Khovanov Homology and Virtual Cobordism

Louis H. Kauffman, UIC [16], and Bar-Natan’s emphasis on cobordisms [2]. We use similar considera- tions [i1n6]o,uarndpaBpare-rN[a1ta0n]’s. emphasis on tangle cobordisms [2]. We use similar considera- tions in our paper [10].

TwoTkweoykmeyomtiovtaivtaintinggiiddeeaassaarereinivnovlvoeldviendfiinndinfigndthiengKhtohveanKovhionvvarniaonvt. inFivrsatriant. First of all, one would like to categorify a polynomial such as K . There are many of all, one would like to categorify a link polynomial su⟨ch⟩ as K . There are many meanminegansintogsthtoethtertmermccaatteeggorriiffyy, ,bubtuhtehreetrheetqhueesqt uisetsotfiinsdtoa wfianydtoaewxp⟨areyss⟩tothexlipnkress the link polynomial as a graded Euler characteristic K = χ H(K) for some homology polynomial as a graded Euler characteris⟨tic⟩ K q=⟨ χ ⟩H(K) for some homology theory associat[e1d6],wanitdhBaKr-Na.tan’s emKhovanovphasis on tang leHomologycobordisms [2]. We quse similar considera- theory associatedtiownsiitnhour⟨Kpap⟩e.r [10]. ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ The bracket pTowlyonkoeymmiaoltiv[a7ti]ngmiodedaeslarfeorinvtholeveJdoinnefisndpionglytnheoKmhioavlan[4ov, i5n,va6ri,a1nt7. ]Fiisrstusually described by thoef aellx, poanenswioounld like to categorify a link polynomial such as K . There are many The bracket pmoelayningos tmo tihae lter[m7c]atmegooridfye, blutfhoerettheequJeost nisetosfipndoalwy⟨anyo⟩tomexiparelss[t4he,li5nk, 6, 17] is usually polynomial as a graded Euler characteristic K = χ H(K) for some homology described by the expansion ⟨ 1⟩ q⟨ ⟩ theory associated with K =. A + A− (4) ⟨ ⟨ ⟩ ⟩ ⟨ ⟩ ⟨ ⟩ The [7] model for the Jones poly1nomial [4, 5, 6, 17] is usually and we have = A + A− (4) described by the ex⟨pansion⟩ ⟨ ⟩ ⟨ ⟩ 2 2 1 K = ( =AA A+−A−) K (4) (5) and we have ⟨ ⃝⟩ ⟨ ⟩− ⟨ −⟩ ⟨⟨ ⟩⟩ and we have 3 = ( A )22 2 2 (6) K⟨ K⟩ = (=−( AA ⟨ A−⟩A) −K ) K (5) (5) ⟨ ⃝⟨⟩ ⃝⟩ −− −− ⟨ ⟩ ⟨ ⟩ = ( 3A3) (6) =⟨ ( ⟩ A−− ) ⟨ ⟩ (7) ⟨ ⟩ − ⟨ 3 ⟩ = ( A3 ) (6) = ( A− ) (7) Letting c(K) denote the numb⟨er of⟨⟩cro⟩ ssin−gs in⟨th⟨e⟩ dia⟩gram K, if we replace K c(K) 1 ⟨ ⟩ by A− K , aLnedttinthgec(nKr)edpelnaocteethAe nbuymberqo−f cr,ostshinegsbirnatchke edtiawgraimll Kbe, ifrewwe reipttleacne iKn the fol- ⟨ ⟩ c(K) − 1 3 ⟨ ⟩ lowing form: by A− K , and then replace A=by( q−A,−the b)racket will be rewritten in the fol- (7) lowing form⟨ : ⟩ ⟨ ⟩ −− ⟨ ⟩ = = q q (8) (8) ⟨ ⟩ ⟨ ⟨ ⟩ ⟩⟨ −⟩ −⟨ ⟨ ⟩⟩ 1 1 Letting c(K)wditeh not=e (tqh+eq−nu).mIt ibs ueserfuol tfo ucsreothsissfionrmgosf tihne btrahceketdstiaategsurmamfor tKhe s,akief we replace K with = (q +q⟨−⃝⟩). It is useful to use this form of the bracket state sum for the sake ofct(hKe⟨⃝)gr⟩ading oifnthtehgeraKdihngovinatnhoe vKhhoovamnoovlohogmyo(lotgoyb(teo1bdeedsecsrcirbibeedd bbeelolwo)w. W).eWsheallschoanltilnuceontinue ⟨ ⟩ by A− K , atonrdefetrhtoetnhe rsmepooltahicnges lAabelbedyq (orqA−1 i,n ttheeorbigrinaacl bkraectkewt foirlml ublaetionr)eaws ritten in the fol- ⟨ ⟩ −1 − lowintgo freofremr t:o theBs-msmoooothhiinnggs.sWleabsheolueld fqurt(hoerrnAote−thatinwethusee tohreiwgienll-aklnobwrancckonevtenftoiornmouf leant-ion) as B-smoothings.haWncedsshtaoteuslwdhfeurerathneernhnanocteedtshtaatet hwaseaulasbeeltohfe1 woreXll-oknneaocwh nof citos ncovmepnotnioennt of en- 1 hanced states wloohpesr.eWaenthennrheaganrcdethde svtaalute oh=fathsealoloapbqe+l oq−f q1asotrheXsumonofethaecvhalouef oiftsa ciorcmleponent (8) labeled with a 1 (the va⟨lue is q⟩) adde⟨d to the⟩va−lue of⟨a circl⟩e labeled with an X (the 1 1 loops. We thenvarleuge1aisrqd−th).eWveaclouueldohfavtehceholoseonpthqe m+orqe−neutarasl tlahbeelsuofm+1ofantdhe1vsaolutheatof a circle with = (q +q− ). It is useful to use this form of the brac−ket state sum for the sake labeled with a 1 (the value is q) added t+o1the value of a circle labeled with an X (the ⟨⃝⟩ 1 q +1 1 of thevagluraedisinq−g i)n. WtheecoKuhldohvaavneochvohseonmthoelmo⇐gor⇒ye (nteou⇐tbr⇒ael ldabeeslcsroifb+ed1 abnedlow1)s.oWthaet shall continue and 1 − to refer to the smoothings labele+d1 q (1or A− in the original bracket formulation) as q q− +1 1 1X, ⇐⇒⇐⇒ −⇐⇒⇐⇒ B-smoothings. Wbuet, ssinhcoe aunladlgefburaritnhvoelvrinngo1taendtXhantatuwralely uapspeeartshlaeterw, wee ltalk-ektnhios fwormn ocfonvention of en- hanceadndstates whleabreelinagnfroemnthheabengcinenidng.state has a label of 1 or X on each of its component 1 q− 1 X, 1 loops. We then regard the value of ⇐th⇒e l−oop⇐q⇒+ q− as the sum of the value of a circle labelebdut,wsinthceaan1a(lgtheberavainlvuoelviisngq1) andddXedntaotutrhalelyvaaplpueearsoflaaterc, iwrcelteakleabtheilsefdorwmiothf an X (the 1 valuelaisbeqli−ng )fr.oWm ethecobueglidnnhiangv.e chosen the more neutral labels of +1 and 1 so that 5 − q+1 +1 1 ⇐⇒ ⇐⇒ and 1 q− 5 1 X, ⇐⇒ − ⇐⇒ but, since an algebra involving 1 and X naturally appears later, we take this form of labeling from the beginning.

5 Cubism The bracket states form a category. How can we obtain topological information from this category? [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 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 polynomial [7] model for the [4, 5, 6, 17] is usually described by the expansion

1 = A + A− (4) ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ and we have

2 2 K = ( A A− ) K (5) ⟨ ⃝⟩ − − ⟨ ⟩

= ( A3) (6) Exploration: ⟨Examine⟩ the− Bracket⟨ ⟩ Polynomial

for Clues.3 = ( A− ) (7) Let ⟨c(K) =⟩ number− of crossings⟨ ⟩on link K. -c(K) -1 Letting c(K) denote the nFormum bAe r o f cros sandin greplaces in t hAe byd i-qa g r .am K, if we replace K c(K) 1 ⟨ ⟩ by A− K , and then repThenlace theA skeinby relationq− , tforhe bra cwillke t will be rewritten in the fol- lowing form⟨ : ⟩ be replaced− by: = 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 (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 en- hanced 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 Use enhanced states by labeling each loop with +1 or -1.

+1 -1

= +

-1 q + q + - = + = q + q-1

+ + =

+ - = qq + + qq-1 + q -1 q + q -1 q -1 - + -1 + = qq + 2 + (qq)

- - -1 2 + = (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 : 1 Forq reasons− that will1 soonX become, apparent, we let K-1 be=⇐ denoted⇒ −( by1)⇐ Xn⇒B and(s) q+1j( sbe) denoted by 1. but, since an algebra involving 1 and X⟨naturally⟩ s appears− later, we take 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 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 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 − +1 1 1 withwithaa11(the(thevvaluealueisisqq))aaddedddedttootthehevvaluealue ooff a qcircle labeledlabeled+1wwithithanan1XX(the(thevvaaluelueisisq−q−).)W. We ecouldcouldhahvaeve chosenchosenthethemmoreoreneutralneutrallabelslabelsofof+1+1andand 11 so that⇐⇒ ⇐⇒ −− and +1 qq+1 1 +1+1 11 q⇐−⇒ ⇐⇒1 X, ⇐⇒⇐⇒⇐−⇒ ⇐⇒ andand but, since an algebra involving 1 and X1 naturally appears later, we take this form of labeling from the q−1 1 X, q− 1 X, beginning. ⇐⇒ −− ⇐⇐⇒⇒ bubt,ut,sincesinceananalgebraalgebraininvvoolvinglving 11 andand XX naturallynaturally appearsappears laterlater,,wewetaktakeethisthisformformofoflabelinglabelingfromfromthethe beginning. beginning.To see how the Khovanov grading arises, consider the form of the expansion of this version of the brackTeto polynonmialsee how the Khoinvenhancedanov gradingstates.arises,Wceonsiderhave thetheformulaform of theas aexpansionsum oveorfetnhancedhis versionstatesof thes : 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 : bracket polynonmial in enhanced states. We haKve=the formula( 1)naBs (assum)qj(so)ver enhanced states s : K ⟨= ⟩ ( 1)nB−(s)qj(s) s nB (s) j(s) ⟨K⟩ = s (−!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 To see how the Khovanov grading ariswherees, connsiBde(sr )thise ftheormnumberof the exopfaBns-typeion ofsmoothingsthis in s, λ(s) is the number of loops in s labeled 1 minus thewherenumbernB(s)oisf ltheoopsnumberlabeledof BX-type, andsmoothingsj(s) = nBin(ss), +λ(sλ)(sis).theThisnumbercan beof lreoopswrittenin s labeledin the follo1 minuswing form: version of the bracket polynonmial in enhtheancnumbered stateso.fWloopse havlabelede the foXrm,uandla asj(ass)u=m nB(s) + λ(s). This can be rewritten in the following form: over enhanced states s : the number of loops labeled X, and j(s) = nB(s) + λ(s). This can be rewritten in the following form: i j i j ij ij nB (s) j(s) K =K =( 1) (q dim1) q( dim) ( ) K = !( 1) q ⟨ ⟩ i −j ij C ⟨ ⟩ − ⟨K ⟩ = i ,j (−i1),j q dim(C ) s ! ⟨ ⟩ i ,j −! C ij n (s) = number of B-smoothings! in the state s. where nB(s) is the number of B-type smwhereoothingwseindsefine, λ(s) iijs tothebenuthembBelinearr of loospanps in(over k = Z/2Z as we will work with mod 2 coefficients) of where we defineCij to be the linear span (over k = Z/2Z as we will work with mod 2 coefficients) of s labeled 1 minus the number of loops lwhereabeled Xwe, adnefined j(s) =C ntoBbe(s)the+ λlinear(s).=T numberhsipans can (ofov +1er loopsk = minusZ/2Z numberas we ofwill -1 loops.work with mod 2 coefficients) of thethesetsetofofenhancedenhancedC statesstateswithwithnBn(s)(=s)i=andi jand(s) =j(sj.)Then= j.theThennumberthe numberof such statesof suchis thestatesdimensionis the dimension be rewritten in the following form: the set ofij enhanced states with n (sB) = i and j(s) = j. Then the number of such states is the dimension dim( ij ). CBij = module generated by enhanced states dimdim( Cij)).. iCj ij with i =n B and j as above. K = !( W1e)wCqoulddimlik( e )to have a bigraded complex composed of the ij with a differential ⟨ ⟩ − C C ij i ,jWWee wwouldouldliklikeetotohahvaevaebaigradedbigradedcomplecomplex composedx composedof theofijthewith a dwithifferentiala differential ∂ : ij i+1 jC. C where we define ij to be the linear span (over k = Z/2Z as we will work with mod∂ : C∂ij :−→ijC i+1 j. i+1 j. C The differential should increase the homologicalC gr−ading→ Ci by 1 and preserve the quantum grading j. Then 2 coefficients) of the set of enhanced states with nB(s) = i and j(s) = j. Then the C −→ C number of such states is the dimensionThedTheiwem(dcouldififijferentialferential). writeshouldshouldincreaseincreasethethomolohe homologicalggicalradinggradingi by 1 iandbyp1reservand epreservthe quantume the quantumgrading j.grThenading j. Then C j i ij j j wewe couldcould wwriterite K = q ( 1) dim( ) = q χ( • ), ij ⟨ ⟩ j − i C ij j C j We would like to have a bigraded complex composed of the with aKdiffe=ren!tjialq !i j( 1) dim(i ) =!ijj q χ( • )j, j j C ⟨ ⟩K = q − ( 1) Cdim(j ) = C q χ( • ), where χ( • ) is the Euler characteristic⟨ ⟩ j of ithe subcomple− x • Cforj a fixed value oCf j. ∂ : ij i+1 Cj. ! j ! i C ! j C where−→ Cχ( j) is the Euler characteristic!of the!subcomplex j for a fix!ed value of j. • j • j whereThis formulaχ(C • )wisouldthecEonstituteuler characteristica categorificationof theof tsheubcomplebrackCet polynomial.x • for aBelofixedw, vwealueshallofseej. how the The differential should increase the homooriginallogical grCKhoadinvanog i bvy 1difanferdentialpreserv∂e tisheuniquelyquan- determined by the reCstriction that j(∂s) = j(s) for each tum grading j. Then we could write This formula would constitute a categorification of the bracket polynomial. Below, we shall see how the enhanced state s. Since j is preserved by the differential, these subcomplexes j have their own Euler ThisoriginalformulaKhovanowouldv differconstituteential ∂ isa cateuniquelygorificationdeterminedof theby tbrackhe restrictionet polynomial.thatC• j(∂sBelo) =wj,(wes) forshalleacseeh how the j icharacteristicsij andj homologyj . We have j K = ! q !( 1originalenhanced) dim( Kho)state= !vanos.qSincevχ(dif• ferj),isentialpreserv∂edisbyuniquelythe differential,determinedthese bsubcompley the restrictionxes • hathatve theirj(∂so)wn=Eulerj(s) for each ⟨ ⟩ − C C C j i j j j j enhancedcharacteristicsstateands.homologySince j .isWpreserve have edχ(byH(the• ))dif=ferential,χ( • ) these subcomplexes • have their own Euler C C C j characteristics and homologyj . We have j j where χ( • ) is the Euler characteristic owheref the suHb(comjp)ledenotesx • fother a fihomologyxed value ofof thejχ. (Hcomple( • ))x =jχ. (We• c)an write C C• C C C• C j j jj j j This formula would constitute a categowhererificatiHon(of• th)edenotesbracketthepolyhomologynomial. BofelotheKw, c=ompleχ(Hq(xχ•(•H)).(W=•e ))χcan.( write• ) C ⟨ ⟩ CC C C we shall see how the original Khovanov differential ∂j is uniquely determined by the !j j jj where H( • ) denotes the homologyKof =the compleq χ(Hx( •• )). .We can write restriction that j(∂s) = j(s) for each enThehanclasted sformulaCtate s. Seixpressesnce j is ptheresbrackervedetbypolynomial⟨the⟩ as a gradedCC Euler characteristic of a homology theory j j differential, these subcomplexes • have their own Euler characteristics and homol- ! j j C associated with the enhanced states of the brackK =et stateqsummation.χ(H( • ))This. is the categorification of the ogy. We have The last formula expresses the bracket polynomial⟨ ⟩ as a graded EulerC characteristic of a homology theory jbracket polynomial.j Khovanov proves that this homologyj theory is an invariant of knots and links (via the χ(H( •associated)) = χ( •with) the enhanced states of the bracket state! summation. This is the categorification of the C ReidemeisterC moves of Figure 1), creating a new and stronger invariant than the original Jones polynomial. j bracket polynomial.j Khovanov proves that this homology theory is an invariant of knots and links (via the where H( • ) denotes the homology oThef the lastcompformulalex • . Wexpressese can writethe bracket polynomial as a graded Euler characteristic of a homology theory C associatedReidemeisterwithCmovthees ofeFnhancedigure 1), creatingstates ofa nethew abracknd strongeret stateinvsariantummation.than the Thisoriginalis Jtheonescpolynomial.ategorification of the j j 6 K = ! q χ(H( • )). ⟨ ⟩ bracket polynomial.C Khovanov proves that this homology theory is an invariant of knots and links (via the jReidemeister moves of Figure 1), creating a new6 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 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 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 the form of the expansion of this version of the i j ij bracket polynonmial in enhanced states.K W=e ha!ve(the 1formula) q daisma(sum o)ver enhanced states s : ⟨ ⟩ − C K =i ,j ( 1)nB (s)qj(s) ⟨ ⟩ s − ij ! whwhereere wne(ds)eisfinthee numberto obfeBt-typehe lismoothingsnear spanin(so,vλe(rs)kis=theZnumber/2Z aofsloopswe winislllabeledwork1wminusith mod B C 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 To see how the KhovWanewhereovwgroaduwinledgdalefineriiksees,tcooijnhstoiadvebereththeae fbolinearirgmroafdtshepanedexc(pooavmnesripoknleo=xf tZhcio/s2mZpasosweedwillof twhorke withij wmodith 2a coefdiffficients)erentiaolf version of the bracket polynonmial in enhancCed states. We have tWanted:he form udifferentialla as a su actingm in the form the set of enhanced states with nB(s) = i and j(s) = j. Then the number ofCsuch states is the dimension over enhanced states s : ij dim( ). nB (s) j(s) ij i+1 j 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 number of loops labeled X, and j(s) = nB(s) + λ(s)C.toT hdecrease−is→caCn by 1. be rewritten in the followtuinmg fgorrma:ding j. Then we could write The differential should increase the homological grading i by 1 and preserve the quantum grading j. Then we could write i j ij j i ij j j K = !( 1) q dim( ) j i ij j j ⟨ ⟩ − K C=K!= q !q (( 11)) dimdim( ( ) =) =q!χ( •q ),χ( • ), i ,j ⟨ ⟩ ⟨ ⟩ −− C C C C j !j i!i !j j ij j j where we define to be twherehe lineχar( sp•an) is(ovtheer kEuler= Zcharacteristic/2Z as we wilofl wtheorkswubcompleith mod x • for a fixed value of j. 2 coefficients) ofCthe set of enhanceCd stajtes with n (s) = i and j(s) = j. Then the C j where χ( • ) is theBEuler characteristic of the subcomplex • for a fixed value of j. number of such states is theThisdimeformulansion diwmould( ij)c.onstitute a categorification of the bracket polynomial. Below, we shall see how the C C 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• characteristics and homology. We have we shal∂l:seije howi+t1hje. original Khovanov differential ∂ is uniquely determined by the C −→ C j j restriction that j(∂s) = j(s) for χe(aHc(h •en))h=anχc(e•d )state s. Since j is preserved by the The differential should increase the homological grading i by 1 and preserjveCthe quan- C tum grading j. Then wedciofwhereufledrwerniHteti(al,jt)hdenotesese suthebchomologyomplexeofs the•complehavex thje.iWr eowcannwriteEuler characteristics and homol- C• C C• ogy. We have j j j i ij j j K = q χ(H( • )). K = ! q !( 1) dim( ) = ! q χ( • ), j j ⟨ ⟩ − C C χ⟨ (H⟩ ( j• )) = χC ( • ) 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 restriction that j(∂s) = j(s) for each enhanced state s. Since j is preservejd 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 the homology of the complex • . We can write C This is the categorificCation of the bracket polynomial. Khovanov proves that this ho- 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 ho- 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

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 differentialof the shouldsubcomple increasex theC• homologicalfor a fixed value of j. C ij i+1Cj original Khovanov differential ∂gradingis uniquely i by 1 and∂determined leave: fixed the bquantumy. the rgradingestriction 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∂ istheuniquelyhomoloThengicaldetermined weg rwouldading haveibbyy 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 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 that the existence of a bigraded complex of this type allows us to further if (s, τ) denotes a pair consisting in warnite:enhanced state s and site τ of that state with τ of type A, then ⟨K⟩ = qj (−1)idim(Cij ) = qj χ(C• j ), X X X we consider all enhanced states s′ obtained from sj byi smoothing atj τ and relabeling only those loops that where χ(C• j ) is the Euler characteristic of the subcomplex C• j for a fixed are affected by the resmoothing. Call vthisalue of setj. Sincoe jfisepnhancedreserved by thestatesdifferentiaSl, th′[ess,e suτbc]o.mThenplexes havwee shall define the partial their own Euler characteristics and homology. We can write differential ∂τ (s) as a sum over certain elements in S′[s, τ], and the differential by the formula ⟨K⟩ = qj χ(H(C• j )), X j

where H(C•∂j )(dsen)ot=es the hom∂oloτg(ysof)this complex. Thus our last for- mula expresses the bracket pτolynomial as a graded Euler characteristic of a homology theory associat!ed with the enhanced states of the bracket state summation. This is the categorification of the bracket polynomial. Kho- The boundary is a sum of partial differentials with the sum over all type A sites τ invans.ov pItrovthenes that tremainshis homology tthoeorsy eeis anwhatinvariantareof knotthes andpossibilitieslinks, for ∂τ (s) so that correspondingcreating a new and tostr oresmoothingsnger invariant than tonhe o therigin astates.l Jones polynomial. j(s) is preserved. We explain the differential in this complex for mod-2 coefficients and leave it to the reader to see the references for the rest. The differential is defined via the algebra A = k[X]/(x2) so that X2 = 0 with coproduct Note that if s′ S′[s, τ], then nB(s′) ∆=: An−→BA(s⊗)A+defi1ne.d bThusy ∆(X) = X ⊗ X and ∆(1) = 1 ⊗ X + X ⊗ 1. ∈ Partial differentials (which are defined on an enhanced state with a chosen site, whereas the differential is a sumXof 2 t h e s=e m0appings) are defined on j(s′) = enach e(nhsa′n)ced+staλte(ssa′n)d a=site1κ +of tynpe A(isn )tha+t stλate(.sW′)e .consider statBes obtained from the given state by smBoothing 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 From this we conclude that j(s) = j(sin′g)le ifloopandat κ, oonlyr we diviidfe aλlo(opsa′t)κ i=nto tλwo(dsis)tinct lo1op.s.RecallIn the case that of amalgamation, the new state s acquires the label o−n 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 theλpa(rstia)l d=iffere[nstia:l is+de]scribed[sby:the m]ultiplication in the algebra. If one circle becomes two circle−s, then we−apply the coproduct. Thus if the circle is labelled X, then the resultant two circles are each labelled X where [s : +] is the number of loops incorresspolabelednding to ∆(X+1) = ,X[⊗sX:. If th]e isorgithenal cirnumbercle is labelled o1 fthelnoops labeled 1 (same as we take the partial boundary to be−a sum of two enhanced states with − labeled with X) and j(s) = nB(s) + lλab(els1).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 Proposition. The partial differentialsto∂veτri(fysd)irearectly thuniquelyat the square ofdeterminedthe differential mapbyping ithes zero canonditiond that j(s′) = j(s) that it behaves as advertised, keeping j(s) constant. There is more to say for all s′ involved in the action of the apartialbout the nadifture foerentialf this construcontion wtheith reespnhancedect to Frobeniustates algebras.s This unique form of the partial differential can be described bayndttheanglefollocobordwingisms. Seestructures[Kh, BN, BN2] of multiplication and comultiplication on the algebra A = k[X]/(X2) where k = HZer/e 2weZconforsidermbigod-2raded cocmoefplexefis cients,Cij with heoighrt k(hom=oloZgicalfor integral coefficients. 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 1. The element 1 is a multiplicativeeverunity enhanacnded staXte, 2the=inde0x.j is equal to the number of components

8 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 I

II

III

Figure 1: Reidemeister Moves

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 somAe site. Thus ifd(s, τ) denotes a pair c-1onsisting in an enhanced state s and site τ of that state with τ of type A, then we cAonsider 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)Aas a sum overΔcertain elements in S′[s, τ], and the differential by the formula A-1 ∂(s) = ! ∂τ (s) τ with the sum over all type A sites τ in s.mIt then remains to see what are the possibilities -1 for ∂τ (s) so that j(s) is pAreserved. A

Note that if s′ S′[s, τ], then n (s′) = n (s) + 1. Thus ∈ B B

Figurje(s2′:) S=andBd(lse′P)o+inλt(ss′a)n=d1S+tanteB(Ssm) +ooλt(hsi′)n.gs

From this we conclude that j(s) = j(s′) if and only if λ(s′) = λ(s) 1. Recall that − λ(s) = [s : +] [s : ] the relationships between Frobenius algebras an−d the−surface cobordism category. The proof ofwhinevrear[sia:n+ce] iosfthKe hnoumvabneor vofhloompsoliongsylawbeiltehdr+es1p, e[sct: to] tihs ethRe neuidmebmereoisfteloropmsoves − (respectlianbgelegdrad1in(sgamcheaansglaebse)lewdiwllitnhoXt )baengdijv(esn) =henrBe.(sS)e+eλ[(1s2)., 1, 2]. It is remarkable that this versio−n of Khovanov homology is uniquely specified by natural ideas about adjacenPcyroopfosittaiotens. iTnhethpearbtiraalcdkifefterpeonltyianlso∂mτ i(asl).are uniquely determined by the condition that j(s′) = j(s) for all s′ involved in the action of the partial differential on the en- RemarkhaoncnedInsttaetegrsa. Tl hDisifufneirqeunetfioarlms.ofCtheoopasretialndioffredreernitniagl cfaonrbtehedecscrorisbseidnbgys tihne tfhole- link lowing structures of multiplication and comultiplication on the algebra A = k[X]/(X2) diagram K and denote them by 1, 2, n. Let s be any enhanced state of K and let where k = Z/2Z for mod-2 coeffic·ie·n·ts, or k = Z for integral coefficients. ∂i(s) denote the chain obtained from s by applying a partial boundary at the i-th site 1. The element 1 is a multiplicative unit and1X2 = 0. of s. If the i-th site is a smoothing of type A− , then ∂i(s) = 0. If the i-th site is 2. ∆(1) = 1 X + X 1 and ∆(X) = X X. ⊗ ⊗ ⊗

7 m Δ

F G H

Figure 3: Surface Cobordisms

9 Bracket states 0 C form a category that assembles C1 itself into a chain complex.

Levels in the chain 2 C complex are direct sums of modules corresponding to C3 states with a constant number of B smoothings.

Enhanced States Plus Boundary Requirement Yields Frobenius Algebra. Checking Order Compatibility

We have arrived at the Frobenius algebra, but there is still work to be done to see the invariance under ambient isotopy of knots and links. Cubism Again Categorification and the Morse Dream

(flattening a higher category)

Dror’s Canopoly An abstract categorical analog of a chain complex. That can be taken up to chain homotopy. The maps are additive combinations of surface cobordisms. We examine this question as though we had not seen the Frobenius algebra.

Schematic Four-Tube Relation The dot can be taken to represent an algebra element x.

From 4Tu to Frobenius Algebra

Algebra from 4Tu - Guaranteed to Produce Link Homology Lee’s Algebra

Facts: s max (K) = smin (K) + 2

s(K) = s min (K) + 1 A-State: s(K) = 1 - (#loops) + (# crossings) = 2genus(Seifert(K)) For positive knot all loops labelled x.

Virtual studies stabilized knots in thickened surfaces.

Figure 4: Surfaces and Virtuals

We have the Theorem 1 [17, 24, 19, 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 virtual diagram on the left and an “abstract knot diagram” [38, 3] on the right. The abstract knot diagram is a realization of the knot on the left in a thickened surface with boundary and it is obtained by making a neigh- borhood of the virtual diagram that resolves the virtual crossing into arcs that travel on separate bands. The virtual crossing appears as an artifact of the projection of this surface to the plane. The reader will find more information about this correspondence [17, 24] in other papers by the author and in the literature of virtual knot theory.

4 Flat Virtual Knots and Links

Every classical knot or link diagram can be regarded as a 4-regular plane graph with ex- tra structure at the nodes. This extra structure is usually indicated by the over and under crossing conventions that give instructions for constructing an embedding of the link in three dimensional space from the diagram. If we take the flat diagram without this extra structure then the diagram is the shadow of some link in three dimensional space, but the weaving of that link is not specified. It is well known that if one is allowed to apply the Reidemeister moves to such a shadow (without regard to the types of crossing since they are not specified) then the shadow can be reduced to a disjoint union of circles.

5

Virtual knots are all oriented (signed) Gauss codes taken up to Reidemeister moves on the codes.

Virtual crossings Figure 6 — Signed GaussareC artifactsodes of the planar Now consider the effect of changing these signs.diagram.For example let

Figure 6 — Signed Gauss Codes g = O1 + U2 + O3 − U1 + O2 + U3 − . Now consider the effect of changing these signs. For example let

Then g is a signedg =GOa1u+sUs2c+oOd3e− aU1n+dOa2s+ UF3i−gu. re 6 illustrates, the corresponding diagramTheinsgfios arcsiegndedtGoauhssacvoedevanirdtaus Failgucrero6 sillsuisntragtses,itnheocrordreesprontdoinag commodate the change in orienditagartamioins f.orcTedhtoehacvoe vdiretusaltcraosnsidngsginhoradveretotahcoemmsaodmateethue nchdanegrelying (unsigned) Gauss in orientation. The codes t and g have the same underlying (unsigned) Gauss code Oc1odUe O21OU23OU3U11OO2U23U, b3ut,gbcuortresgpocndosrtroeasvpirtouanl dknsottwohiale tvreiprrteusenatls knot while t represents the clatshseicclaasslicatlrtereffooiil.l. Carrying this approach further, we define a virtual knot as an equivalence class of oriented Gauss codes under abstractly defindeed fiRneideemeister moves Carforrythinesegcotdhesi—s waitphpnroomaencthionfouf rvtirhtuealrc,rowsseings. The virtauavl cirotssuinagsl knot as an equivalence class obfecoomrieeanrtitfaecdts oGf a apluansasr recporedseentsatiuonnodf tehre viartbuasltkrnaotc. tTlhye mdoveefisentsed Reidemeister moves for these codes—with no men1t2ion of virtual crossings. The virtual crossings become artifacts of a planar representation of the virtual knot. The move sets

12 There exist infinitely many non-trivial K with unit Jones polynomial. Bracket Polynomial is Unchanged when smoothing flanking virtuals.

Z-Equivalence to the handling of classical knot diagrams. Many structures in classical knot theory generalize to the virtual domain.

In the diagrammatic theory of virtual knots one adds a virtual crossing (see Figure 1) that is neither an over-crossing nor an under-crossing. A virtual crossing is repre- sented by two crossing segments with a small circle placed around the crossing point.

Moves on virtual diagrams generalize the Reidemeister moves for classical knot and link diagrams. See Figure 1. One can summarize the moves on virtual diagrams by saying that the classical crossings interact with one another according to the usual Rei- demeister moves while virtual crossings are artifacts of the attempt to draw the virtual structure in the plane. A segment of diagram consisting of a sequence of consecutive virtual crossings can be excised and a new connection made between the resulting free ends. If the new connecting segment intersects the remaining diagram (transversally) then each new intersection is taken to be virtual. Such an excision and reconnection is called a detour move. Adding the global detour move to the Reidemeister moves completes the description of moves on virtual diagrams. In Figure 1 we illustrate a set of local moves involving virtual crossings. The global detour move is a consequence of moves (B) and (C) in Figure 1. The detour move is illustrated in Figure 2. Virtual knot and link diagrams that can be connected by a finite sequence of these moves are said to be equivalent or virtually isotopic. Generalized Reidemeister Moves for Virtual Knots and Links

planar isotopy vRI

vRII RI

RII vRIII B

RIII mixed RIII A C

Figure 1: Moves

Another way to understand virtual diagrams is to regard them as representatives for oriented Gauss codes [8], [17, 18] (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

3

We say that K is concordant to K` K ~c K` if there exists a cobordism from K to K` of genus 0.

A virtual knot is said to be slice if it is concordant to the .

VS

D

VS

g = (1/2)(-r + n + 1) = (1/2)(-3 +4 + 1) = 1.

Seifert Cobordism for the Virtual Stevedore and for a correspondingD positive diagram D.

VS Figure 21: Virtual Stevedore CobVSo risd theis virtualm S estevedoreifert Surface and bounds another surface of genus zero.

g = (1/2)(-r + n + 1) = (1/2)(-3 +4 + 1) = 1. D Seifert Cobordism for the Virtual Stevedore and for a correspondingD is a positive positive diagram D. virtual diagram and Figure 21: Virtual Stevise dNOTore C slice.obordism

g = (1/2)(-r + n + 1) = (1/2)(-3 +4 + 1) = 1.

Seifert Cobordism for the Virtual Stevedore and for a corresponding positive diagram D.

Figure 21: Virtual Stevedore Cobordism Seifert Surface

VS

VS

VS

VS on a torus. VS on a torus.

Figure 22: Virtual Stevedore oFnigauTroeru2s2: Virtual Stevedore on a Torus

VS17 on a torus. 17 Figure 22: Virtual Stevedore on a Torus

17 We now observe that for any classical knot K, there is a surface bounding that knot in the four-ball that is homeomorphic to the Seifert surface. One can construct this surface by pushing the Seifert surface into the four-ball keeping it fixed along the boundary. We will give here a different description of this surface as indicated in Figure 19. In that figure we perform a saddle point transformation at every crossing of the diagram. The result is a collection of unknotted and unlinked curves. By our interpretation of surfaces in the four-ball obtained by saddle moves and isotopies, we can then bound each of these curves by discs (via deaths of circles) and obtain a surface S(K) embedded in the four- ball with boundary K. As the reader can easily see, the curves produced by the saddle transformations are in one-to-one correspondence with the Seifert circles for K, and it easy to verity that S(K) is homeomorphic with the Seifert surface F (K). Thus we know that g(S(K)) = (1/2)(−r + n + 1). In fact the same argument that we used to analyze the genus of the Seifert surface applies directly to the construction of S(K) via saddles and minima.

Now the stage is set for generalizing the Seifert surface to a surface S(K) for virtual knots K. View Figure 20 and Figure 21. In these figures we have performed a saddle transformation at each classical crossing of a virtual knot K. The result is a collection of unknotted curves that are isotopic (by the first classical ) to curves with only virtual crossings. Once the first Reidemeister moves are performed, these curves are identical with the virtual Seifert circles obtained from the diagram K by smoothing all of its classical crossings. We can then Isotope these circles into a disjoint collection of circles (since they have no classical crossings) and cap them with discs in the four-ball. The result is a virtual surface S(K) whose boundary is the given virtual knot K. We will use the terminology virtual surface in the four-ball for this surface schema. In the case of a virtual , we have that the knot bounds a virtual surface of genus zero. But with this construction we have proved the

Lemma. Let K be a virtual knot, then the virtual Seifert surface S(K) constructed above has genus given by the formula g(S(K)) = (1/2)(−r + n + 1) where r is the number of virtual Seifert circles in the diagram K and n is the number of classical crossings in the diagram K.

Proof. The proof follows by the same argumeTnt that we already gave in the classical case. Here the projected virtual diagram gives a four-regular graph G (not necessarily planar) whose nodes are in one-to-one correspondence with the classical crossings of K. The edges of G are in one-to-one correspondence with the edges in the diagram that extend from one classical crossing to the next. We regard G as an abstract graph so the the vSeifertirtual Circlescrossings disappear. The argument then goes over verbatim in the sense that G with two-cells attached to the virtual Seifert circles is a retract of the surface S(K) constructed by cobordism. The counting argment for the genus is identical to the classical case. This completes the proof. //

Remark. For the virtual stevedore in Figure 21 we have the interesting phenomenon that there is a Seifert Surface much lower genus surface that can be produced by cobordism t h a F(T)n the virtual Seifert surface. In that same figure wHeathere have illus tDye,rated aAarondiagram KaestnerD with the sandame pLK,roje cproveted dia gtheram as the virtual stevedore, but D has allfollowingpositive cro sgeneralizationsings. In this case ofwe Rasmussen’scan prove [2] th aTheorem,t there is no virtual surface for this diagram D ogivingf four-b theall g efour-ballnus less tha genusn 1. In f aofct ,aw epositivehave the fvirtualollowing knot.result which is proved in [2]. This Theorem is a generalization of a corresponding result for classcial knots due to Rasmussen [16].

Theorem [2]. Let K be a positive virtual knot (all classcial crossings in K are positive), then the four-ball genus g4(K) is givenFbiygtuhreef1o8rm: uClalassical Seifert Surface

g4(K) = (1/2)(−r + n + 1) = g(S(K))

where r is the number of virtual Seifert circles in the diagram K and n is the number of classcial 14 crossings in this diagram. In other words, that virtual Seifert surface for K represents its minimal four-ball genus.

The virtual Seifert surface for positive virtual K 3.2 Propertrepresentsies of the V ithertua minimall Steved ofour-ballre’s Kno tgenus of K. We first Thepoint Theoremout that the visir tprovedual steve dbyore generalizing(V S) is an exa mbothple th Khovanovat illustrates the viability of our theory. We prove that V S is not classical by showing that it is represented on a surface of geus one and no smaller.andThe rLeeeade rhomologyshould note t htoe d ivirtualfference bknotsetween andrepre sgeneralizingentation of a vi rtual knot or link on a surface (as an embtheeddi nRasmusseng into the thic kinvariantened surfac eto) a nvirtuald the pr eknots.vious subsection’s work on spanning surfaces.

The technique for finding this surface genus for the virtual stevedore is to use the bracket expansion on a toral representative of V S and examine the structure of the state loops on that surface. See Figure 22 and Figure 23. Note that in thes Figures the virtual crossings correspond to parts of the diagram that loop around the torus, and do not weave on the surface of the torus. An analysis of the homology classes of the state loops shows that the knot cannot be isotoped off the handle structure of the torus. See [6, 23] for more information about using the surface bracket.

Next we examine the bracket polynomial of the virtual stevedore, and show as in Figure 24 that it has the same bracket polynomial as the classical figure eight knot. The technique for showing this is to use the basic bracket identity for a crossing flanked by virtual crossings as discussed in the previous section. This calculation shows that V S is not a connected sum of two virtual knots. Thus we know that V S is a non-trivial example of a virtual slice knot. We now can state the problem: Classify virtual knots up to concordance. We will discuss this problem in this paper, but not solve it. The reader

15 Remarks on Generalizing Khovanov Homology, Lee Homology and Rasmussen Invariant Mod 2 Khovanov Homology for Virtuals is OK. diagram in Figure 10. For more on sour-sink structures and virtual knots see Maturov [Man05] [MI13].

Figure 10: Source-sink orientation

Remark 2.1. This can be done arbitrarily but we will assume the canonical source-sink orientation given in Figure 11.

Figure 11: Canonial source-sink orientation

Remarks on GeneralizingUsing the (ca nKhovanovonical) source- sHomology,ink orientations wLeee pla ce cut loci on the semi-arcs of the knot diagram whenever neighboring source-sink orientations diagram in Figure 10. For more on sour-sink structures and virtual knots Homologydisagre eand. Fig uRasmussenre 12 illustrates tInvarianthis process for a two-crossing virtual knot. see Maturov [Man05] [MI13]. Note that the number of cut loci is fixed for any particular diagram and diagram in Figure 10. For more on sour-sink structures and virtual knots choice of source-sink orientations on the crossings. However, as we will see, see Maturov [Man05] [MI13]. the number of cut loci may change when a Reidemeister move is performed.

Figure 10: Source-sink orientation

Remark 2.1F.igThureis1c0a:nSboeudrocen-esianrkbiotriaerniltyatbiuotnwe will assume the canonical source-sink orientation given in Figure 11. Remark 2.1. This can be done arbitrarily but we will assume the canonical source-sink orientation given in Figure 11.

Figure 12: Cut loci for a two-crossing virtual knot

FigurCanonicale 11: Canonia lSource-Sinksource-sinSkouorrcie-ns tiOrientationnatkioonrientations were originally introduced by Naoko Kamada

UsinFgigtuhree (1c1a:noCnaincaoln)iaslousorcuer-csei-nskinokrioerniteanttiaotniosnwe place cut loci on the 17 semi-arcs of the knot diagram whenever neighboring source-sink orientations Usingditshaegr(ecea.nFonigicuarle) 1s2ouilrlcues-tsriantkesortiheinstaptrioocnessswfeorpalactwe oc-uctrolsosciingonvitrhtueal knot. semi-arcsNoofttehtehkantotthdeiangurmambewr hoefnceuvterlonceiigihsbfioxriendgfsooruracney-spinakrtoicriuelnatratdiioangsram and disagree.chFoiigcueroef1s2ouilrlcues-tsrinatkesortihenistaptrioncessosnfotrhea ctrwooss-cinrogss.inHgowveirvteura,laksnwoet.will see, Note thatthethneumnubmerboefr couftcluoctilmocaiyischfiaxnegde wfohreannay Rpeairdteicmuelaisrtedriamgorvaemisapnedrformed. choice of source-sink orientations on the crossings. However, as we will see, the number of cut loci may change when a Reidemeister move is performed.

Figure 12: Cut loci for a two-crossing virtual knot

SFoiugurcree-s1i2n:kCouritenlotcaitifoonrsawtewroe-corroigssininagllyviirntutraoldkuncoetd by Naoko Kamada

Source-sink orientations were originally i1n7troduced by Naoko Kamada

17 Cut Locus Involution

a a

The Frobenius algebra controlling the Khovanov homology differentials has an order two function a a that is applied whenever an algebra element is moved across a cut locus. x = - x 1 = 1 Transportation past cut loci 1 bars the corrsponding elements and makes the square commute integrally.

x + x = -x + x = 0. multiply and add

x 1 11 x 1 x + + x 1 Transport Along with the bar operation and local coeffient transport there are other issues that demand extra care. We leave these for the reader to find in our paper.

Khovanov Homology, Lee Homology and a Rasmussen Invariant for Virtual Knots Heather A. Dye, Aaron Kaestner, Louis H. Kauffman arXiv:1409.5088

Barring Operations for the Lee Algebra

Flattened K K Lee Algebra rg=gr = 0 rr = r gg= g r+g= 1 D(r) = 2r g g D(g) = 2g r r r = g g = r

K with canonical source sink Seifert state labelled orientations and cut loci with Lee algebra is a non-trivial cycle. Another Example of a Virtual Lee Cycle

r g g

r g r

g

r

g r genus = (1/2)(-r + n +1) = (1/2)(-2 + 5 +1) r g g = 2.

r g r

g

r

g r There are many questions about virtual knot cobordism and its relations with Khovanov homology and with variations on Khovanov homology. (See particularly the papers of William Rushworth.) We are presently working on cobordism and KhoHomology of knotoids, where a knotoid is a diagram with endpoints, not necessarily in the same region. Knotoids are taken up to Reidemeister moves that do not involve the endpoints. Thank you for your attention!

VS I E

-8 -4 4 8 = = = A - A + 1 - A + A

The knot VS has bracket polynomial equal to the bracket polynomial of the classical figure eight knot diagram E. This implies that VS is not a connected sum.

Figure 12: Bracket Polynomial of the Virtual Stevedore Knot More about the Virtual Stevedore’s Knot

VS

VS on a torus.

Figure 13: Virtual Stevedore Knot on a Torus

9 Virtual Stevedore

AABA AABB is not AAAA AAAB classical.

ABBB ABAA ABAB ABBA Definition of Virtual Ribbon: A knot is said to be virtually ribbon BABB BAAA BAAB BABA if it can be sliced using only saddles and deaths of BBBA BBBB BBAA BBAB trivial circles. No births allowed. (There may be a way to define this concept that is closer to ‘ribbon singularities’ Figure 14: Torus States for Virtual Stevedore Knot as in classical knots)

10

Rotational Virtual Knots Disallow VR1.

The Virtual Stevedore is Rotationally Slice This talk has been a walk through Khovanov Homology and finding the relevant Frobenius algebras from the categorical chain complex and its associated 4Tu relation. We find the 4Tu relation naturally in looking for relations on surface cobordisms that render the theory invariant under isotopy of links.

There are many questions! Other directions involve the question of an appropriate homotopy context for Khovanov Homology. Work of Turner, Everitt, Lipshitz an Sarkar shows how to produce spectra whose homotopy is Khovanov Homology and spaces whose homology is Khovanov homology. Simplicial Stucture of Khovanov Homology

Work of Chris Gomes and LK shows how to use Dold-Kan construction to make homotpy spectra for link homologies.

Dold-Kan Functor: Chain Compexes -----> Homotopy Simplical Objects Moral: We can work simplicially by using the cube category with a trivial 0-th partial boundary. One can then make spaces whose homology is Khovanov homology by forming the geometric realization of the corresponding simplicial objects. By adding degeneracies to the simplicial objects (as in ,<012> ----> <0112>) one can do homotopy theory with them. In particular one can realize spaces whose weak homotopy type corresponds to the chain homotopy type of the Khovanov complex. A related insight (due to Jozef Przytycki) is that Khovanov homology from the cube category can be understood to be the cohomology of the nerve of that category with coefficients via the Frobenius algebra functor.

We have been exploring the cohomology of categories.