantum3 Qformulationuantum oStatisticsf the state sum.andLettheq beJoneson thePunitolynomialcircle in the complex plane. (This is equivalent to letting the original bracket variable A be on the unit circle and equivalent to letting the JWonese nowpolynmialuse the enhancedvariablestatet besummationon the unitfor ctheircle.)brackLetet polynomial(K) denotewith tvheariablecompleq toxgivvectore a space C withquantumorthonormalformulationbasisof sthe statewheresum.s runsLet qobeveonr thetheenhancedunit circlestatesin the cofomplethe dxiagramplane. (ThisK. Theis vector spaceequiv(alentK) istotheletting(finitethe{|originaldimensional)! } bracket vHilbertariable Aspacebe onforthe unitour circlequantumand eformulationquivalent to lettingof the Jones polynomial.theCJones polynmialWhile it isvariablecustomaryt be onfortheaunitHilbertcircle.)spaceLet to(Kbe) denotewrittenthewithcomplethex lettervectorHspace, we do not with orthonormal basis s where s runs over the enhancedC states of the diagram K. The vector follow that convention h{|ere,! }due to the fact that we shall soon regard (K) as a chain complex space (K) is the (finite dimensional) Hilbert space for our quantum formulationC of the Jones and take iCts homology. One can hardly avoid using for homology. polynomial. While it is customary for a HilbertRECALLING:spaceHto be written with the letter H, we do not follow that convention here, Adue Quantumto the Statisticalfact that Modelwe forshall Khovanosoonv Homolgregardy (K) as a chain complex Withandq ontaketheits unithomologycircle,. Owene candefinehardlya andunitaryav theoid Brackusingtransformationet Polynonmial.for homology. C Let C(K) denote a HilberH t space with basis |s> where s runs over the With q on the unit circle, we defineenhanceda unitaryU states: of(transformation Ka knot) or link diagram(K) K. We defineC a unitary− transf→ormation.C U : (K) (K) by the formula C −→ C U s = ( 1)i(s)qj(s) s by the formula | ! − | ! for each enhanced state s. Here i(s)Uqand iss chosen=j(s( on) 1)arethei (unitsa)qs jcir(dscle)efineds in the in the previous section of this paper. | ! complex− plane. | ! for each enhanced state s. Here i(s) and j(s) are as defined in the previous section of this paper. Let Let ψ = s . ψ| =! ss .| ! | ! s !| ! The state vector ψ is the sum over the basis s!tates of our Hilbert space (K). For convenience, The state vector| !ψ is the sum over the basis states of our Hilbert space (K).CFor convenience, we dowenotdo notnormalizenormalize| ! ψψ totolengthlength oneoneininthetheHilbertHilbertspacespace(K).(KWe).thenWCehathenve thehave the | | !! C C Lemma.Lemma.TheTheeveavluationaluationoofftthehe brackbracketetpolynomialpolynomialis giisveginvbeynthebyfollothewingfolloformulawing formula

4 To see how the Khovanov grading arises, consider the form of the expansion of this version of the bracket polynonmial in enhanced states. We have the formula as a sum over enhanced states s : nB (s) j(s) K = !( 1) q ! " − s where nB(s) is the number of B-type smoothings in s, λ(s) is the number of loops in s labeled 1 minus the number of loops labeled X, and j(s) = nB(s) + λ(s). This can and be rewritten in the following form: j(∂ s ) = j( s ) | ! i j| ! ij K = !( 1) q dim( ) ! " − C for each enhanced state s. In the next section,i ,j we shall explain how the boundary operator is constructed. and ij where we define to be the linear sjp(∂ans )(=ovj(esr )k = Z/2Z as we will work with mod C | ! | ! 2. Lemma.2 coByeffidefiningcientsfor) oeachfUthenhancede: se(tKostatef)es.nInhathencnee(xdKt ssection,)taastesweinwshallitheth enxplainpreB(svioush)ow=theiboundarysection,and joperator(s)via= j. Then the number of suischconstructed.statesCis the d−im→enCsion dim( ij). i(Cs) j(s) 2. Lemma. By defining U : (K) (K) as in the previous section, via UC s −=→ (C 1) q s , We would like to have a bigrade|d!comp−lex compose|d !of the ij with a differential With U s = ( 1)i(s)qj(s) s , C we have the following basic relationship| !between− U| !and the boundary operator ∂ : we have the following basic relationship∂ : ijbetween Uiand+1 tjhe. boundary operator ∂ : C −→ C U∂ + ∂U = 0. The differential should increase theUh∂om+o∂loUgic=al 0g.rading i by 1 and preserve the quantum grading jPr.oofTh. eThisn wfolloe cwsouatldoncewThisrfi rommeanste the thatd efinitionthe unitaryo ftransfU andormationthe fact that ∂ preserves j and increases i to i + 1. // U acts on the homology so that

Proof. This follo3. wsFrom atthisonceLemma fweromconcludej thethatU:d H(C(K))efinitionthe ioperator -----> H(C(K))Uoiactsjf Uonandthe homologythej factof j (Kthat). We∂ preserves j and K = ! q !( 1) dim( ) = ! q χ( • C), increases i to i + 1can. //regar! d H"( (K)) = Ker(∂)/I−mage(∂) as a nCew Hilbert space on whichC the unitary operator U acts.C In tjhis way, tihe Khovanov homology and its relationshipj with the Jones polynomial has a natural quantum context. 3. From this Lemmaj we conclude that the operator U acts on thej homology of (K). We where χ( •4. F)orisa fitxheed vEaluueleofrjc, haracteristic of the subcomplex • for a fixed value of j. C ,j i,j C C can regard H( (K)) = Ker(∂)/Image(∂• )=asi a new Hilbert space on which the unitary C C ⊕ C operatorThUis acts.formuIlaniswathissoubcompleuldwcayoxn,ofsttheit(uKt)Khoewitha cthevaanotboundaryegovrihomologyficoperatoration ∂o. fConsequentlythande britsac, kwrelationshipeetcanpospeaklynoofmialw. ithBeltheow,Jones ,jC ij the homology H( • ). Note that the dimension of is equal to the number of enhanced polynomialwe shhasall saeenaturalstateshowsthwithquantume oir=iCgi(isn) aandl conteKj =hoj(vsa)x.nConsequentlyt.ov differ,Cewnethiavle ∂ is uniquely determined by the restriction that j(| ∂! s) = j(s) for each enhanced state s. Since j is preserved by the K = qj(s)( 1)i(s) = qj ( 1)idim( ij) % ! j− − C 4. For a fidxiefdferveanltuiael,otfhejs,e subcomplexess • have thjeir oi wn Euler characteristics and homol- !C ! ! ogy. We have j ,j ,j j i,j ,j = q χ•( • =) = iq χ(H( • )). C j C j χ!j(HC( • ))⊕!j=Cχ( • ) Here we use the definition of the EulerCcharacteristic ofCa chain complex is a subcomplex ofj (K) with the boundary operator ∂. jConsequently, we can speak of where H( • ) denotes the homology of the complex • . We can write ,jC ,j i ijij the homology HC( ). Note that the χd(imension• ) = ( 1) odimf ( ) isC equal to the number of enhanced • C − C C i C states s with i = i(s) and j = j(s). Consequently! j , wje have and the fact that the Euler characteristicK = !of qtheχcomple(H(x is•equal)).to the Euler characteristic | ! ! " C of its homology. The quantum amplitudej associated with the operator U is given in terms of theKEuler=characteristicsqj(s)of( the1)Khoi(vsa)no=v homologyqjof the(link 1)K. idim( ij) % ! − j ,j − C The last formula expressses theKbr=acψkUetψpo=lynqojχm(Hia( l•ai(sKa)))g. raded Euler characteristic of ! % ! % | | ! ! C! a homology theory associated with the enh!aj nced states of the bracket state summation. This is the categorification of jthe bra,jcket polynjomial. Kh,jovanov proves that this ho- = q χ( • ) =7 q χ(H( • )). mology theory is an invariajnt of knCots and linj ks (via theCReidemeister moves of Figure 1), creating a new and str!onger invariant tha!n the original Jones polynomial. Here we use the definition of the Euler characteristic of a chain complex

,j i ij χ( • ) = ( 1) dim( ) C − C !i 6 and the fact that the Euler characteristic of the complex is equal to the Euler characteristic of its homology. The quantum amplitude associated with the operator U is given in terms of the Euler characteristics of the Khovanov homology of the link K.

j ,j K = ψ U ψ = q χ(H( • (K))). % ! % | | ! C !j

7 and j(∂ s ) = j( s ) | ! | ! for each enhanced state s. In the next section, we shall explain how the boundary operator is constructed.

2. Lemma. By deﬁning U : (K) (K) as in the previous section, via and C −→ C j(∂ s ) = j( s ) | ! i(s|) ! j(s) for each enhanced state s. UIn thes ne=xt s(ection,1)we shallq explains ,how the boundary operator is constructed. | ! − | ! we have the follo2. Lemma.wing basicBy deﬁningrelationshipU : (K) b(Ketween) as in theUpreviousand section,the boundaryvia operator ∂ : C −→ C U s = ( 1)i(s)qj(s) s , U|∂! +−∂U = 0| .! we have the following basic relationship between U and the boundary operator ∂ :

Proof. This follows at once from the dUeﬁnition∂ + ∂U = 0.of U and the fact that ∂ preserves j and increases i to i +Pr1oof. .// This follows at once from the deﬁnition of U and the fact that ∂ preserves j and increases i to i + 1. //

3. From this Lemma3. Fromwethis concludeLemma we concludethat thatthetheoperatoroperator U Uacts actson the ohomologyn the homologyof (K). We of (K). We can regard H( (K)) = Ker(∂)/Image(∂) as a new Hilbert space on which theC unitary C can regard H( (K)) = KC er(∂)/Image(∂) as a new Hilbert space on which the unitary C operator U acts. In this way, the Khovanov homology and its relationship with the Jones operator U acts.polynomialIn thishaswaya natural, thequantumKhovconteanoxvt. homology and its relationship with the Jones polynomial has4. aFonaturalr a fixed valquantumue of j, context. ,j i,j • = C ⊕iC 4. For a fixed valueisoafsubcomplej, x of (K) with the boundary operator ∂. Consequently, we can speak of ,jC ij the homology H( • ). Note that the d,jimension of i,jis equal to the number of enhanced states s with i =Ci(s) and j = j(s).•Consequently= i ,Cwe have | ! C ⊕ C is a subcomplex of (K) withK the= boundaryqj(s)( 1)i(s) = operatorqj ( 1)i∂dim. Consequently( ij) , we can speak of % ! s − − C ,jC ! !j !i ij the homology H( • ). Note that the dimension of is equal to the number of enhanced j ,j j ,j C = q χ( • ) = q χ(HC( • )). states s with i = i(s) and j = j(s). ConsequentlyC , wC e have | ! !j !j Here we use the definition of the Euler characteristic of a chain complex jThis(s) shows hoiw(s ) as a quantumj amplitude i ij K = q contains( 1) inf,jormation= abouti theq homologij ( y. 1) dim( ) χ( • ) = ( 1) dim( ) % ! s −C − j C i − C ! !i ! ! and the fact that the Euler characteristic of the complex is equal to the Euler characteristic j ,j j ,j of its homology.=The quantumq χamplitude( • ) =associatedq withχ(Hthe(operator• ))U. is given in terms of the Euler characteristics of theCKhovanov homology of the linkC K. !j !j j ,j K = ψ U ψ = q χ(H( • (K))). % ! % | | ! C Here we use the definition of the Euler characteristic!j of a chain complex

,j 7 i ij χ( • ) = ( 1) dim( ) C − C !i and the fact that the Euler characteristic of the complex is equal to the Euler characteristic of its homology. The quantum amplitude associated with the operator U is given in terms of the Euler characteristics of the Khovanov homology of the link K.

j ,j K = ψ U ψ = q χ(H( • (K))). % ! % | | ! C !j

7 be a unitary operator that satisfies the equation U∂ +∂U = 0. We do not assume a second grading j as occurs in the Khovanov homology. However, since U is unitary, it follows [22] that there is a basis for in which U is diagonal. Let = s denote this basis. Let λs denote the eigenvalue of U correspondingC to s so that U s =B λ {|s .!Let} α be the matrix element for ∂ so that | ! | ! s| ! s,s! be a unitary operator that satisfies the equation U∂ +∂U = 0. We do not assume a second grading ∂ s = α s! j as occurs in the Khovanov homology. However, since U is unitary, it follows [22]|that! there is as,s! | ! s! basis for in which U is diagonal. Let = s denote this basis. Let λs denote the eigen!value of U correspondingC to s so that U s =B λ {|s .!Let} α be the matrix element for ∂ so that | ! where| ! ss!|runs! ovs,ser! a set of basis elements so that i(s!) = i(s) + 1. be a unitary operator that satisfies the equation U∂ +∂U = 0. We do not assume a second grading ∂ s = αs,s s! j as occurs in the Khovanov homology. |H!owever, since! |U !is unitary, it follows [22] that there is a Lemma. Ws!ith the above conventions, we have that for s! a basis element such that αs,s! = 0 be abasisunitaryfor operatorin whichthatU issatisfiesdiagonal.theLetequation= Us!∂ +denote∂U =this0. Wbasis.e do notLet assumeλ denotea stecondhe eigengradingvalue | ! " C then λBs ={| !} λs. s wherej asofsoccurs!Urunscorrespondinginovtheer aKsethovtoaonofsbvasissohomologythatelementsU s. H=osoλw! evsthater.,Letsiince(sα!)U=beisi(unitarythes) m+atrix1,.it folloelementws [22]for ∂thatso thatthere is a | ! | ! s| ! − s,s! basis for in which U is diagonal.PrLetoof.=Notes denotethat this basis. Let λs denote the eigenvalue C B∂ s {|= !} α s! Lemma.of U correspondingWith the abotoves conso thatventions,U s =wλe hsav. eLetthatαs,s!forbe sthe! amatrixbasiselementelementforsuch∂ sotthathat αs,s = 0 | s! s,s|! ! U∂ s = U( α ! s! ) = α λ s! | ! | ! | ! !s! | ! s,s! " s,s! s! then λs! = λs. | ! | ! | ! s! s! where−s runs over a set of basis elements∂ s so= thatαi(s,ss )s=! i(s) + 1. ! ! Proof. Note !that | ! !! | ! while !s! U∂ s = U( αs,s s! ) = αs,s λs s! Lemma. With the above conventions, we have! that for s! a basis! ! element such that αs,s! = 0 where s! runs over a set of basis| !elementssso that|i(s!!) = i|(ss!) + 1. | ! ∂U s = ∂λs" s = αs,s λs s! . then λs = λs. !! !! ! ! − | ! | ! | ! whileProof. Note that !s! Lemma. With the above conventions, we have that for s! a basis element such that αs,s! = 0 | ! ! " then λ = λ . U∂∂UsSinces= U=( ∂Uλ∂αs s,ss+! s∂=! U) ==αs,s0α,!s,sλthes! λss!cs.onclusion! of the Lemma follows from the independence of the elements s! s | !| ! | !| ! | |! ! − s! s s! Proof. Note that in the!basis for!the! !Hilbert space. // SincewhileU∂+∂U = 0, the conclusion of the Lemma follows from the independence of the elements U∂ s = U( αs,s! s! ) = αs,s! λs! s! | !∂U s = ∂λs s| =! αs,s! λs s! . | ! in the basis for the Hilbert space. // | ! !s! | ! !s! | ! 0 In this way wes! see that eigenvalues will propagate forward from with alternating signs ac- while ! Since U∂+∂U = 0, the conclusion of the Lemma follows from the independence0 of the elements C In this way we see that eigenvalues∂Ucordings will= ∂λpropagatetos =the aαforwppearanceλards! from. of successiwith alternatingve basissignselementsac- in the boundary formulas for the chain in the basis for the Hilbert space.|//! s| ! s,s! s| ! C cording to the appearance of successicompleve basisx. Velementsarious!s! statesin the ofboundaryaffairsformulasare possiblefor theinchaingeneral, with new eigenvaluues starting at some compleSinceInx.thisUV∂a+wrious∂ayUw=statese 0see, thethatofconclusionafeigenfairsvaluesarekofpossiblethewillLemmapropagatein follogeneral,forwws fromardwithfromtheneindependencew0 eigenwith alternatingvaluuesof thestartingselementsigns ac-at some k for k > 0. The simplestC state of affairs would be if all the possible eigenvalues (up to multi- inforthecordingk b>asis0.toforThethetheasppearanceimplestHilbert sspace.tateof successiofC// affairsve basiswouldelementsbe if allin thetheboundarypossibleformulaseigenvalues0for the(upchainto multi- Cplicationcompleby x. 1V)ariousfor Ustatesoccurredof affairsinplicationare0 sopossiblethatbiyn general,1) forwithUneoccurredw eigenvaluuesin startingso thatat some In thisk wa−y we see that eigenvaluesC will propagate−forward from 0 with alternatingC signs ac- for k > 0. The simplest state of affairs would be if all the possibleC eigenvalues (up to multi- cordingC to the appearance of successi0ve basis0 elements0 in the boundary formulas for the c0hain 0 plication by 1) for U occurred in so that = = λ complex. Various− states of affairs areCpossibleC in g⊕eneral,λCλ with new eigenvaluues starting atCsome ⊕ Cλ k 0 = 0 whereforλ runsk > 0o.vTheer allsimplestthe distinctstate ofeigenaffairsvalueswouldofbλeUiλf allrestrictedthe possibleto eigen0, andvalues0 is(upspannedto multi-by all 0 0 C where0 λC runs⊕ Cover all the distinct eigenλ values of U restricted to , and is spanned by all plication0 by 1) for U occurred in so that C C λ s inwherewithλ runs−U sove=r allλ thes . dLetistinctuCseigentakeEigenspacevaluesthe0 furtherof PicturU restrictede assumptionto 0,thatandfor0 iseachspannedλ asbaboy allve, the C C | ! C | ! | ! s in with U s = λCs . LetCλ us take the further assumption that for each λ as above, the subcomples inxes0 with U s = λ s . Let us take t0he further0 assumption that for each λ as above, the | ! C = λ λ | ! | ! |subcomple! C xes | ! | ! 0subcomple1C x⊕esC2 n λ• : λ λ +λ ( 1)nλ C C 0−→ C−1 −→ C2 −→ · · ·nC − 0 00 1 2 n where λ runs over all the distinctλ• : λeigenvaluesλ of U+λrestricted (to1)nλ , and• : λ is spanned by all n C C −→ C− −→ C −→ · · · C − C λ C λ λ +λ ( 1) λ haves in=0 withλ λ•Uass their= λdirects . Letsum.us takeWtithhe furtherthis assumptionassumptionatbouthatCfortheeachCchain−λ→ascompleaboC−ve,−x,the→defineC −→ · · · C − Chave ⊕=C λ • as their direct sum. With this assumption about the chain complex, define ψ |=! C Cs as ⊕before,C| λ! with| ! s running over the whole basis for . Then it follows just as in the subcompleψ s= xess as before, with s harunningve ove=r the wholeλ λ• basasistheirfor . Thendirectit follosum.ws justWasithin thethis assumption about the chain complex, define | ! | ! | ! s | ! | !| !0 C1 ⊕2 C n C C beginningbe"ginningof thisof thissectionsectionthatthatλ• : λ λ +λ ( 1)nλ " C C ψ−→=C− −s→sC as−→before,· · · C − with s running over the whole basis for . Then it follows just as in the | ψψ!UUψψ == | !λχλχ((HH(C(C•))λ•.)). | ! C have = λ λ• as their direct sum.be& &ginning| | |W| ith!! thisofassumptionthis sectionλ aboutthatthe chain complex, define C ⊕ C " λλ ψ = s s as before, with s running over the!whole basis for . Then it follows just as in the Here| !Hereχ denotesχ|denotes! the theEulerEulercharacteristic|characteristic! ofof thethe homologyhomology. .TheTChepointpointis, tis,hat tthatψhisUformulathisψ formula=for λχfor (H(Cλ•)). beginning" of this section that & | | ! ψ U ψψ Utakψestakexactlyes exactlythetheformformwewehadhadforforthethe sspecialpecialcasecaseofofKhoKhovanovanov homologyv homology(with(witheigen-eigen-λ & | | ! i j ψ U ψ = λχ(H(Cλ•)). ! & | |values! (i j1) q ), but here the formula occurs just in terms of the eigenspace decomposition of the values ( 1) q ), but here the formula& |occurs| ! justλ in terms of the eigenspace decomposition of the unitary t−ransformation U in relationHereto theχ chaindenotes! complethex. ClearlyEulertherecharacteristicis more work to beofdonethe homology. The point is, that this formula for unitaryHeret−ransformationχ denotes the EUulerin characteristicrelation to theofchainthe homologycomplex.. ClearlyThe pointthereis, tihats mtorehis formulawork toforbe done here and we will return to it in a sψubsequentU ψ papertake.s exactly the form we had for the special case of Khovanov homology (with eigen- ψ U ψ takes exactly the form we had for the special case of Khovanov homology (with eigen- here and we will return to it in a subsequent& | | !paperi .j v&alues| | (! 1)iqj), but here the formulavaluesoccurs( just1)inqterms), bouftthheereeigenspacethe formuladecompositionoccursofjustthe in terms of the eigenspace decomposition of the unitary t−ransformation U in relation to the chain− complex. Clearly there is more work to be done unitary transformation10 U in relation to the chain complex. Clearly there is more work to be done here and we will return to it in a subsequent paper. here and w10e will return to it in a subsequent paper.

10 10 One advantage of the expression of the bracket polynomial via enhanced states is that it is now a sum of monomials. We shall make use of this property throughout the rest of the paper.

3 Quantum Statistics and the Jones Polynomial

We now use the enhanced state summation for the bracket polynomial with variable q to give a quantum formulation of the state sum. Let q be on the unit circle in the complex plane. (This is equivalent to letting the original bracket variable A be on the unit circle and equivalent to letting the Jones polynmial variable t be on the unit circle.) Let (K) denote the complex vector space with orthonormal basis s where s runs over the enhancedC states of the diagram K. The vector space (K) is the (ﬁnite{| dimensional)! } Hilbert space for our quantum formulation of the Jones polynomial.C While it is customary for a Hilbert space to be written with the letter H, we do not follow that convention here, due to the fact that we shall soon regard (K) as a chain complex and take its homology. One can hardly avoid using for homology. C H With q on the unit circle, we deﬁne a unitary transformation

U : (K) (K) C −→ C by the formula U s = ( 1)i(s)qj(s) s | ! − | ! for each enhanced state s. Here i(s) and j(s) are as defined in the previous section of this paper. and Let j(∂ s ) = j( s ) | ! | ! ψ = s . for each enhanced state s. In the next|section,! s we| !shall explain how the boundary operator is constructed. ! The state vector ψ is the sum over the basis sSUMMARtates Yof our Hilbert space (K). For convenience, | ! We have interpreted the bracket polynomial as a C we do not normalize ψ to length onequantumin amplitudethe H byilbert making a spaceHilbert space( C(K)K) . We then have the 2. Lemma. By defining| ! U : (Kwhose) basis is the( Kcollection) as ofin enhancedthe prestatesCvious of the section, via C −→ C bracket. Lemma. The evaluation of the brackThiset spacepolynomial C(K) is naturalli(yiss intepr) gij(vetedse) n as bthey the following formula Uchains space= for( the1) Khovanoqv homologs y , associated| ! with− the bracket polynomial.| ! K = ψ U ψ . we have the following basic relationship between U and the boundary operator ∂ : The homolog$ y! and the$ unitar| y |transf!ormation U Proof. speak to one another via the formula Ui(∂s)+j(∂s)U = 0. i(s) j(s) ψ U ψ = s! ( 1) q s = ( 1) q s! s $ | | ! s $ | − | ! s − $ | ! !s! ! !s! ! Proof. This follows at once from the definition of U and the fact that ∂ preserves j and = ( 1)i(s)qj(s) = K , increases i to i + 1. // s − $ ! ! since 3. From this Lemma we conclude that the operator U acts on the homology of (K). We s! s = δ(s, s!) C can regard H( (K)) = Ker(∂)/Imag$ | e!(∂) as a new Hilbert space on which the unitary C whereoperatorδ(s, s!)Uisacts.the KIroneckn this weraydelta,, the equalKhovanoto 1vwhenhomologys = s!andanditsequalrelationshipto 0 otherwise.with the// Jones polynomial has a natural quantum context. 4 4. For a fixed value of j, ,j i,j • = C ⊕iC is a subcomplex of (K) with the boundary operator ∂. Consequently, we can speak of ,jC ij the homology H( • ). Note that the dimension of is equal to the number of enhanced states s with i =Ci(s) and j = j(s). Consequently,Cwe have | ! K = qj(s)( 1)i(s) = qj ( 1)idim( ij) % ! s − − C ! !j !i

j ,j j ,j = q χ( • ) = q χ(H( • )). C C !j !j Here we use the deﬁnition of the Euler characteristic of a chain complex

j ,j K = ψ U ψ = q χ(H( • (K))). % ! % | | ! C !j

7 Questions We have shown how Khovanov homology fits into the context of quantum information related to the Jones polynomial and how the polynomial is replaced in this context by a unitary transformation U on the Hilbert space of the model. This transformation U acts on the homology, and its eigenspaces give a natural decomposition of the homology that is related to the quantum amplitude corresponding to the Jones polynomial. The states of the model are intensely combinatorial, related to the representation of the knot or link. How can this formulation be used in quantum information theory and in statistical mechanics?! a smoothing of type A, then ∂i(s) is given by the rules discussed above (with the same signs). The compatibility conditions that we have discussed show that partials commute in the sense that ∂i(∂j(s)) = ∂j(∂i(s)) for all i and j. One then defines signed boundary formulas in the usual way of algebraic topology. One way to think of this regards the complex as the analogue of a complex in DeRahm cohomology. Let dx1, dx2, , dxn be a formal basis for a Grassmann algebra so that dxi dxj = { dx dx· ·S· tartin}g with enhanced states s in C0(K) (that is, state with al∧l A-type − i ∧ j smoothings) Define formally, di(s) = ∂i(s)dxi and regard di(s) as identical with 1 ∂i(s) as waesmhaovotehipngreovfiotyupselyA,rethgeanr∂die(ds)iitsignivCen b(Ky th)e. Irnulegsednisecruasls,edgiavbeonvea(nweithnhthaenced state k s in C (Ksa)mewsiitghnsB). -Tshme ocoomthpaintibgislitaytcolondciatitoinosntshait1we

s dx 1 dx where s is any enhanced state of the link K. The partial boundaries have written iit∧a·b· o· ∧ve. ikThis gives a new chain complex associated with the link K. Whether iatrse hdoeﬁmneodloingtyhecsoanmteaiwnasy nasewbeftoorepoanldogthiecagloibnaflobromunadtairoynfoarmbouluatisthjuestlains kweK will be have written it above. This gives a new chain complex associated with the link K. the subjecWt ohefthaersuitbs sheomquoleongtypcoanptaeirn.s new topological information about the link K will be the subject of a subsequent paper.

3 Th3e DTihcehDroicmhraomtiactiPcoPloylynnoommiiaallanadntdhethPoettPs oMtotsdeMl odel We deﬁneWthe ededﬁincehthroe mdicahtriocmpaDichroticlypnoolymnomaticoimailaal ass ffPooollllooynomialwws:s: Z[G](v, Q) = Z[G!](v, Q) + vZ[G!!](v, Q) Z[G](v, Q) = Z[G!](v, Q) + vZ[G!!](v, Q) Z[ G] = QZ[G]. • # where G! is the result of deletiZng[an edGge]fr=omQGZ, w[Ghil]e.G!! is the result of contracting that same edge so that its end-nod•es#have been collapsed to a single node. In the second where G! eiqsutahtieonr,esurelptroesfednteslaetgirnapghawnitehdogne nforodemanGd ,nowehdiglees,Gan!!d is thGe rreepsruesletnotsfthceontracting that same deidsjgoientsuon•tiohnaot fittsheesnindg-lne-ondodeesghrapvhewbiethetnhecgorlalpahpGse. d to a s•i#ngle node. In the second equation, represents a graph with one node and no edges, and G represents the • • # disjoint union of the single-node graph wit1h0 the graph G.

G G’ G’’ 10 Delete Contract In [8, 9] it is shown that the dichromatic polynomial Z[G](v, Q) for a plane graph can be expressed in terms of a bracket state summation of the form

1 = + Q− 2 v (9) { } { } { } with 1 = Q 2 . {!} Here Z[G](v, Q) = QN/2 K(G) { } where K(G) is an alternating link diagram associated with the plane graph G so that the projection of K(G) to the plane is a medial diagram for the graph. Here we use the opposite convention from [9] in associating crossings to edges in the graph. We set K(G) so that smoothing K(G) along edges of the graph give rise to B-smoothings of K(G). See Figure 4. The formula above, in bracket expansion form, is derived from the graphical contraction-deletion formula by translating ﬁrst to the medial graph as indicated in the formulas below: Precursor:

Z[ ] = Z[ ] + vZ[ ]. delete contract Z[R K] = QZ[K]. " Here the shaded medial graph is indicated by the shaded glyphs in these formulas. The medial graph is obtained by placing a crossing at each edge of G and then connecting all these crossings around each face of G as shown in Figure ?. The medial can be checkerboard shaded in relation to the original graph G, and encoded with a crossing structure so that it represents a link diagram. R denotes a connected shaded region in the shaded medial graph. Such a region corresponds to a collection of nodes in the original graph, all labeled with the same color. The proof of the formula Z[G] = QN/2 K(G) then involves recounting boundaries of regions in correspondence with the loo{ps in th}e link diagram. The advantage of the bracket expansion of the dichromatic polynomial is that it shows that this graph invariant is part of a family of polynomials that includes the Jones polynomial and it shows how the dichromatic polynomial for a graph whose medial is a braid closure can be expressed in terms of the Temperley-Lieb algebra. This in turn reﬂects on the sturcture of the Potts model for planar graphs, as we remark below.

It is well-known that the partition function PG(Q, T ) for the Q-state Potts model in statistical mechanics on a graph G is equal to the dichromatic polynomial when

1 v = eJ kT 1 − where T is the temperature for the model and k is Boltzmann’s constant. Here J = 1 according as we work with the ferromagnetic or anti-ferromagnetic models (see ±[3] Chapter 12). For simplicity we denote 1 K = J kT

11 G

K(G)

Figure 4: Medial Graph, Checkerboard Graph and K(G)

so that v = eK 1. − We have the identity P (Q, T ) = Z[G](eK 1, Q). G − The partition function is given by the formula

KE(σ) PG(Q, T ) = ! e σ where σ is an assignment of one element of the set 1, 2, , Q to each node of the graph G, and E(σ) denotes the number of edges o{f the·g·r·aph}G whose end-nodes receive the same assignment from σ. In this model, σ is regarded as a physical state of the Potts system and E(σ) is the energy of this state. Thus we have a link diagrammatic formulation for the Potts partition function for planar graphs G.

P (Q, T ) = QN/2 K(G) (Q, v = eK 1) G { } − where N is the number of nodes in the graph G.

This bracket expansion for the Potts model is very useful in thinking about the physical structure of the model. For example, since the bracket expansion can be expressed in terms of the Temperley-Lieb algebra one can use this formalism to express the expansion of the Potts model in terms of the Temperley-Lieb algebra. This method clariﬁes the fundamental relationship of the Potts model and the algebra of Temperley and Lieb. Furthermore the conjectured critical temperature for the Potts model occurs

12 Partition Function Recall that the partition function of a physical system has the form of the sum over all states s of the system the quantity

exp[(J/kT)E(s)] where J = +1 or -1 (ferromagnetic or antiferromagnetic models) k = Boltzmann’s constant T = Temperature E(s) = energy of the state s Potts Model In the Potts model, one has a graph G and assigns labels (spins, charges) to each node of the graph from a label set {1,2,...,Q}. A state s is such a labeling. The energy E(s) is equal to the number of edges in the graph where the endpoints of the edge receive the same label. For Q = 2, the Potts model is equivalent to the Ising model. The Ising model was shown by Osager to have a phase transition in the limit of square planar lattices ( in the the 1940’s). In [8, 9] it is shown that the dichromatic polynomial Z[G](v, Q) for a plane graph can be expressed in terms of a bracket state summation of the form

1 = + Q− 2 v (9) { } { } { } with 1 = Q 2 . {!} Here Z[G](v, Q) = QN/2 K(G) { } where K(G) is an alternating link diagram associated with the plane graph G so that the projection ofInK[8(, 9G] i)t istoshotwhnethpatlathne edicihsroamamticepodlyianolmdiailaZg[Gra](mv, Qf)ofrortahpelangergarapphh . Here we use can be expressed in terms of a bracket state summation of the form the opposite convention from [9] in associating 1crossings to edges in the graph. We set = + Q− 2 v (9) K(G) so that smoothing K(G) a{lon}g e{dge} s of th{e g}raph give rise to B-smoothings of with K(G). See Figure 4. The formula above, in b1 racket expansion form, is derived from = Q 2 . the graphical contraction-deletion form{!ul}a by translating first to the medial graph as Here indicated in the formulas below: Z[G](v, Q) = QN/2 K(G) { } where K(G) is an alternating link diagram associated with the plane graph G so that the projection of K(GZ) [to the ]pl=ane Zis [a medi]al+diavgrZam[ for th]e. graph. Here we use the opposite convention from [9] in associating crossings to edges in the graph. We set K(G) so that smoothing K(G) along edges of the graph give rise to B-smoothings of K(G). See Figure 4. The Zfor[mRula abKove], i=n brQackZet [eKxpa]n.sion form, is derived from the graphical contraction-deletion"formula by translating first to the medial graph as Here the shadeidndimcateed iinatlhegfroarmpuhlasisbeilnowd:icated by the shaded glyphs in these formulas. The medial graph is obtained by placinZg[ a c] r=oZss[ in]g+avtZe[ac]h. edge of G and then connecting all these crossings around each face of G as shown in Figure ?. The medial can be checker- Z[R K] = QZ[K]. board shaded in relation to the original "graph G, and encoded with a crossing structure Here the shaded medial graph is indicated by the shaded glyphs in these formulas. The so that it represmeenditasl garaplhiniskobdtaiianegdrbaympla.ciRng adceronssointgeast eaacchoedngneeofcGteadndsthheandcoendnecrteinggiaolln in the shaded medial graph. tSheusecchroassirnegsgairoounndceoacrhrefascepoofnGdass sthoowan icnoFligluercet?i.oTnheomfedniaol cdaen sbeicnhetchkeer-original graph, board shaded in relation to the original graph G, and encoded with a crossing strucNtur/e2 all labeled withso thhaet itsraepmreseenctsoalloinrk.dTiaghraemp. Rrodoenfotoesfatchonenefcoterdmshuadlead Zreg[iGon ]in=the sQhaded K(G) then involves recoumnetdiniagl grbapohu. Snudcharaireegsioonfcorreregspioonndsstoina ccoollercrtieosnpofonnodeesninctheeworiigtihnalthgreaplho, o{ps in th}e link all labeled with the same color. The proof of the formula Z[G] = QN/2 K(G) then diagram. The aindvvolavensItnrae[cg8oeu, n9ot]infigttbhioseusnhbdoarrwaiencskothferatetgetihoxnepsdianincchsorrioroemsnpaootnidfcetpnhcoeelywdnitohicmthhiearllooZom{p[sGaint]i(tcvh}e,pQlion)klyfonroampliaanleigsrathphat it shows that tdhiaicsgarangmrb.aeTpheehxapdirvneasvnstaeagrdeiaionnf ttheerimbsrsapcokafertatexbopraafncsakioenftaosftmathtieeldysiucmhorofmmpaattoiioclnpyonolyfontomhmeiiafaollirssmththatat includes the it shows that this graph invariant is part of a family of polynomials that includes the Jones polynomJoianelsapnoldynoitmsiahl aonwd ist shoowws hotwhetheddiicchrroommaticaptoiclynpoomliayl1nfoor amgriaaplhfwohrosae mger-aph whose me- = + Q− 2 v (9) dial is a braid cdliaolsisuarberacidacnlosbuere ecaxnpbereexspsreesdse{dinintte}errmms o{sf tohfe}TtehmepeTrleeym-L{pieebrallg}eeybr-aL. Tiehibs inalgebra. This in turn reflects on the sturcture of the Potts model for planar graphs, as we remark below. with turn reflects on the sturcture of the Potts model for plan1 ar graphs, as we remark below. It is well-known thaThet the parpatitionrtition functionfunction P G ( Q , = T ) Q f ofor2r .t hthee Q -state Potts model in statistical mechanQ-stateics on a Pgottsraph modelG is e qonua lat ogra{t!hphe }d Gic hisro givmaentic bpyo thelyn omial when dichromatic polynomial It is well-knoHwerne that the partition funct1ion PG(Q, T ) for the Q-state Potts model vZ=[GeJ]k(Tv, Q1) = QN/2 K(G) in statistical mechanics on a graph G is equal−to the dich{roma}tic polynomial when where whwereheTreis Kthe(tGem)piesraatunreafloterrthneamtinodgeliannkd kdiiasgBroaltmzmaasnsno’scciaotnesdtanwt.iHthertehJe =plan1e graph G so that according as we work with the ferromagnet1ic or anti-ferromagnetic models (see ±[3] the projection of K(G) to the pJlaknTe is a medial diagram for the graph. Here we use Chapter 12). For simplicity we dveno=te e 1 the opposite convention from [9] in as−sociating crossings to edges in the graph. We set J = +1 or -1 (ferromagnetic1 or antiferromagnetic models) where T is the teKm(pGe)rsaotuthraet fsomrotohtheinmg oKd(eGKl)=aanlJodnkg eidsgBesoolftzthme agnranp’hsgcivoenrsisteantot.BH-semroeoJthi=ngs of1 K(G). See Fki g=u Boltzmann’re 4. Thes fconstantormulakaTbove, in bracket expansion form, is derived fro±m according as wethwe ogrrakphwiciatlhTc =ot nThemperaturteracfteiornro-demelaetgionneftoicrmourlaabnytit-rafnesrlraotimngafigrnstetoicthme omdeedilasl (gsraepeh [a3s] Chapter 12). Forinsdiimcapteldiciintythewfeordmeunlaostbeelow:11 Z[ ] =1Z[ ] + vZ[ ]. K = J Z[RkTK] = QZ[K]. " Here the shaded medial graph is indicated by the shaded glyphs in these formulas. The medial graph is obtained by placing a crossing at each edge of G and then connecting all these crossings around each face1o1f G as shown in Figure ?. The medial can be checkerboard shaded in relation to the original graph G, and encoded with a crossing structure so that it represents a link diagram. R denotes a connected shaded region in the shaded medial graph. Such a region corresponds to a collection of nodes in the original graph, all labeled with the same color. The proof of the formula Z[G] = QN/2 K(G) then involves recounting boundaries of regions in correspondence with the loo{ps in th}e link diagram. The advantage of the bracket expansion of the dichromatic polynomial is that it shows that this graph invariant is part of a family of polynomials that includes the Jones polynomial and it shows how the dichromatic polynomial for a graph whose medial is a braid closure can be expressed in terms of the Temperley-Lieb algebra. This in turn reflects on the sturcture of the Potts model for planar graphs, as we remark below.

It is well-known that the partition function PG(Q, T ) for the Q-state Potts model in statistical mechanics on a graph G is equal to the dichromatic polynomial when

11 In [8, 9] it is shown that the dichromatic polynomial Z[G](v, Q) for a plane graph can be expressed in terms of a bracket state summation of the form

Z[ ] = Z[ ] + vZ[ ].

Z[R K] = QZ[K]. " Here the shaded medial graph is indicated by the shaded glyphs in these formulas. The G medial graph is obtained by placing a crosGsing at each edge of G and then connecting all these crossings around each face of G as shown in Figure ?. The medial can be checkerboard shaded in relation to the original graph G, and encoded with a crossing structure so that it represents a link diagram. R denotes a connected shaded region in the shaded medial graph. Such a region correspondGs to a collection of nodes in the original graph, G all labeled with the same color. The proof of the formula Z[G] = QN/2 K(G) then involves recounting boundaries of regions in correspondence with the loo{ps in th}e link diagram. The advantage of the bracket expansion of the dichromatic polynomial is that it shows that this graph invariant is part of a family of polynomials that includes the Jones polynomial and it shows how the dichromatic polynomial for a graph whose medial is a braid closure can be expressed in terms of the Temperley-Lieb algebra. This in turn reﬂects on the sturcture of the Potts model foK(G)r planar graphs, as we remark below.

It is well-known that the partition function PG(Q, T ) for the Q-state Potts model in statistical mechanFiicgsuoren4a: gMraepdhiaGl GirsaepqK(G)hu,aCl htoectkherdbiocahrrdomGartaipchpoalnydnKom(Gia)l when K(G) 1 K(G) v = eJ kT 1 so that − where T is the tFeimgupreer4a:tuMreefdoiarlthGermapohd,eCl havenc=dkkeerKibsoBaor1dl.tzGmranpnh’sancdonKst(aGn)t. Here J = 1 Figure 4: MeFdigiuarleG4:rMaepdhia,l CGrhaepchk, Cerhbecok−aerrbdoaGrdrGarpahphaanndd KK((GG) ) ± accordinWgeahsavweethwe oidrekntwitiyth the ferromagnetic or anti-ferromagnetic models (see [3] Chapter 12). For simplicity we dePno(tQe , T ) = Z[G](eK 1, Q). G − so thaTthe partition function is given by the form1ula so that Let vK==eKJ 1. −kKT so that v = e 1.KE(σ) We have the identity PGK(Q, T ) = !− e v = e 1.K σ We have the identity PG(Q, T ) = Z[G](e 1, Q). − −K The pwahrteirtieoσn fiusnacntiaosnsigsngmiveenntPboGyf(tohQnee,fToelr)emm1=u1elaZnt[oGf]t(hee set 11,,Q2,). , Q to each node of the We have the identity −{ · · · } ThegpraaprhtitGio,nafnudncEti(oσn)isdegniovteens bthyethneumfobremrKuolfa edges of the graph G whose end-nodes PG(Q, T ) = Z[G](e KE(1σ), Q). receive the same assignmePnGt (frQo,mTσ).=In!thisem−odel, σ is regarded as a physical state of the Potts system and E(σ) is the energy oσf this staKteE. T(σh)us we have a link diagrammatic PG(Q, T ) = ! e The partition functfioormnuilsatigonivfoernthbeyPotthtsepafrotirtimonufulanction for planar graphs G. where σ is an assignment ofFoorne planareleme ngrat ophsf th eGs σwete ha1,v2e, , Q to each node of the { · · · } graph G, and E(σ) denotes the numberNo/f2 edges of the grapKh G whose end-nodes where σ is an assignmPenGt(oQf,oTn)e=elQement KofK(GthE)e(s(σeQ)t, v1=, 2e, , Q1) to each node of the receive the same assigPnmGen(tQfr,omTσ). =In th!is m{oedel, σ}is regarded a−s a physical state of graph G, and E(σ) denotes the number of edges o{f the·g·r·aph}G whose end-nodes the PwothtserseysNtemis athnednEu(mσb)eisr otNhfe n=eo ndneeumberrsgiynotfh tσeofhig snodesrastpahteG. ofT. h G.us we have a link diagrammatic forremceuilvaetiothnefosarmthee aPsostitgsnpmaretintitofnrofumncσti.oInnftohrisplmanoadreglr,aσphis Gre.garded as a physical state of the PotTtshsiyssbteramckaentdeExp(aσn)siiosnthfeorentheergPyoottfsthmisodsetalties. vTehruysuwseefuhlavine athliinnkkindgiaagbraomutmthaetic where σ is an assignment of one elemenNt /o2 f the set 1, 2K, , Q to each node of the formulation for thePPGo(Qtts, Tpa)r=titiQon funKct(ioGn)fo(Qr p, vlan=aregrap1h)s G. physical structure of the model. Fo{r exam}ple, s{ince the·−b·r·acket }expansion can be ex- graph G, and E(σpr)esdseedninoteersmsthofethneuTemmbpeerrleyo-fLieebdaglgeesbraoofnethcean gusreatphhis fGormwalihsmostoeexepnreds-snodes where N is the numberPof n(Qod,eTs i)n=theQgNra/p2hKG.(G) (Q, v = eK 1) receive the same asthseigexnpmanseionntoffrtohmeGPoσtt.s Imnodtehliisn tmermo{ds oefl,thσe}Tiesmrpeergleayr-dL−eiedb aalgsebarap. hTyhissimcaetlhosdtate of clarifies the fundamental relationship of the Potts model and the algebra of Temperley the Potts system aTnhdisEbr(aσck)etisextphanesieonneforgr ytheofPotthtsismsotdaetleis. vTehryuussewfuel ihnatvhienkainlginabkodutiathgerammatic whearnedNLieisb.thFeurntuhmermbeorreofthneocdoensjeincttuhreedgrcariptihcaGl .temperature for the Potts model occurs formulation forphthyseicaPlosttrtusctpuraerotfittihoenmofudenl.cFtioor nexfamorplpe,lsaincaerthgerbarpachksetGex.pansion can be ex- presseTdhins tberramcskeotf ethxepTanemsiopenrlfeoyr-Lthieeb Palogtetsbrma oondeelcaisn uvseerythuissefofurml ainlistmhintokienxgpraesbsout the thpeheyxspicaanlsisotnruocftuthree PoofttshemmoNdoed/lei2nl.teFromrseoxfatmhe1p2lTee,msipnecrleeyth-LeKibebraaclkgebt reax.pTahnissimonetchaond be ex- clarifiesPthe f(uQnd,amTe)nta=l reQlationshipKof(tGhe)Pot(tQs m,ovde=l aned the alg1eb)ra of Temperley pressed iGn terms of the Temperl{ey-Lieb a}lgebra one can us−e this formalism to express anthdeLeixebp.anFsuirothneormf tohree Pthoettcsomnjoecdteulreind tcerritmicsalotfetmhpeeTraetmurpeefroleryth-Le iPeobttaslgmeobdreal. oTchciusrsmethod where N is the ncluarmifibesetrheofunndoadmeesntianl rtehlaetiognrsahpiphoGf th.e Potts model and the algebra of Temperley and Lieb. Furthermore the conjectured12critical temperature for the Potts model occurs This bracket expansion for the Potts model is very useful in thinking about the physical structure of the model. For example, s1i2nce the bracket expansion can be expressed in terms of the Temperley-Lieb algebra one can use this formalism to express the expansion of the Potts model in terms of the Temperley-Lieb algebra. This method clarifies the fundamental relationship of the Potts model and the algebra of Temperley and Lieb. Furthermore the conjectured critical temperature for the Potts model occurs

12 In [8, 9] it is shown that the dichromatic polynomial Z[G](v, Q) for a plane graph can be expressed in terms of a bracket state summation of the form

1 = + Q− 2 v (9) { } { } { } with 1 = Q 2 . In [8, 9] it is shown that the dichromatic po{l!y}nomial Z[G](v, Q) for a plane graph Here can be expressed in terms of a braTheorckeem:t stZat[eG]s(uv,mQm) =atQioNn/2oKf (thGe) form In [8, 9] it is shown that the dichromatic poly{ nomi}al Z[G](v, Q) for a plane graph where K(G) is an altewherrnatein K(G)g lin isk and ialternatingagra1m as slinkoc iassociatedated with withthe plane graph G so that can be expressed in terms of a bracket state2 summation of the form = the+ medialQ− gravph of G and (9) the projecti{on of K} (G){to the}plane is a med{ial dia}gram for the graph. Here we use the opposite convention from [9] in associating cros1sings to edges in the graph. We set 2 with K(G) so that smoothing K(G)=along edges+ofQth−e grvaph give rise to B-smoothings of (9) { } { 1 } { } K(G). See Figure 4. The form=ulaQabo2v.e, in bracket expansion form, is derived from with the graphical contraction-deletion formula by translating first to the medial graph as {!} 1 indicated in the formulas below: 2 Here For the Potts Model the =criticalQ point. is at {N!1/2} Z[fGor](Tv,wQh)eZ=n[ QQ] =2 vZ[=K](1+G. Wv)Ze[ se]e. clearly in the bracket expansion that this value of T Here − { } N/2 where K(G) is an alternating clionrkredspiZaog[nGrda]sm(vtZo,a[RQsas)pooKc=ii]naQ=teoQdfZsw[yKimtK]h.m(tGhete)ryploafntehegrmaopdheGl wshoertheatthe value of the partition function does not dep"end upon t{he desi}gnation of over and undercrossings in the associated the projection of KH(eGre)thetoshtahdeedpmleadniael girsapah ims ienddiicaatleddbiaygthreasmhadfeodrgtlyhpehsginrathpehse. foHrmeurelasw. Teheuse where K(G) is an altekrnnoattionrglilninkk. TdihaigsrcaomrraessspoocniadtsetdowaisthymthmeeptlraynbeegtwraepehnGthesopltahnaet graph G and its dual. the opposite convenmteiodinalfgrroapmh is[9ob]taiinneadsbsyopclaiacitnignagcrcorsosisnsgiantgeascthoedegdegofeGs ianndtthhengcorannpehct.inWg aellset the projectthieosne corofssKing(sGar)outondtehaechpfalcaenoef Gisaas smhoewdniianlFdigiuarger?a. mThefomredtihael cagnrabepchh.ecHkeer-re we use K(G) so that smoothing K(G) along edges of the graph give rise to B-smoothings of the opposibtoearcdosnhvadeendtiinonrelfartWoiomnetofi[9trh]setionarniagasinlsyaolzcgeiraahptiohnwGg,ocanruodrsehsneicnougdreisdsttwoiciteshdlaegcaerdossiisnninggthtsoetrutghcrteuarpKehh.oWvaensoevt homology looks when K(G). See Figure s4o.thTathiterefporersmenutslalianbkodviaeg,raimn. Rbrdaecnkoteets aexcopnannecsteiodnshafdoerdmre,giosndinetrhievsehdadferdom K(G) so that smoothignegnKer(aGliz)eadlotongtheedcgoenstoefxtthoef gthraepdhicghivroemriasteictopoBly-snmoomoitahli.n(gTshoisf is a different approach the graphKica(lGc)o.nStermaeceFdtiiiaoglnug-radepeh4l.e.StuTiochhneafrfoeogrrimomnuuclloaarreabsbypontvrdeas,ntoisnalacbtorilnalecgcktifieontrsoetfxnptooadnetsshiienontmhefeoodrrimigain,laiglsgrrdaapephrhi,vaesd from all labeled withttohetshaemeqcuoelosrt.ioTnhetphraoonf tohf ethemfoertmhuolda Zo[fGS] =toQsiNc/2[1K5](Go)r [th1e4n], but see the next section for a { } indicated tihnethgerafpohrimicnvauolllvacesosnbretecrloaoucwntditio:nisngc-budoseusnleidotainroienosfoffSorrteomgisouinclsa’sinbacypoprtrrreaospnaoscnlhdaettinoncegcwafiitterhsgtthoetroilofyothpinseginmtthheeedliidaniklcghrraopmhaatisc polynomial.) We then indicated idniagthraemf.oTrhme audalvasaskntbaqegueleooswftti:hoenbsraackbeot uextptahnesiorneloafttihoendsichhirpomoaftiKc phoolyvnaonmoiavl ihs tohmat ology and the Potts model. It is it shows thatZth[is gra]ph=invZar[iant i]s +parvt oZf [a fam]i.ly of polynomials that includes the Jones polynominalaatnudriatlZshtoo[wasshk]ow=suthcZehd[iqchureo]sm+tiaotivcnZpso[slyinnocme].iathl feoraadgjraacpehnwchyosoefmset-ates in the Khovanov homology dial is a braid clcosourrreecsapnobne dexsprteossaedninadtejramcseonfcthye Tfoemr peenrleeyrg-Leiteibcalsgteabtreas. Tohfistihne physical system described by turn reflects on theZs[tuRrctureKof]th=e PQottZs m[Kode].l for planar graphs, as we remark below. the Pot"ts mZo[dRel, aKs w] =e sQhaZll[Kde]s.cribe below. Here the shaded medial graph is indicated by "the shaded glyphs in these formulas. The It is well-known that the partition function PG(Q, T ) for the Q-state Potts model medial graHpehreisthoebtsaihniansdteaedtidsbtimcyaelpmdlaieaclhinagngrFiacaosprohcnrtihoasisgissrnaipdpnhuigcrGapattioesesdeeaqbcuwyahletethodnetghoseewhdoaicafdhdeGroodmpagtanltyiydcepptthholaseynnninocmttohhiaenelrnsweebhcrefanoticrnmkgeutallealxs.pTanhseion as indicated below. medial graph is obtained by placing a crossing at each edge of G and then connecting all these crossings around each facWe oefcGallatshsishotwwon-vinarFiaig1bulereb?ra.cTkheet emxepdaniasliocnanthbeeρc-hbercakcekre-t. It reduces to the Khovanov v = eJ kT 1 board shatdheedseincrroeslsaitniogns atorotuhnedvoeeraricgsihionfnaalcogefrotahfpeGhbGaras,csakhneodtwaens−niacnofFudinegcdutriwoeni?th.oTfahqcerwmohseesdinniaρgl icsstarneuqbcuteuaclrhetoecokneer-. so that it rbeoparredsesnhtasdwaehdelrieinnTkrideslitaahtegitoreanmmptoe.raRtthuerdeoefornriogthtieensmaloagdcerloaapnnhnd ekGci,steBadnoldstzhemanadcneno’dds ecrodengswtiaointnth. iHanecrterhoJes=ssihnag1dsetdructure so that it raecpcroerdsienngtassawleinwkordkiawgitrhamthe. fRerrdomenagontetsicaocr oanntni-efecrtreomdasghnaedticedmoredegliso(nseien±[3t]he shaded medial graph. Such a region corresponds to a collection of no[des]in=th[e or]iginqaρl[grap] h, (10) medial graCphhap.tSeru1c2h). aForresgimiopnliccitoyrwreesdpeonontde s to a collection of nodes in −the original graph, all labeled with the same color. The proof of the formula Z[G] = QN/2 K(G) then all labeled with the same color. The proof of t1he formula Z[G] = QN/2 K(G) then with K = J { } involves recounting boundaries of regions in correspoknTdence with the loops in 1t{he link} involves recounting boundaries of regions in corresponden[ ce]w=ithq t+heql−oo.ps in the link diagram. Tdihaegraadmv.anTthaegaedovfatnhteagberaocfktheteebxrpacaknestioenxpoafntshieondoicfhtrhoemdaict"hicropmolaytnicopmoilaylniosmthiaaltis that it shows that this graph invariaWnet icsanparretgoarfdathfiasmeixlpyaonfsipoonlaysnaonmiinatlesrmtheadtiianrcylbuedtewsetehnethe Potts model (dichromatic it shows that this graph invariant is part o1f1a family of polynomials that includes the Jones polyJonnoemsipaollaynndomit isahloawndspihtooslwhyontwohmes hidaoilcw)hartnhodemtdahitecihctorpopomollyaontgiocicmpaoilalbylrnfaoocrmkaeiatg.lrfWaoprhhaewngrhρaop=she1wmhweoe-sehamvee-the topological bracket dial is a brdaiiadl icsloasburraeidcacnlobsuereexcepaxrnpeasbsneesedioxinpnrietnessrKmehdsoinovaftnethromevsTfooermfmtph.eWrTleheyme-nLpeierlbeya-lLgeiebbraa.lgTehbirsa.inThis in turn reflecttusrnonretflheecststuorncttuhreesotufrtchteurPeootftsthmeoPdoetltsfomropdlealnfaorrgprlaapnhasr,garsapwhes,raesmwa1erkrebmelaorwk.below. qρ = Q− 2 v − It is well-Iktnisowwneltlh-kantotwhenpthaarttittihoenpfaurntictitoionnfuPnGct(iQon, TP) (fQor, tTh)e fQor-sthtaeteQP-sottattsemPoodtteslmodel and G 1 in statisticianlsmtaeticshtiacnalicms eocnhangicraspohnGa girsaepqhuGalitsoetqhueadl itcohtrhoemdaictihcropmolaytnicop1moilaylnwomheianl when q + q− = Q 2 , 1 1 v = eJ kvT = e1J kT 1 we have the Pott−s model−. We shall return to these parametrizations shortly. where T iws thheereteTmipsetrhaetuterme fpoerrathtueremfoodr ethleanmdokdeilsaBnodlktzims aBnonlt’zsmcoannst’asncto.nHstearnet.JH=ere J1 = 1 Just as in the last section, we have ± accordingacacsowrdeinwgoarskwweitwhotrhkewfeitrhrotmheagfenrertoimc aogrnaentitci-oferrraonmti-afgernreotmicagmnoedtieclsm(osdeeels±[3(]see [3] nB (s) j(s) Chapter 1C2)h.aFpoterrs1im2)p. lFicoirtysimwpeldiceintyowtee denote [K] = !( ρ) q − s 1 1 K = J K = J where nB(s) isktThe numkTber of B-type smoothings in s, λ(s) is the number of loops in s labeled 1 minus the number of loops labeled X, and j(s) = nB(s) + λ(s). This can be rewritten in the following form: 11 11 i j ij [K] = !( ρ) q dim( ) − C i ,j

j i ij j j = ! q !( ρ) dim( ) = ! q χρ( • ), − C C j i j

13 1 for T when Q 2 v = 1. We see clearly in the bracket expansion that this value of T 1− forcoTrrweshpeonndQs−to2 avp=oin1t.oWf seymseme ectlreyarolfythine tmheodberlawckheetreexthpeanvsailoune othfatthtehpisarvtiatliuoen ofufnTc- cortiroenspdoonedssntoo1tadeppoeintdoufpsoynmthmeedtreysiogfnathtieonmoofdoevl ewrhaenrdeuthnedevraclruoessoifngthseinpathrteitaisosnofcuiantce-d fortiToknnwdohoteeonsrnQlion−tkd.2evTphe=insdc1uo.prWroenseptshoeneeddsceltseoiagarnlsayytimionnmtohefetroybvrbeaerctakwneedteenuxntphdaeenrpcsliraoonnsestighnragatsptihhniGsthvaeanaldsusietosocdifauTtaeld. corkrneospt oonrdlisntko. aTphiosinctoorrfessypmonmdsettroyaosfytmhemmeotrdyeblewtwheereentthheevpallauneeogfrathpehpGaratintdiointsfudnuca-l. 1 tion dfooersTWneowtfihdresentpaQenn−adly2uzvpeo=hnot1wh.eoWudereshsieeguenracistliteoiacnrsloylefaiondvitnehgreatbonrdtahcuekneKdtheeroxcvpraoannsossviinohgnostmhinaotltohthgeiysalsvosaoolkcusieawtoehfdeTn knot cgooeWrrnlreieenrsfiakprl.oiszTtneadhdnsistatoolcytoazhreepreohcsiopnwnottnoeoxfdutssroythofmetamuhreseiystdrtmiiyccmhosrfleoettmharydeaitmbnicegotpdwtooeelltyehwnehotKemhreheiaoptlvh.laae(nnTveohaviglsuhreiaospomahfodtGhliofefgaepnyraedlrnotiitottsaikopdsnpuwrfaouhla.neccnh- gentioeorntahldeiozqeeudsentsootittohdneeptcheoannndtetuhxpetoomnftethhteheodddeiscoihgfrnSoamttoiosaitncico[fp1oo5lv]yeonrroam[n1di4au]l,.nbd(Tuetrhcsirseoeissstiahnedgisnfefinexrttehsneetcaatsipsoponrcoifaaotcrehda toWkdtehniseoficturqsosutsreialosintnniaokloy.nfzTStehhtaoihsnsoicwctoh’rsoeraeumpsrppheorteonhuadorcsdihsttotiofcascSsaltyeotemasgdimcoinre[igt1fry5tyoi]nbtgoehrttewh[Ke1e4edhn]io,cvthbharuenotopmslvaehnteieoctmhgperooallnpyoehngxoGytmlsoaieanocldkt.)isoiWtnws edhfoutehranela.n gendeisracasulkiszsqeiudoentsotoiofthnSestocasobinoct’uestxatthpoepfrroethalaecthidotinocshchraioptmeogafotKricifhypoiovnlagynntohovemhdioaicml.hor(olTomhgiyastaiicsndpaotdhliyefnfPeorometntisatlma.)poWpdreeola.thcIhtenis to athskenaqqtuuuWeerassetltiifitooornnsasttsaahkbnasoanuluyctthzhteehqehmuoreewestlthaioootuindorsnhosseifhnuiScrpietsoottsifhciecKs lah[ed1oaj5vda]aicnneognorcvt[oy1hto4ohf]me,sKotbaluhtoetogsvsyeaineannotthdvheehthKonemhePoxoovtloatstgnseyocmvtliooohodnokemsfl.oowrIlohtageisny disncautgcsuoesrinraorelenrstaopoliofaznSsedkdtosstsotuioccth’hasenqacuapodepnsjrattoeicoaxenctnhsocfystoitnfhoccearedteetihncgheorrragoidfemyjtaiaicnctiegscntatpchtoyeelsyodonficfoshmttarhtoieeamslp.aih(ntTyicsthhiipcesaoiKllsyshanyoosdvtmieaffmnieaorldev.)nehstWocarmepibpotehrlodoeangbcyhy askcoqrtuohreeestshPtpieootnnqtdsusemasbttooiodouaentnl,thaadesnjwraetcehleaesnthicmoaylnelsftdohheoripsdecnoroiefbfrKegSebhtoteoislcviocawsnt[.ao1tv5e]shoormf [to1hl4eo]g,pyhbuyatsnisdceaetlhtsehyPestonetemtsxtmdseoescdcteriiolb.neIdtfoibsrya nattuhredaPilsotcotutssassmikoonsduoecflh,Satqosusweicse’tsisohanpaspllsrdoinaeccsechrttihobeecabtdeejlgaoocwrei.nfyciyngofthsetadteicshirnomthaetiKc hpolvyannoomvihaol.m) Woleogthyen correasspkoFnqodursesttthoiiosannpsuaarbdpojoauscteethwneceyrnefoloawrtioeanndseohrpgitpeytoiefctKsatnhaootetvhsaenorofbvrthacoekmpeothlyeoxsgipcyaanalnssidoyntshteaesmPiondtdtesisccmartioebdeedbl.eblIoytwis. the PnWoaFttetosucrrmaatllholitdtsohepials,suaktrwspsoowus-cevehaswrqhiueaaeblnlsleotdiwoebnsrcascrdsikobinepetctbeyxeteplhtoaewanans.doiojtanhceterhnebcryρa-ocbfkreasttcaketexetsp.aiInntsrtiheodenuKcahessoivntoadnitchoaveteKhdohmboveolaloonwgoy.v Wecvoecrarsleilostpnhoiosnfdttwsheoto-bvraaanrcikaedbtjlaeacsbearnacfcyuknefcottreioxenpnaeonrfgsqeiotwinchtsehtnaetρeρs-isborefaqctuhkaeeltp.tohItyosrneiecd.aulcseysstoemthdeeKschroibveadnobvy vFertohsrieotPnhoiostftpstuhmrepobodrseaelc, kwaesetwanesowsahfaauldnl ocdpteitsocnyreiobtfeaqnbeowltohhweenr. bρriascekqeut aelxtpoaonnsieo.n as indicated below. [ ] = [ ] qρ[ ] (10) We call this two-variable bracket expansion the ρ−-bracket. It reduces to the Khovanov For this purpose we now [adop]t=ye[t ano] theqrρb[rack]et expansion as indicated be(l1o0w). versiownitohf the bracket as a function of q when ρ−is equal to one. We call this two-variable bracket expansion the ρ1-bracket. It reduces to the Khovanov [ ] = q + q− . wivthersion of the bracket as aTof uanalncyzetio Khon o"vanof q vw homologhen ρ yi,s wee qadoptual tao neown e. [ ] = [ ]brackqetρ[ 1 ] (10) We can regard this expansion as a[n in]te=rmq−e+diaqr−y b.etween the Potts model (dichromatic " polynomial) and the topologica[l bra]c=ket[. W]henqρρ[= 1]we have the topological bra(c1k0e)t witWhe can regard this expansion as an intermediar−y between the Potts model (dichromatic 1 expansion in Khovanov form. W[ he]n= q + q− . powlyinthomial) and the topological b"racket. When ρ = 1 we have the topological bracket 1 1 expansion in Khovanov form.WhenWh erhon[ = 1,] w=e haqv+e theq2 topological. We can regard this expansion as an intermqeρdi=aryQb−et−wveen the Potts model (dichromatic brack"−et in Khovanov form. polynomial) and the topological bracket. When ρ 1= 1 we have the topological bracket We can regard this expansioWhenn as an intermediar2y between the Potts model (dichromatic and qρ = Q− v expansion in Khovanov form. When 1 polynomial) and the topological br−acket. W1 hen ρ = 1 we have the topological bracket q + q− = Q 2 , andexpansion in Khovanov form. When 1 2 1 we have theqρ P=ottsQ 1model.− v 2 we have the Potts model. We sha−lql r+etuqr−n to=thQes1e,parametrizations shortly. qρ = Q− 2 v and − we have the Potts model. We shall return to the1se parametrizations shortly. Just as in the last section, we have 1 and q + q− = Q 2 , 1 1 nB (2s) j(s) Just as in the last section, we[Kha]v=eq +!q−( ρ=) Q , q we have the Potts model. We shall return to th−ese parametrizations shortly. s we have the Potts model. We shall return to tnhBes(se)paj(rsa)metrizations shortly. [K] = !( ρ) q − Juwsht earseinnBth(es)laisstthseecntiuomnb, ewreohf aBv-etypse smoothings in s, λ(s) is the number of loops in s labJuesletdas1imn tihneuslatshtesneuctmiobne,rwoef lhoaovpes labeled X, and j(s) = n (s) + λ(s). This can nB (s) j(s) B where nB(s) is the number [oKf B] =-ty!pe s(moρo)thingsqin s, λ(s) is the number of loops in be rewritten in the following form: − nB (s) j(s) [K] =s !( ρ) q s labeled 1 minus the number of loops label−ed X, and j(s) = nB(s) + λ(s). This can s i j ij be rewritten in the following fo[rKm]: = !( ρ) q dim( ) where nB(s) is the number of B-type smoo−things in s, λC(s) is the number of loops in i ,j s labewlehder1e nmBin(us)s itshethneunmumbebreorfolfoBop-tsylpaebesmleodoXth,inagnsdinj(ss,)λ=(s)nis(tsh)e+nuλm(bse).r oTfhliosocpasnin [K] = ( ρ)iqjdim( ij) B s labeled 1 minus the number of lo!ops labeled X, and j(s) = nB(s) + λ(s). This can be rewritten in the following forjm: i − ij C j j = ! q !( iρ,j) dim( ) = ! q χρ( • ), be rewritten in the following form−: C C j i i j ijj [jK] = !(i ρ) q ijdim( ) j j = ! q ![K(] =ρ) d−im( (ρ)i)qj=di!mC ( qij)χρ( • ), −i ,j ! C C j i − j C i ,j 13 j i ij j j = ! q !j ( ρ) dimi ( ) i=j ! q χρj( • ),j = ! q !− ( ρ) diCm( ) = ! q χCρ( • ), j i − 13 C j C j i j

13 13 1 for T when Q− 2 v = 1. We see clearly in the bracket expansion that this value of T corresponds to a1 point of symmetry of the model where the value of the partition func- 2 tfiornTdoweshennotQd−epevn=d u1p.oWn ethseeedecslieganrlaytiionntohfe obvraecrkaentdexupnadnesricornosthsiantgthsiisnvtahleueasosfocTiated kcnoorrteosrpolinndks.tTohaispocionrtroefspsyomndmsettoryaosfytmhemmetordyebl ewthweerenthtehevaplluaenoefgthraeppharGtitaionndfiutsncd-ual. tion does not de1pend upon the designation of over and undercrossings in the associated for T when Q 2 v = 1. We see clearly in the bracket expansion that this value of T knot or link. −This corresponds to a symmetry between the plane graph G and its dual. corresWpoenfidrssttoanaaplyozienthoofwsyomurmheeturryisotifctshleemadoindgeltowthheereKthhoevvaanlouve hoofmthoelopgayrtiltoioknsfuwnhce-n tiognendeoWreaeslifinzreosdtt datonepatlheynezdecouhnoptwoenxottuhorefhdteheuesriidsgtiniccahstrilooemandaointfigcovtpoeortlhyaennKodmhuoinavdla.enr(ocTvrhohisossmiisnogalosdgiinyfftleohroeeknastswsaophcpeirnaotaecdh kntgooettnohereralqilniuzkee.sdtTitoohnitshtechoacronrnetthsepexotmnodfesthteooddaiocshfyrmSotmmoasetiticcry[p1bo5ley]tnwooreme[ni1a4tl.h],(eTbphulitasnsieseagrtdhaipeffhenrGeexnttanaspedcpitrtiosoandcuhfaolr. a dtoiscthuessqiounesotifonSttohsainc’tshaepmpreothaocdh toof cSattoesgicor[i1f5y]inogr t[h1e4]d,icbhurtosmeeattihcepnoelxytnsoemctiiaoln.)fWoreathen adsiWkscqeuusfiserisosttnioaonnfsaSlaytbozoseiucht’sothwaepoprrueolraahtciehountrosihsctiaipctesogfloerKaifdhyioinnvggatntohoetvhdheiocKhmrhooomlvoagtnyicoavpnohdloytmnhoeomlPoioaglty.t)slWomoeokdtsheweln.hIetnis genanasetkuraqrlauilzeetsotdioatnosksthasbeuoccuhotnqthtueeexsrtetoiloafntitsohnessidnhicicpehotrhfoeKmahadotijvcaacnpeoonvlcyyhnooommfosiltaoalgt.ey(sTaihnndisthtiheseaKPdhoitoftvsfeamrneoondvteahl.poIpmtriosalocghy tocntoharteruerqsapuloetosntdaiosskntostuhacanhnaqtdhujeeasctmieonentcshyosifdnocroefetnSheetoragsdeicjtaicc[1esn5tca]ytoeosrf[os1ft4att]he,esbipunhttyhsseeiecKathhloesvynasneteoxmvt shdeocemtsiocorlnoibgfeyodr bay ditschcoeurrsPesosiotptnosnmodfsoSdtoteolas,niacs’asdwajaepcpsehrnoacalylcdfhoetrsocercniaebtreeggeboteircliofswytai.ntegstohfetdhiechprhoymsicaatilcspyostleymnodmesicarl.i)beWdebythen askthqeuPeostttisomnsodaeblo,uats twhe srhealalltidoensschriibpeobfeKlohwo.vanov homology and the Potts model. It is naturaFlotrothasisk psurcphoqseuewsteionnoswsiandcoeptthyeeatdajancoethnecryborfacstkaettesexinpatnhseioKnhaosvainndoivcahtoemd obleolgowy . coWrreescFpaoolrlnttdhhsiisstoptwuaronp-ovasaderjiwaacbeelnnecobywrafacodkroeeptnteeyxrepgtaeantniscoiotshnteatrthebesraρoc-fkbetrhtaeecxkppehat.ynsIsiticoraneldasusycisentsedmitcoatdheedescbKreihlbooewvd.anboyv thveWePresoictoatsnllmothfoitsdhetewl,boar-savcawkrieeatbsahlesablalrfaducenkseccttriieobxnepoabnfeslqioown.htheen ρρ-bisraecqkueat.l Itot roednue.ces to the Khovanov version of the bracket as a function of q when ρ is equal to one. For this purpose we now ado[pt y]e=t an[ oth]er bqrρac[ ket]expansion as indicated belo(1w0.) [ ] = [ ] −qρ[ ] (10) We call this two-variable bracket expansion t−he ρ-bracket. It reduces to the Khovanov with vewrsiitohn of the bracket as a function of q when ρ is e1qual to one. [ ] = q + q1− . [ "] = q + q− . " We can regard this expansion a[s an] i=nte[rme]diarqyρb[ etw]een the Potts model (dichrom(1a0t)ic We can regard this expansion as an intermedia−ry between the Potts model (dichromatic polynomial) and the topological bracket. When ρ = 1 we have the topological bracket wiptholynomial) and the topological bracket. When ρ = 1 we have the topological bracket eexxppaannssiioonn in KKhhoovvaannoovvffoormrm. .WWhehnen 1 [ ] = q + q− . " 1 1 2 We can regard this expansion as an intqeρrqmρ=e=dQiaQ−ry2−vbevtween the Potts model (dichromatic −− polynomial) and the topological bracket. When ρ = 1 we have the topological bracket aanndd 1 1 expansion in Khovanov form. When 1 1 2 q q++q−q−==QQ2 , , 1 wweehhaavvee tthe PPoottttssmmooddeel.l.WWeeshshalallrlerteuqtruρnrn=tottoQhet−hse2svpeapraamraemtrieztaritzioantisosnhsosrthloy.rtly. − and JJuusstt aas inn tthheellaassttsseecctitoionn, ,wweehahvaeve 1 1 q + q− = Q 2 , nB (s) j(s) [K] = !( ρ) nB (qs) j(s) [K] = !−( ρ) q we have the Potts model. We shall retusrn to−these parametrizations shortly. s where nB(s) is the number of B-type smoothings in s, λ(s) is the number of loops in whJuerset ansBi(nst)hiesltahset nseucmtiboenr, owfeBh-atvyepe smoothings in s, λ(s) is the number of loops in s labeled 1 minus the number of loops labeled X, and j(s) = nB(s) + λ(s). This can s labeled 1 minus the number of loops labeled X, and j(s) = nB(s) + λ(s). This can nB (s) j(s) be rewritten in the following f[oKrm]:= !( ρ) q be rewritten in the following form: − s i j ij [K] = !( ρ) qi djim( )ij [K] = !−( ρ) q dimC( ) where nB(s) is the number of B-typie,jsmo−othings in sC, λ(s) is the number of loops in i ,j s labeled 1 minus thije number of loops labeled X, and j(s) = nB(s) + λ(s). This can where we define to be thje linear spian of thiej set of enhjanced stjates with nB(s) = i C = ! q !( ρ) dim( ) = ! q χρ( • ), ij bearnedwjr(istt)en=ijn. tThheefnotlhleownuimngbjefor ormf −s:uch sitatesCis thije dimensionj dCim( j ). Now we have = !j q !i ( ρ) dim( ) =j ! q χρ( C• ), expressed this general bracket expa−nsion in terCms of generalized ECuler characteristics j i where i j jij of the complexes: [K] = !( ρ) q dim( ) j − i iCj χρ( • ) =i ,j!(13ρ) dim( ). C − C i 13 j i ij j j These generalized=Eu!ler cqha!ract(erisρti)csdibmec(ome)c=las!sicalqEχulρe(r c•ha)r,acteristics when − C C ρ = 1, and, in that casej, are thie same as the Euler charajcteristic of the homolgy. With ρ not equal to 1, we do not have direct access to the homology.

Nevertheless, I believe that this raises a1s3igniﬁcant question about the relationship of [K](q, ρ) with Khovanov homology. We get the Khovanov version of the bracket polynomial for ρ = 1 so that for ρ = 1 we have

j j j j [K](q, ρ = 1) = ! q χρ( • ) = ! q χ(H( • )). C C j j

Away from ρ = 1 one can ask what is the influence of the homology groups on the coefficients of the expansion of [K](q, ρ), and the corresponding questions about the Potts model. This is a way to generalize questions about the relationship of the Jones polynomial with the Potts model. In the case of the Khovanov formalism, we have the same structure of the states and the same homology theory for the states in both cases, but in the case of the Jones polynomial (ρ-bracket expansion with ρ = 1) we have expressions for the coefficients of the Jones polynomial in terms of ranks of the Khovanov homology groups. Only the ranks of the chain complexes figure in the Potts model itself. Thus we are suggesting here that it is worth asking about the relationship of the Khovanov homology with the dichromatic polynomial, the ρ-bracket and the Potts model without changing the definition of the homology groups or chain spaces. This also raises the question of the relationship of the Khovanov homology with those constructions that have been made (e.g. [15]) where the homology has been adjusted to fit directly with the dichromatic polynomial. We will take up this comparison in the next section.

We now look more closely at the Potts model by writing a translation between the variables q, ρ and Q, v. We have

1 qρ = Q− 2 v − and 1 1 q + q− = Q 2 , and from this we conclude that

q2 Qq + 1 = 0. − " Whence √Q √Q 4 q = ± − 2

14 ij where we define to be the linear span of the set of enhanced states with nB(s) = i and j(s) = j. TheCn the number of such states is the dimension dim( ij). Now we have expressed this general bracket expansion in terms of generalized EuCler characteristics of the complexes: ij where we define to be the linear span of the set of enhanced states with nB(s) = i C j i ij ij and j(s) = j. Then the nuχmbρe(r of•su)ch=stat!es is(the dρi)mednisimon(dim(). ). Now we have C − C C expressed this general bracket expansion in iterms of generalized Euler characteristics of the complexes: j i ij These generalized Euler chaχrρa(c•te)ri=st!ics( bρe)cdoimm(e c).lassical Euler characteristics when C − C ρ = 1, and, in that case, are the same ias the Euler characteristic of the homolgy. With These generalized Euler characteristics become classical Euler characteristics when ρ not equρa=l t1o, a1n,d,winethdatocansoe,tahreatvheesdamireeacstthaecEcuelesrscthoaratchterihstoicmofothloe ghoym. olgy. With ρ not equal to 1, we do not have direct access to the homology. Nevertheless, I believe that this raises a significant question about the relationship Nevertheless, I believe that this raises a significant question about the relationship of [K](qo,fρ[)Kw](qi,tρh) wKithhoKvhaonvaonvovhhoommoololgoyg. yW.e WgetethgeeKthtohvaenoKv hveorsvioannoofvthve ebrrascikoent of the bracket polynompioallynfomr iρal f=or ρ1=so1 stohtahtatffoor ρρ==1 w1ewhaevehave

j j j j [K](q, ρ = 1) = ! q χρ( • ) = ! q χ(H( • )). j C j C j j [K](q, ρ = 1) = !j q χρ( • j) = ! q χ(H( • )). C C j j Away from ρ = 1 oAnwae cya frnomask rho=1,what i sonethe caninfl askuen whatce of isth e homology groups on the coefficients of the ethexp ainfluencension of of[K the](q ,Khoρ), vanoand tvh ehomologcorrespyonding questions about the Away froPmottsρmo=del.1Tohins eisonac awthenay acoefficientstoskgenwerhalaiz tinei qstheuets hexpansiontieonisnaflbouu etofnthceereolaftiotnhsehiphoofmtheoJloongesy groups on the coefficienpotslynoofmtiahlewiethxpthaenPsoitotsnmoodfel[.KIn](thqe, cρa)se, oafntdhetKhheovcaonrorvefsoprmoanlidsmin, gweqhuaevestions about the the same structure of the states and the same homology theory for the states in both Potts mocdaesels., bTuht ins tihse acaswande ao yfcorthteroespondingJgoneens eproal yquestionslniozmeiaqlu( ρeabout-sbtriaoc kntheest e axbpaonusitonthweithreρla=ti1o)nwsehip of the Jones polynomhiavlewexiptrhesstihones fPoor thtes cmoeoffidcieePlnotts.ts oI fnmodel.thtehJeoncesaspoelyonofmtihaleinKtehrmosvoafnroanvksfoofrtmhealism, we have the sameKshtorvuancotvuhroemooflogthy egrosutpast.eOsnlaynthde rtahnekssoaf mtheechhaoinmcoomlpolgexyestfihgeuoreriyn tfhoerPotthtse states in both model itself. Thus we are suggesting here that it is worth asking about the relationship cases, buotf ithne tKhheovcaanosvehoomf otlhoegyJwointhetshepdoiclhyrnomoamticiaplol(yρno-bmriaal,ctkheetρ-ebxrapcakentsainodnthwe ith ρ = 1) we have expProetstssimoondsel fwoitrhotuhtechcaongeifnfig cthieendetfisnoitifonthoef thJeohnoemsolpoogylygrnooupms ioar lchianintsepramcess. of ranks of the This also raises the question of the relationship of the Khovanov homology with those Khovanocvonhstorumctoiolnosgthyatghraoveubpese.n Omandley(et.hg.e[1r5a]n) kwsheoref tthheehocmhoaloingychoasmbpeelnexadejussfitegd ure in the Potts model itsteolfift.dTirehcutlyswwiteh tahreedischurgomgeatsictipnoglynhoemrieal.thWaetwiitllitsakwe uoprthhisacsokmipnagrisaonbionuthtethe relationship of the Knheoxvt saenctoiovn. homology with the dichromatic polynomial, the ρ-bracket and the Potts modelWwe niothwolouotkcmhoarencgloinseglytaht ethedPeofittns imtioodenl boyfwtrhiteinghaotmranosllaotgioyn bgertwoeuepnstheor chain spaces. This alsovarraiaibsleesqt,hρeanqduQe,svt.ioWne hoafvethe relationship of the Khovanov homology with those 1 constructions that have been made (qeρ.g=. Q[−125v]) where the homology has been adjusted to fit directly with the dichromatic −polynomial. We will take up this comparison in the and 1 1 next section. q + q− = Q 2 , and from this we conclude that We now look more closely at the Potts model by writing a translation between the q2 Qq + 1 = 0. variables q, ρ and Q, v. We have − "

Whence 1 √Q √Q 4 2 q = qρ± = Q−− v − 2 and 14 1 1 q + q− = Q 2 , and from this we conclude that

q2 Qq + 1 = 0. − " Whence √Q √Q 4 q = ± − 2

14 where we deﬁne ij to be the linear span of the set of enhanced states with n (s) = i where we deﬁne ij to be the linear span of the set of enhanced states with n B(s) = i and j(s) = j. TheCn the number of such states is the dimension dim( ij). NoBw we have and j(s) = j. TheCn the number of such states is the dimension dim( ij). Now we have expressed this general bracket expansion in terms of generalized EuCler characteristics expressed this general bracket expansion in terms of generalized EuCler characteristics ooffththeeccoommpplleexxeess:: j i ij χρ( •j ) = !( ρ)i dim( ij ). χρ(C• ) = !( −ρ) dim( C ). C i − C i TThheesseeggeenneerraalliizzeedd EEuulleerr ccharactteerriissttiiccss bbeeccoommeeccllaassssicicaallEEuulelerrcchharaarcatcetreirsitsictiscswwhehnen ρρ==11,,aanndd,,iinntthhaatt ccaassee,, aarre the ssaammee aasstthheeEEuulleerrcchhaarraaccteterirsistitcicooffththeehhomomologlyg.yW. Withith ρρnnootteeqquuaallttoo11,,wwee ddoo nnoott have ddiirreeccttaacccceessssttootthheehhoommoolologgyy. .

NNeevveerrtthheelleessss,, II bbeelliieevvee that thhiiss rraaiisseessaassiiggnniiﬁﬁccaannttqquueesstitoionnaabboouut tththeererlealtaiotinosnhsihpip ooff[K[K]]((qq,,ρρ))wwiitthh KKhhoovvaannoov homoollooggyy.. WWee ggeetttthheeKKhhoovvaannoovvvveersrisoionnofofthtehebrbarcakcektet ppoolylynnoommiiaallffoorrρρ == 11 ssoo tthhat for ρρ == 11 wweehhaavvee

jj jj jj j j [[KK]]((qq,,ρρ == 1) = !!qq χχρρ(( •• ))==!!qq χχ((HH(( •• ))).). CC CC jj jj

AAwwaayyffrroommρρ == 11 oonnee ccaan ask wwhhaatt iiss tthhee iinnflfluueenncceeooffththeehhoommoolologgyygrgoruopuspsononthtehe ccooeefffificciieennttssooff tthhee eexxppaannssiion of [[KK]]((qq,,ρρ)),, aannddtthheeccoorrrreessppoonnddininggqquuesetsitoinosnsabaobuotutthtehe PPoottstsmmooddeell.. TThhiiss iiss aa wwaayy to genneerraalliizzeeqquueessttiioonnssaabboouuttththeererelalatitoionnshshipipofofthteheJoJnoenses ppoolylynnoommiiaallwwiitthh tthhee PPoottttss modeell.. IInn tthhee ccaassee oofftthheeKKhhoovvaannoovvfoformrmalailsimsm, ,wwe ehahvaeve ththeessaammeessttrruuccttuurree ooff tthhee sstates aanndd tthhee ssaammeehhoommoolologgyyththeeooryryfoforrththeestsattaetsesininbobtohth ccaasseess,,bbuuttiinn tthhee ccaassee ooff tthhe Joneess ppoollyynnoommiiaall((ρρ--bbrraacckkeetteexxppaannsisoionnwwitihthρρ==1)1)wwe e hhaavveeeexxpprreessssiioonnss ffoorr tthhee ccoefficiieennttss ooff tthheeJJoonneessppoolylynnoommiaiallinintetermrmssoof frarnaknsksofofthtehe KKhhoovvaannoovvhhoommoollooggyyggrroouupps. Onllyy tthhee rraannkkssoofftthheecchhaaininccoommpplelexxeessfifigguurereininthtehePoPtotstts mmooddeelliittsseellff..TThhuusswwee aarree ssuggessttiinngg hheerreetthhaattiittiisswwoorrththaasskkininggaabboouut ththeererlealtaiotinosnhsihpip ooffththeeKKhhoovvaannoovv hhoommoollooggy withh tthhee ddiicchhrroommaattiiccppoolylynnoommiaial,l,ththeeρρ-b-brarcakcektetanadndthtehe PPoottstsmmooddeellwwiitthhoouutt cchhaanngging thee ddeefifinniittiioonnoofftthheehhoommoolologgyyggrorouuppssoor rchcahianinspsapcaecse.s. TThhisisaallssoorraaiisseesstthhee qquueessttiioon of thhee rreellaattiioonnsshhiippoofftthheeKKhhoovvaannoovvhhoommoolologygywwithiththtohsoese ccoonnsstrtruuccttiioonnsstthhaatt hhaavvee bbeeen maddee ((ee..gg.. [[1155]]))wwhheerreeththeehhoommoolologgyyhhasasbebeenenadajdujsutsetded totofifittddiirreeccttllyywwiitthh tthhee ddiicchhromattiicc ppoollyynnoommiiaall..WWeewwililltatakkeeuuppththisiscocommpaprairsiosnonininthtehe nneexxttsseeccttiioonn.. We now look more closely at the Potts model by writing a translation between the We now look more closely at the Potts model by writing a translation between the variables q, ρ and Q, v. We have variables q, ρ and Q, v. We have Tracking Potts 1 qρ = Q− 21v −qρ = Q− 2 v − and 1 and 1 2 q + q− 1 = Q 1, q + q− = Q 2 , and from this we conclude that and from this we conclude that whence 2 q "Qq + 1 = 0. q2 − Qq + 1 = 0. − " Whence Whence √Q √Q 4 q = √Q± √Q− 4 q = ±2 − 2 14 14 and 1 √Q √Q 4 = ∓ − . q 2 andand 1 √Q √Q 4 Thus 1 =√Q ∓√Q −4 . v 1 1 q=4/Q ∓ 2 − . ρ = = v(− ± ! −q ) 2 ThTushu−s √Qq 2 v 1 1 4/Q ρ = v = v(−1 ±!!1 −4/Q ) and For physical applications, Q is a positive inρte=ger−g√reQatqe=r tvh(a−n o±r equ2a−l to 2.)Let us 1 √Q √Q 4 −√Qq 2 begin by analyzing the P=otts m∓odel−at .carnidticality (see discussion above) where ρq = 1. q F2or physical applications, Q is a1 po√siQtive√iQnte4ger greater than or equal to 2. Let us For physical applications, Q is a po=sitiv∓e inte−ge. r grea−ter than or equal to 2. Let us ThusThen begin by analyzing the Potts model aqt criticali2ty (see discussion above) where ρq = 1. v begin b1y ana1l1yzTihn4u/gsQt√heQPAtot tcriticalitys√mQode Plotts4at c meetsritica Kholityvano(seev atd ifsourcu scolorssion above) where− ρq = 1. Then ! v 1 1 4/Q ρ = = vρ(−=± =− − ) ± ρ−and= imaginar. = yv (temperatur− ± ! −e! ) − −√QqThen − qa2nd 2 −1√Qq √Q 2√Q 4 ρ = 11 =√Q−√√QQ ±4√Q −4 . q= ∓ − 2. FFoor prhtyhseicKal hapopvliacantoiovnsh, oQmisoalopogsyiti(vietsinEteugelregrrceahFtaeorrrapthhcaytsneicoralriaseptqρpilucic=asal)titootno−s2,q.aQLp=iesptaeu−pasosritdi2veirine±tcegtelrygriena−tetrhthea.npoar erqtuiatlioton2. Let us begin by analyzing the Pot−ts mqodel at criticality (s2ee discussion above) where ρq = 1. beginfubynacntaiolynzinwgethwe Paonttts model at criFtiocarltithye(sKeehTodhuivssacnusosviohnoamboCriticality:ovleo)gwyh(eirtes Eρuqle=r c1h. aracteristics) to appear dir−ectly in the partition Then − v 1 1 4/Q Then ForfuthnectKiohnowvaenwoavnhtomology (ρit=s Euler c1=hva(r−a√cQ±te!ri√stQ−ics)4 to) appear directly in the partition 1 √Q √Q 4ρ = 1. −ρ√=Qq = − ± 2 − . ρ = f=un−ction±we wa−nt . − q ρ = 1.2 − q 2 Thus we want FFoorrpthheyKsichaolvanpopvlihcSupposeaotmioonlos,gQy (thatiitss aE uploeρsrictih=vaerai1cntt.eergisetricgsr)etaotaeprpthearndoirreecqtluyainl ttohe2p.aLrteittiuosn For the Khovanov homology (its Euler chTahraucstewriestiwbcesag)nitnfotubnaycptaipnoenalawyrzedinwigraenthctetlPyoitntstmhoedpeal rattictiroitincality (see discussion above) where ρq = 1. ρ = 1. − function we want Thus we2w=aTnhten!Q !QThen42. = Q Q 4. − ± − 1 !√Q !√Q 4 ρ = 1. Thus we want 2ρ == −= −Q ±± Q−−.4. So −2q=! Q√!2 Q√ 4. Squaring both sides and colSleqcutairninggtbeormthss,idweseafinnddcothllaetct4ingQt−er=m−s!, w±e±fiQ!nd−tQh−at 4 4.QS=quar√inQg√Q 4. Squaring Thus we want For tShequKahrionvgabnootvh hsiodmesoalnodgyco(li−ltescEtiunlgertecrhma∓sr,awcteefirinsdticths)atto4 ap−Qpe=−ar di√reQct√ly∓Qin th4e. pSaqrutaitriionn−g once more, and collectinSgqutoenarrcmiensgm,boworeteh, afisnindddecstohalnlaedtctcihonelgleotcenAndtrimlnygs need,ptweorsem sQfiisb n,=idw l4i tetandhyfiat nfeotd Kh r t e=hρ oa-1.−nt=l4y1pQ∓oisss=iQbil=i√t−yQ4fo.√r ρQ= 14.isSQqu=ari4n.g 2 = !Q !Q fun4c.toinocnewmeowrea,natnd collecting terms, we find that the only pos−sibility for∓ρ = 1 is Q = 4−. √ Returning to the equatio−onnfcRoeertm±uρro,nriwen,ge−antsodeRtechetoeutrhlneliaenqtgcutttaoihnttihigosenetwqefuroaimtlrilosρnb,,fowewr ρese,awfisteneissedρefiet=tehthhda1aat.tttwhtthihshiweseilwonl nbiewlllysaebtpiestfioaessdkastweiibhse√fiinleiwt4dye wt=faokhere√ρn4 w=e1taikseQ =4 =4. Squaring both sides and collecting terms, we find that 4 Q = √Q√Q 4. Squaring Retu2rn. iTnhgistoTchatuhns−ewb2ee. Tqwdhuaionsa∓tnctaienoibnne fdtohnree −ρipn,athrweapemarseaemtreeitzrtiahzatatiiotonnt,h,ainasdntwdheitnlhtlheebnpeatrhstieatitopinsaffiurnteicdttioownnwhfileul nhnacvwetieontawkeill√h4av=e once m2or.e,Tahndiscoclalenctbineg tdeormnse, wine fithn−dethpaatrtahemoentlryi−Kpzhoaosvtsaiinoboinvli,toyapofnolodrgiρctah=l eten1rmitss.hQeHo=pwea4vre.tr,itniootenthaftuwnitchttihoisnchwoiciel,lqh=ave1 and so − Khovanov topological terms. H2o=weveQr, notQe th4a.t with this choi−ce, q = 1 and so 2. This can bve/√doQn=e iρnq t=h1eimppalrieasmthaettvr−i=z!at2i.o±Tnh!,usaenK−d t1h=en 2th, aendpsaortition function will have RetuKrnihnogvtoanthoeveqtuoaptioonlofogriρc,awl etes−eremthsa.t thHisowwilel vbe rs,atnisofi−etde wthheant wweittahket√hi4s−=choice,K−q =− 1 and so − Khvo/v√anQov=tSoqpuρoaqrlion=ggbioc1tahilsmidtpeesrliKamendssc.tohllaHetcotvinwg=etevrmers2, .wnTeofihtnuedsthtehata4t wQ i1=th=−√thQis2√,Qcahnod4i.csSeoq,uaqring= 1 and so 2. vTh/i√s cQan b=e donρeqin=the1paimrampeltireizsattihona,tavnd=th−en t2h.e Tpahrtuitsioen functio1n=will h−2av,eeaKn=d s1o. −− ∓ − − − √ once more, and collecting terms, we find that t−he oKnly possibility for ρ = 1 is Q = 4. − Khovanov topolo−gical terms. Howevv/er, nQote=thatRρweq−tiut=rhnint1hgitsiomcthhpeoleiicqeeus,attiqho−na=ftovr ρ=,1wae−nsde2es.KtohTathtuhiss weill be sa1tis=fied w2he,nawnedtaskoe √4 = K− From this we see that in or−der to−haveρ ==1 at cr1it.ica−lity, we nee−d a 4-state Potts model v/√Q = ρq = 1 implies that v = 2. Thus e 21. T=hKis c2a,nabneddosnoe in the parametrization, and then the partition function will have − − − ewith−im=aginar1y .temperature variableKK = (2n−+ 1)iπ. It is worthwhile considering the −Khovanov topological terms. Howeever, n=ote th1at.with this choice, q = 1 and so K Potts mode−ls at imaginary temperature values. For example the Lee-Yang Theorem e Fr=om 1th. isvw/√eQse=e thρaq t=in1 iomrpdlieers thoathva=ve ρ2.=Thu1s−aeKt cri1ti=cali2t,ya,nwd seo need−a 4-state Potts model − [13] s−hows that under certain circ−umstances the−zeros o−n the partition function are on From this we see that in ordweirthtoimhaagvienaρtrhye=utneim1t ciparcetlercainrtuithtreieccovamalpirtlieyax,bplwaeneeK. Wn=e teakd(e2tanhe4+p-res1ste)anittπcea.lcPIutolaitstiotwsn aomsratonhidwndeihclailtieoncoofnsidering the From this we see that in order to have ρe=K =1 a1t.criticality, we need a 4-state Potts model From this we see that in order to have ρ =Po1ttast cmriotidcaellisttyha,ewtneeiemdnefaoegrdifnuarat4hr-eysrtiantteveemsPtipogtaettsrioamntuoofrdteehelvPao−lttusemso.deFl woirtherexaal amndpcloemtphleex vLaleues-Yforang Theorem with imaginary temperatwuritehviamraiagbinlearKyittse=pmarap(me2ertnearts+.ure1v)aiπri.abItleisKw=ort(h2wn +hil1e)icπo.nIstiidsewrionrgthtwhheile considering the with imaginary temperature variable K [=13(2]nsh+o1w)Fisrπot.mhIatthtiissuwwneodrseetherwtchhaetirlitenacoirondnecsritdrocehuraivmnegsρtth=aen1caetscrtithicealzitye,rwoesnoeend ath4e-stpataerPtoittisomnodfuelnction are on PottsPmoottdselsmaot dimealgsinaatryimtemagpeirnaatPurroyettvtsaelmmueops.deerFalowstriutaherxtiemaimamvgpiaanlealgruiytnheteeasmr.LypeFreta-eotYumraenpvegaexrriTaaabhmtleueorpKreele=mva(t2lhnuee+sL1.)ieπFe.o-ItYrisaewxnoargtmhwTphhilleeecotohrneseidmLereineg-tYheang Theorem the unit circle inNtohwewceogombpaclkeaxndpcloannseid.erWρ =e t1awkiethotuhteinpsirsteinsgeonntccriatilccaluitlya. Ttihoenn waeshaavne indication of [13] [s1ho3w]ssthhaotwunsdethr acetrtuainndceirrcucme[sr1tta3an]icnseshcotihwrecszuetPmrhootasstst1toamu=nnontdchdeveele/ssrp(aqatc√trhietQmiret)aitogaszoininentahfruaycotnitscretmciooupnemratstruhetraeonvnpcaleauserstt.ihtFeioorzneexrfaomuspnloecntthieothLneeea-pYraearntgoitTnihoenorefmunction are on the unit circle in the complex plane. Wetthaekenteheedprf[eo1s3re]nfstuhcorawt−hlsceuthrlaaittniuovnndeesartsicgaernatatiiinnodcniicrcoautfimotsnhtaeonfcPesotthteszmeroosdoenlthwe iptahrtirtieoanlfuanncdtiocnoarme opnlex values for the unit circle in the comtphleeuxnpitlacnirec.tlheeWiunneitthtcaiercklceeointmhthpeelcepoxmreppsleleaxnnpletan.ceWa. WlceeuttaalkaketetiQhotehnper√easpQser√neatQcsnaelcniu4ntladctiaioclncaaustilaoantniinodonicfaatisonaonf indication of the need for further investigation of theiPtsotptsarmamodeetlewrsit.h real and complexv =valuqesQfo=r − ∓ − . the need for further invetshteignaeteiodnfoortfhfetuhnreetehdPefrooritfntusvrtehmesrtoingdvaeesttiliogwantioiotn−fhotf!rhteheeaPPlooatttsntsmdomdceool2dwmeitplh lwreeaixlthanvdraecloaumlepaslenxfdovacrluoems fporlex values for its parameters. its parameters. its parameters. its paraNmowetewrse.gFroombathciskwaensede tchoatnsider ρ = 1 without insisting on criticality. Then we have Now we go back and consider ρ = 1 without insisNtinowg wone gcoribtiaccakliatnyd. cTohnesindewr ρe =ha1vew2ithoQut in√sisQti√ngQon c4riticality. Then we have 1 = v/(q√Q) so that eK = 1 + v = − ∓ − . 1 = v/(q√Q) so that Now−we1g=o bva/c(qk√aQn)dsocthoant sider ρ = 1 without2insisting on criticality. Then we have − − Now we go back and consider ρ =Fr1omwthistwheoguettthienfosliloswtiinnggforomnulacs froirteiKca: lFiotryQ. =T2hweenhavwe eeKh=avie. For Q = 1 = v/(q√Q) so that Q Q√Q√QQ√4Q 4 ± Q √Q√Q3, we4have eK = v 1=v√3=i . qFoqr!QQQ===4 w−e ha∓ve eK = −1. Fo. r Q > 4.it is easy to verify 1 = v/(q√Q)v s=o thq atQ = −− ∓ − . − ±2 −! − ∓2 − − − ! 2 − 2− From this we see that Q √Q√Q 4 From this we see that From this we see thQat v√=Q√qQ!Q =4 − 15 ∓ − . − 2 Q √Q√2Q 4 v = q!Q = − ∓ eK = 1 +−v = . − ∓ − . K 2 −Q √Q√Q 4 2 2 Q 2 √Q√Q 4 e = 1 + v F=rom−thi∓s we see t−hat. eK = 1 + v = − ∓ − . 2From this we get the following formulas for eK : For Q2= 2 we have eK = i. For Q = From this we see that K 1 √3i K ± K 3, we have e = − ±2K . For Q = 4 w2e haveQe =√1Q. F√orQQ > 44it is easy to verify From this we get the following formulas for e : For Q = 2 we havKe e = i. For Q = K − K K 1 √3i From thKis we get the foellow=in1g±+forvm=ulas f−or e ∓: For Q =−2 w.e have e = i. For Q = 3, we have e = − ± . For Q = 4 we have e = 1.2For QQ> 4 i√t isQea√syQto ver4ify 2 ± 2 K −K 1 √3i 15 K e 3=, w1e h+avve e= =−− ±2 ∓ . For Q =−4 we. have e = 1. For Q > 4 it is easy to verify From this we get the followin2g formulas for eK : For Q−= 2 we have eK = i. For Q = 15 K 1 √3i K ± 3, we have e = − ± K . For Q = 4 we have e K= 1. For Q > 4 it is easy to verify From this we get the following formulas for 2e : For Q = 2 we h1a5ve e =− i. For Q = K 1 √3i K ± 3, we have e = − ±2 . For Q = 4 we have e = 1. For Q > 4 it is easy to verify − 15

15 and 1 √Q √Q 4 = ∓ − . q 2 Thus and v 1 !1 4/Q 1 √Q √Q 4 ρ = = v(− ± − ) = ∓ − . −√Qq 2 q 2 and Thus 1 F√orQphy√sQical 4applications, Q is a positive integer greater than or equal to 2. Let us bve=gin by∓ana1lyz−ing1. the4P/oQtts model at criticality (see discussion above) where ρq = 1. ρ = q = v(−2 ± ! − ) − −√TQhqen 2 Thus √ √ v 1 1 4/Q 1 Q Q 4 For physical applicatioρn=s, Q is a p=osvi(ti−ve i±nt!eger−greate)r thρan=or equ=al t−o 2. Le±t us − . −√Qq 2 − q 2 begin by analyzing the Potts model at criticality (see discussion above) where ρq = 1. For the Khovanov homology (its Euler cha−racteristics) to appear directly in the partition Then For physical applications, Q is a positive integer greater than or equal to 2. Let us begin by analyzing the Potts modfeul1natcctiroitnic√awliQetyw(saee√ntdQiscuss4ion above) where ρq = 1. ρ = = − ± − . − Then − q 2 ρ = 1. 1 √Q √Q 4 ρ = = − ± − . For the Khovanov homology (its ET−uhqluesr cwhearwacatnetr2istics) to appear directly in the partition function we want For the Khovanov homology (its Euler characteristics) to appear direct2ly=in the!paQrtition!Q 4. ρ = 1. − ± − function we want Squaring both sides and collecting terms, we find that 4 Q = √Q√Q 4. Squaring Thus we want once mρo=re1, .and collecting terms, we find that the only−possibi∓lity for ρ =−1 is Q = 4. 2 = Q Q 4. Thus we want Ret−ur!ning±to!the−equation for ρ, we see that this will be satisfied when we take √4 = Squaring both sides and collectin2g=t2e.rmT!sh,Qiws ecafi!nnQdbethda4to.4ne Qin =the p√aQra√mQetriz4at.iSoqnu,aarnindgthen the partition function will have − ± − − ∓ − onceSqmuaorrien,gabnodthcsoidlleescatnindgcotellremctsin,−Kgwhteormfivans,ndwotevhafittnotdphtoehlaootng4liycQaplo=stesirbm√ilsiQt.y√HfQoorwρe4=v. Se1qr,uiansroiQntge=th4a.t with this choice, q = 1 and so − ∓ − Retuornncienmgotore,thaendeqcoulaleticotinngfoterrρm,svw, w/e√esfieQnedt=hthaattttρhhiqes o=wnill1yl pibmoespssialbitieilsistfiytehfdoartwρvh=e=n1wise2Q.taTk=heu4√s. 4eK= 1 = 2, and so − 2.RTehtuirsncinagntboethdeoenqeuaintiothnefopraρr,amweetsreiezathtiaot−nt,hiasnwdiltlhbeen stahteisfipeadrtiwtihoenn fwuenct−atikoen√w4il=l have− − −Khov2a.nTovhistocpanolboegdicoanletienrmthse.paHraomweetrviezar,tionno,teantdhtahtewn tihthe ptharistitciohnofiucnec, tqion=ewKill =h1avaend1.so −Khovanov topological terms. However, note that with this choice, q = 1 and so v/√Q = ρq = 1 implies that v = 2. Thus eK 1 = 2, and so − − v/√Q = ρq = 1 implies that v = 2. Thus eK 1 = 2, and so − − − From−− this we se−e−that −in−order to have ρ = 1 at criticality, we need a 4-state Potts model K with iemK a=gina1ry. temperature variable K = (2n + 1)iπ. It is worthwhile considering the e = −1. Potts mode−ls at imaginary temperature values. For example the Lee-Yang Theorem From this we see that in order to have ρ = 1 at criticality, we need a 4-state Potts model From this we see that in order to[h1a3ve] ρsh=ow1 asttchriatitcualnitdy,ewr ecenreteadina 4c-isrtcatuemPosttasnmcoedsetlhe zeros on the partition function are on withwiimthaigmiangairnyartyemtempepreartautruerevvaarriiaabbllee K ==((22nn++11)i)πi.πI.t Iist iwsowrtohrwthhwilehciloencsiodnesriindgerthineg the PottPsomttsomdeoldsealst aitmiamgaignianrayrytetemmppteehrraeattuurneeitvvcaaliluruecesls.e. iFnForothreexeacxmoapmmleppltleheexthLpeeleaL-nYeeea.n-YWgaTenhgteaoTkrhemetohreempresent calculation as an indication of [13][1s3h]oswhoswthsatthautnudnedrecrecretratainincciirrtcchuuemmnsetaaenndcceefsostrhthefeuzreztrehoresorosniontnvheethspteaigrptaiattriiotointnifouonnfcftuihonenctPaioroentotasnrme oondel with real and complex values for the uthneituncitrccliercilne itnhethceocmompplelexxppllaainntese..pWWareeattmaakkeeettehtrhesep. prerseesnetnctaclcaulclautiloantioasnaansianndiicnadtiiocnaotifon of the tnheeendefeodrfofur rftuhrethreirnivnevNosetsiwtgig aconsideratitoionnooff ttrhohee PP=o o1ttt tswithoutsmmoodedle insistingwl withitrhea ronlea nl dancdomcpolmexpvleaxluvesalfuoers for its paraamndeters. criticality. its parameters. 1 No√wQwe√gQo b4ack and consider ρ = 1 without insisting on criticality. Then we have = ∓ − . 1 =q v/(q√2 Q) so that Now we go back and consider ρ =−1 without insisting on criticality. Then we have 1N=ow wvT/eh(ugqs√o Qba)cskoathnadt consider ρ = 1 without insisting on criticality. Then we have − v 1 !1 4/Q Q √Q√Q 4 1 = v/(q√Q) so that ρ = = v(− ± − ) v = q Q = − ∓ − . − −√Qq Q √Q2√Q 4 − ! 2 v = q!Q = − ∓ − . For physical applicatio−ns, Q is a positiQve inte√g2eQr g√reaQter tha4n or equal to 2. Let us v = qF!roQm=th−is w∓e see that − . From thbiesgiwn beysaeneatlyhzaitng the Po−tts model at criticality (see2discussion above) where ρq = 1. Then − From this we see that 1 √Q √Q 4 K 2 Q √Q√Q 4 K ρ = =2 − Q ±√Q√−Q. 4e = 1 + v = − ∓ − . e = 1 + v−=q − ∓2 − . 2 2 Q 2√Q√Q 4 For the KhovanovehoKmo=log1y+(itsvEu=ler ch−aracte∓ristics) to appe−ar dir.ectly in the partition From thfuisncwtieongwetetwheanftollowing foFrmroumlasthfoirsewKe:gFeot2rtQhe=fo2lwloewhianvge efKorm= ulia.sFfoorrQe=K : For Q = 2 we have eK = i. For Q = ± 3, we have eK = 1 √3i . For Q = 4 we ρha=ve1.KeK = 1. F√o3riQ > 4 it is easy to verify K ± − ±2 3, we have eK = − ± . For Q =K4 we have e = 1. For Q > 4 it is easy to verify From this Twheusgweet twhaentfollowing formulas for e : For−Q 2= 2 we have e = i. For Q = − K 1 √3i K ± 3, we have e = − ± . For Q =2 =4 w!e Qhave!eQ =4. 1. For Q > 4 it is easy to verify 2 − 15± − − Squaring both sides and collecting terms, we find that 4 Q = √Q√Q 4. Squaring15 once more, and collecting terms, we find that the only−possibi∓lity for ρ =−1 is Q = 4. Returning to the equation for ρ, we see th1a5t this will be satisfied when we take √4 = 2. This can be done in the parametrization, and then the partition function will have −Khovanov topological terms. However, note that with this choice, q = 1 and so v/√Q = ρq = 1 implies that v = 2. Thus eK 1 = 2, and so − − − − − eK = 1. − From this we see that in order to have ρ = 1 at criticality, we need a 4-state Potts model with imaginary temperature variable K = (2n + 1)iπ. It is worthwhile considering the Potts models at imaginary temperature values. For example the Lee-Yang Theorem [13] shows that under certain circumstances the zeros on the partition function are on the unit circle in the complex plane. We take the present calculation as an indication of the need for further investigation of the Potts model with real and complex values for its parameters.

Now we go back and consider ρ = 1 without insisting on criticality. Then we have 1 = v/(q√Q) so that − Q √Q√Q 4 v = q Q = − ∓ − . − ! 2 From this we see that 2 Q √Q√Q 4 eK = 1 + v = − ∓ − . 2 From this we get the following formulas for eK : For Q = 2 we have eK = i. For Q = K 1 √3i K ± 3, we have e = − ± . For Q = 4 we have e = 1. For Q > 4 it is easy to verify 2 −

15 and 1 √Q √Q 4 = ∓ − . q 2 Thus v 1 1 4/Q ρ = = v(− ± ! − ) −√Qq 2

For physical applications, Q is a positive integer greater than or equal to 2. Let us and begin by analyzing the Potts model at criticality (see discussion above) where ρq = 1. 1 √Q √Q 4 − = ∓ Th−en . q 2 1 √Q √Q 4 ρ = = − ± − . Thus and − q 2 v 1 !1 1 4/Q√Q √Q 4 ρ = = v(− ± − = ) ∓ − . −√Qq For t2heqKhovano2v homology (its Euler characteristics) to appear directly in the partition Thus function we want For physical applications, Q is a positive integer greater than or equal to 2. Let us v 1 1 4/Q ρ = 1. begin by analyzing the Potts model at criticalityρ(s=ee discussi=onva(b−ove±) w!her−e ρq =) 1. −√Qq 2 − Then Thus we want 1 √Q √Q 4 Foρr =physica=l ap−plicat±ions, Q−is a.positive integer greater than2o=r equa!l toQ2. Le!t uQs 4. begin by an−alqyzing the Pott2s model at criticality (see discussion above)−where ρ±q = 1. − For the Khovanov homTohloegny (its Euler characteSrqisuticasr)intogapbpoetahr dsiirdeectslyainndthceoplalretictitoin g terms, w−e find that 4 Q = √Q√Q 4. Squaring 1 √Q √Q 4 − ∓ − function we want oncρe=more=, a−nd co±llecti−ng .terms, we find that the only possibility for ρ = 1 is Q = 4. ρ = 1. − q 2 Returning to the equation for ρ, we see that this will be satisfied when we take √4 = Thus we want For the Khovanov homology (its Euler characteristics) to appear directly in the partition function we want 2. This can be done in the parametrization, and then the partition function will have 2 = !Q !−Q 4. − ± Kho−vanov toρp=ol1o.gical terms. However, note that with this choice, q = 1 and so Squaring both sides and collecting terms, we find that 4 Q = √Q√Q 4. Squaring K − v/√Q−= ρ∓q = 1 im−plies that v = 2. Thus e 1 = 2, and so once more, and collectTinhgustewrme ws,awnte find that the only pos−sibility for ρ = 1 is Q = 4. − − − 2 = Q Q 4. Returning to the equation for ρ, we see that this will be s−at!isfied±w!hen w−e take √4 = K 2. This can be done Sinquthaerinpgarbaomthetsriidzeastiaonnd, caonldletchteinngthteermpasr,twitieofinnfdutnhcattio4n wQil=l hav√e Q√Q e 4. S=quari1n.g − − Khovanov topologicalotnecremms.orHe,oawnedvceor,llnecottiengthtaetrmwsi,thwethfiisndchthoaictet,heqo=nly−p1osasinbdi∓listoy for ρ =−1 is Q = 4. v/√Q = ρq = 1 impRleietusrtnhiantgvto=the2e.qTuahtuiosFnerKfoormρ1,thw=iesswe2e,etahsnaedtetshotihsawtililnb−oe rsdateisrfiteod hwahvene wρe=tak1e a√t4c=riticality, we need a 4-state Potts model − 2. This can b−e done inwthieth−pairmamaeg−tirnizaatriyont,eamndptehreanttuhreepvaratirtiioanblfeunKctio=n w(i2llnha+ve1)iπ. It is worthwhile considering the − eK = 1. Khovanov topological−tePromtst.s Hmoowdeveelrs, naotteimthaatgwinitahrythitsecmhopiecer,atqur=e va1luaneds.soFor example the Lee-Yang Theorem v/√Q = ρq = 1 implies that v = 2. Thus eK 1 = 2, and so − From this we see that in order to−have ρ = 1 at[c1ri3ti]caslhityo,ww−se ntheeadtau4n-dstea−trecPeorttts−aminodceilrcumstances the zeros on the partition function are on with imaginary temperature variable K = (2nt+he1)uiπn.itItciisrwceKloert=hinwhth1il.ee ccoonmsidpelreinxgpthleane. We take the present calculation as an indication of Potts models at imaginary temperature values. For example the−Lee-Yang Theorem [13] shows that underFcreormtaitnhicsirwceumsesetathnacteisnthotherdezeernrtoeosehodanvfetohρer=pfua1rrtatithicoernritificunanvlciettyiso,twnigeaarneteieoodnna o4-fsttahteePPototsttmsomdeol del with real and complex values for the unit circle in the cowmitphleixmpaglainnea.ryWteemtapkeeratthiutersepprveaasrreianbmtlceeaKltceur=lsa.t(i2onn +as1a)niπin.dIticisatwioonrtohfwhile considering the the need for further invPeosttsigmatoiodnelosfattheimPaogtitnsamryodteeml wpeirtahturerealvaanludecso. mFpolrexexvaamlupelse ftohre Lee-Yang Theorem its parameters. [13] shows that under certainNcoirwcumwsetangcoesbthaeckzeraonsdoncothnespiadretirtioρn=fun1ctiwonitahreouont insisting on criticality. Then we have the unit circle in the complex plane. We take the present calculation as an indication of 1 = v/(q√Q) so that Now we go back atnhdecnoenesdidfeorr ρfu=rth1erwiintvheosutitgiants−iiosntinogf tohne cProittitcsamlitoyd.eTlhweinthwreeahlaavned complex values for 1 = v/(q√Q) so thaitts parameters. − Q √Q√Q 4 Now we go back Qand c√onQsi√deQr ρ =41 without insistinvg o=n critqic!aliQty. T=he−n we h∓ave − . v = q Q = − ∓ − . − 2 1 = v−/(q!√Q) so that 2 − From this we see that From this we see that Q √Q√Q 4 v = q!Q = − ∓ − . 2 Q √Q−√Q 4 2 2 Q √Q√Q 4 eK = 1 + v = − ∓ − . eK = 1 + v = − ∓ − . From this we see that 2 2 K K From this we get the following formulas for e K: For Q = 2 w2e haQve e √=Q√Qi. Fo4r Q = K K K 1 √3i eFrKo=m1t+hivs =we g−et t∓he fol±low−ing. formulas for e : For Q = 2 we have e = i. For Q = 3, we have e = − ± . For Q = 4 we have e = 1. For Q > 4 it i2s easy to verify 2 − K 1 √3i K ± 3, Fwore Qh =a v3,e e = − ± . For Q = 4 we have e = 1. For Q > 4 it is easy to verify From this we get the following formulas for eK : Fo2r Q = 2 we have eK = i. For Q = − K 1 √3i K ± 3, we have e = − 1±5 . For Q = 4 we have e = 1. For Q > 4 it is easy to verify 2 − For Q t>4,ha t eK is real and negative. Thus in all cas1es5of ρ = 1 we find that the Potts model has comp1l5ex temperature values. In a subsequent paper, we shall attempt to analyze Thus we get complex temperature values in all cases wherthee thein flcoefficientsuence of tofhe theKh Poottsvan ov homology at these complex values on the behaviour of model are givthene m diroectldelyf oinr termsreal t eofm Eulerpera tures. characteristics from Khovanov homology.

4 The Potts Model and Stosic’s Categoriﬁcation of the Dichromatic Polynomial

In [15] Stosic gives a categoriﬁcation for certain specializations of the dichromatic polynomial. In this section we describe this categoriﬁcation, and discuss its relation with the Potts model.

For this purpose, we deﬁne, as in the previous section, the dichromatic polynomial through the formulas:

Z[G](v, Q) = Z[G!](v, Q) + vZ[G!!](v, Q)

Z[ G] = QZ[G]. • ! where G! is the result of deleting an edge from G, while G!! is the result of contracting that same edge so that its end-nodes have been collapsed to a single node. In the second equation, represents a graph with one node and no edges, and G represents the disjoint un•ion of the single-node graph with the graph G. The gra•p!h G is an arbitrary ﬁnite (multi-)graph. This formulation of the dichromatic polynomial reveals its origins as a generalization of the chromatic polynomial for a graph G. The case where v = 1 is the chromatic polynonmial. In that case, the ﬁrst equation asserts that the number−of proper colorings of the nodes of G using Q colors is equal to the number of colorings of the deleted graph G! minus the number of colorings of the contracted graph G!!. This statement is a tautology since a proper coloring demands that nodes connected by an edge are colored with distinct colors, whence the deleted graph allows all colors, while the contracted graph allows only colorings where the nodes at the original edge receive the same color. The difference is then equal to the number of colorings that are proper at the given edge.

We reformulate this recursion for the dichromatic polynomial as follows: Instead of contracting an edge of the graph to a point in the second term of the formula, simply label that edge (say with the letter x) so that we know that it has been used in the recursion. For thinking of colorings from the set 1, 2, , Q when Q is a positive integer, regard an edge marked with x as indicating{that t·h·e·colo}rs on its two nodes are the same. This rule conicides with our interpretation of the coloring polynomial in the last paragraph. Then G!! in the deletion-contraction formula above denotes the labeling of the edge by the letter x. We then see that we can write the following formula for the dichormatic polynomial: H e(H) Z[G] = ! Q| |v H G ⊂