Khovanov Homology and Virtual Knot Cobordism Louis H. Kauffman, UIC [16], and Bar-Natan’s emphasis on tangle 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 link 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 bracket polynomial [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 Jones polynomial [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 c<K>ros 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 <K>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 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 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.