"The 1,2-coloured HOMFLY-PT link homology" Dos Santos Santana Forte Vaz, Pedro ; Mackaay, Marco ; Stosic, Marko

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1 "The,-coloured HOMFLY-PT lnk homology" Dos Santos Santana Forte Vaz, Pedro ; Mackaay, Marco ; Stosc, Marko Abstract In ths paper we defne the,-coloured HOMFLY-PT trply graded lnk homology and prove that t s a lnk nvarant. We also conecture on how to generalze our constructon for arbtrary colours. Document type : Artcle de pérodque (Journal artcle) Référence bblographque Dos Santos Santana Forte Vaz, Pedro ; Mackaay, Marco ; Stosc, Marko. The,-coloured HOMFLY-PT lnk homology. In: Transactons of the Amercan mathematcal socety,, no.6, p (0) DOI : 0.090/S Avalable at: [Downloaded 08/0/5 at 05:6: ]

2 TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 6, Number 4, Aprl 0, Pages 09 4 S (00) Artcle electroncally publshed on November 7, 00 THE,-COLOURED HOMFLY-PT LINK HOMOLOGY MARCO MACKAAY, MARKO STOŠIĆ, AND PEDRO VAZ Abstract. In ths paper we defne the,-coloured HOMFLY-PT trply graded lnk homology and prove that t s a lnk nvarant. We also conecture on how to generalze our constructon for arbtrary colours.. Introducton In ths paper we defne the coloured HOMFLY-PT trply graded lnk homology for lnks whose components are labelled or. The pont of ths paper s to generalze Khovanov s [6] constructon of the HOMFLY-PT lnk homology, orgnally due to Khovanov and Rozansky [8]. The Euler characterstc of our trply graded lnk homology s a -varable polynomal whch generalzes the HOMFLY-PT polynomal. We dd not develop a complete recursve combnatoral calculus for t, as n [5] and [9]. However, n Secton we wll ndcate some relatons whch should hold n such a calculus and whch are closely related to the ones n [9]. The,-coloured HOMFLY-PT lnk homology s a trply graded lnk homology, ust as the ordnary HOMFLY-PT lnk homology due to Khovanov and Rozansky [8]. In [4] such a lnk homology was conectured to exst from the physcs pont of vew. We follow the approach usng bmodules and Hochschld homology, as was done for the ordnary HOMFLY-PT lnk homology by Khovanov n [6]. Wth Rasmussen s results for the ordnary HOMFLY-PT homology n mnd, we conecture that, n a certan sense, our lnk homology s the lmt of as yet undefned,-coloured sl(n) lnk homologes, when N goes to nfnty. Note that the graded Euler characterstc of these,-coloured sl(n) lnk homologes wll be the Reshetkhn-Turaev polynomals for these colorngs. Although these sl(n) lnk homologes have not yet been defned, there has been some progress made towards ther defnton n [5, 6]. In those papers the matrx factorzaton approach s followed. One should also be able to defne the,-coloured HOMFLY-PT lnk homology usng matrx factorzatons, and n some sense ths should be equvalent to our approach. For techncal reasons the bmodule approach s slghtly easer, whch s why we have not used matrx factorzatons. As for the Receved by the edtors December, 008 and, n revsed form, July, Mathematcs Subect Classfcaton. Prmary 8G60, 57M7. The authors thank Mkhal Khovanov and Catharna Stroppel for helpful conversatons on the topc of ths paper. The authors were supported by the Fundação para a Cênca e a Tecnologa (ISR/IST pluranual fundng) through the programme Programa Operaconal Cênca, Tecnologa, Inovação (POCTI) and the POS Conhecmento programme, cofnanced by the European Communty fund FEDER. The second author was also partally supported by the Mnstry of Scence of Serba, proect c 00 Amercan Mathematcal Socety Reverts to publc doman 8 years from publcaton

3 09 MARCO MACKAAY, MARKO STOŠIĆ, AND PEDRO VAZ ordnary HOMFLY-PT lnk homology, we use brad presentatons of the lnks and prove nvarance under the bradlke Redemester moves and the Markov moves. In the last secton of ths paper we have sketched how to defne the coloured HOMFLY-PT lnk homology for arbtrary colours and how to prove ts nvarance. The underlyng deas are the same, but the actual calculatons are much harder. One needs a dfferent technque to handle those calculatons for arbtrary colours. In the meantme, Webster and Wllamson [4] have defned the general coloured HOMFLY-PT lnk homology usng geometrc technques and have confrmed our conectures. In a future paper we hope to gve algebrac proofs of our conectures. To compute the,-coloured HOMFLY-PT lnk homology s very hard. In [4] there s a conecture for the Hopf lnk, whch we confrm by our calculatons n Secton 7, but we have not done more complcated calculatons. One road to follow would be the one Rasmussen showed for the ordnary HOMFLY-PT lnk homology [0]. Once the sl(n)-lnk homologes have been defned, there should be spectral sequences from the coloured HOMFLY-PT lnk homologes to the sl(n)-lnk homologes. For small knots these spectral sequences should collapse for low values of N, whch mght make them computable. Another road to follow would be the one ntated by Webster and Wllamson [4]. Ths approach mght gve some results for certan classes of knots, such as the torus knots. Fnally, let us brefly sketch an outlne of our paper. In Secton we recall some basc facts from [9] about the combnatoral calculus of the,-coloured HOMFLY- PT polynomal and defne one verson of part of t that we shall categorfy. In Secton we categorfy ths part of the calculus by usng bmodules. In Secton 4 we defne the,-coloured HOMFLY-PT lnk homology. As stated before, we use a brad presentaton of the lnk. In Secton 5 we prove nvarance of the,-coloured HOMFLY-PT lnk homology under the bradlke Redemester moves II and III. In Secton 6 we prove ts nvarance under the Markov moves. In Secton 7 we have ncluded the calculaton of the,-coloured HOMFLY-PT homology of the Hopf lnk. In Secton 8 we sketch the defnton of the coloured HOMFLY-PT lnk homology for arbtrary colours and conecture ts nvarance under the second and thrd Redemester moves and the Markov moves. Although we have tred to wrte a farly self-contaned paper, some famlarty wth [6, 7, 8, 5, 9] wll probably help the reader n understandng ths paper.. The MOY calculus In ths secton we explan part of the MOY calculus for the,-coloured HOM- FLY-PT lnk polynomal. The reader mght want to compare our relatons to the analogous ones n [9]. As already remarked n the ntroducton ths part of the calculus probably does not gve a complete recursve calculus for the polynomal nvarant. At least we do not know any proof of such a fact. We have smply pcked those relatons that are necessary for provng the nvarance of the polynomal. As t turns out, ther categorfcatons prove the nvarance of the related lnk homology. Before we go on, let us remark that ths secton s merely motvatonal. It s meant to help the reader understand why we set up thngs as we dd. No part of the constructon of the lnk homology or the proof of ts nvarance depends on ths secton. As a matter of fact, we could have put ths secton at the end of our paper as a consequence of the constructon of the lnk homology,.e. we can obtan the

4 THE,-COLOURED HOMFLY-PT LINK HOMOLOGY 09,-coloured HOMFLY-PT polynomal as the graded Euler characterstc of our lnk homology. The calculus uses labelled trvalent graphs, whch we call MOY webs, and s smlar to the MOY calculus from [9] and generalzes the calculus used by Khovanov and Rozansky n [8] to defne trply graded lnk homology. The resolutons of lnk dagrams consst of MOY webs whose edges are labelled by postve ntegers, such that at each trvalent vertex the sum of the labels of the outgong edges equals the sum of the labels of ngong edges. Although the theory can be extended to allow for general labellngs, n ths paper we shall only consder the ones where the labellngs of the edges are from the set {,,, 4}. Frst we ntroduce the calculus for such graphs. Ths s an extenson of the one wth labellngs beng only and (see [8]), and a varant of the one from [9]. We requre the followng axoms to hold: (A) k = k = +t q q, (A) + = +t q +l l= q l, (A) + = + [ ] + +, k k (A4) + = + + k k, (A5) = + q, (A6) = 4 +(q + q 4 ), (A7) = 4 + q. In ths paper we use the followng (non-standard) conventon for the quantum ntegers: [n] =+q q (n ) = qn q.

5 094 MARCO MACKAAY, MARKO STOŠIĆ, AND PEDRO VAZ We defne the quantum factoral and the bnomal coeffcents n the standard way: [ ] n [n]! [n]! = [][] [n], = m [m]![n m]!. Although we have defned the axoms (A)-(A) for arbtrary, and k, nths paper we shall only need them n the cases where the ndces, and k are from the set {, }. Snce ths secton s only motvatonal, we also nclude a sketch of the -varable,-coloured HOMFLY-PT lnk polynomal based on the above dagrammatc calculus, although the defnton s necessarly ncomplete. The polynomal would be defned for lnk proectons whch have the form of the closure of a brad. All postve and negatve crossngs between strands labelled would be resolved n three dfferent resolutons. The bracket would be defned usng the followng relatons: = q 6 q + 4 (.) = q 8 4 q 8 + q 6,. The resolutons of the postve and negatve crossngs between strands labelled and would be gven by = q + (.), = q 4 q 4 One would obtan the resolutons, when the labels and are swapped, by rotaton around the y-axs. Fnally, the case of the crossngs when both strands are labelled would be the same as n [8]:. = q + (.) = q q,.

6 THE,-COLOURED HOMFLY-PT LINK HOMOLOGY 095 Assumng that axoms (A)-(A7) could be extended such that any MOY web can be evaluated, we could defne a polynomal D for each lnk dagram D wth components labelled by and, usng the resolutons n (.), (.) and (.). Analogously as n [9], t can be shown that t would be nvarant under the second and the thrd Redemester moves and would have the followng smple behavour under the frst Redemester move: =, when the strand s labelled and = t q =, = t q when the strand s labelled. To obtan a genune knot nvarant, we would therefore have to multply the bracket by the followng overall factor: I(D) =( tq) n + +n +s (D) n + +n +s (D) D, where n + and n denote the number of postve and negatve crossngs, respectvely, between two strands labelled and where s (D) denotes the number of strands labelled for =,.. The categorfcaton of the MOY calculus In ths secton we show whch bmodule to assocate to a web and that these bmodules satsfy axoms (A)-(A7) up to somorphsm. The proof that (A) and (A) are also satsfed wll be gven n Secton 6 after we have explaned the Hochschld homology. We only explan the general dea and work out the bts whch nvolve edges wth hgher labels and whch have not been explaned by Khovanov n [6]. Let R = C[x,...,x n ] be the rng of complex polynomals n n varables. We ntroduce a gradng on R by defnng deg x = for every =,...,n. Ths gradng s called the q-gradng. For any partton,..., k of n, letr k denote the subrng of R of complex polynomals whch are nvarant under the product of the symmetrc groups S S k. For starters we assocate a bmodule to each MOY-web (see Secton ). Suppose Γ s a MOY web wth k bottom ends labelled by,..., k and m top ends labelled by,..., m. Recall that + + k = + + m holds and let R have exactly that number of varables. We assocate an R m R k -bmodule to Γ. We read Γ from bottom to top. To the bottom edges we assocate the bmodule R k. When we move up n the web we encounter a I-shaped or a n-shaped bfurcaton. The I-shape wll always correspond to nducton and the n to restrcton; e.g. f we frst encounter a I-shaped bfurcaton whch splts nto 0 and,thenwetensorr k on the left wth R 0 over k R k. We wll wrte ths tensor product as R 0 k k R k. If we frst encounter a n-shaped bfurcaton wth onts and, then we restrct the left acton on R k to R + k. Note that the latter acton goes unnotced

7 096 MARCO MACKAAY, MARKO STOŠIĆ, AND PEDRO VAZ untl tensorng agan at some next I-shaped bfurcaton, n whch case we tensor over the smaller rng, or untl takng the Hochschld homology n a later stage. As we go up n the web use ether nducton or restrcton at each bfurcaton. Ths way an R m R k -bmodule assocated to Γ s obtaned, whch we denote by Γ. Note that f we have two MOY-webs, Γ and Γ, such that the bottom of Γ (labelled by,..., k ) can be glued onto the top of Γ, then Γ Γ = Γ R k Γ. The dentty web n Fgure whose edges are labelled by,..., k wll always k Fgure. The dentty web x k be denoted by x k and y k = R k. The dumbbell web n Fgure whose + Fgure. The dumbbell web v outer edges are labelled and wll always be denoted by v and o = R + R. Snce the bmodules that we use are graded, we can apply a gradng shft. In the text and when usng small symbols we denote a postve shft of k values appled to a bmodule M by M{k}. When we use MOY-type pctures we denote the same shft by q k. Note that we also do not put a hat on top of these pctures to avod too much notaton and unnecessarly large fgures. Note that for our constructon we frst need to choose a heght functon on the web. However, t s easy to see that the bmodule does not depend on the choce of ths heght functon. Now that we know whch bmodule to assocate to a MOY web, we wll show some drect sum decompostons for bmodules assocated to certan MOY webs. These decompostons are necessary to show that our bmodules ndeed categorfy the MOY calculus and also to show that the lnk homology s nvarant, up to somorphsm, under the Redemester moves. Lemma.. Let U be the dgon n (A). Then we have [ ] + = g +. Note that the quantum bnomal can be wrtten as a sum of powers of q. By [ ] + g +

8 THE,-COLOURED HOMFLY-PT LINK HOMOLOGY 097 we mean the correspondng drect sum of copes of g +, where each copy s shfted by the correct power of q. Proof. Note that R as an R + -(b)module s somorphc to HU(+) (G(, +)), the U(+) equvarant cohomology of the complex Grassmannan G(, +). Therefore the Schur polynomals π k k n the frst varables, for 0 k k, form a bass of R as an R + -(b)module (see [], for example). Alternatvely we can use G(, + ) and obtan that the Schur polynomals π l l n the last varables form a bass of R as an R + -(b)module for 0 l l. Ths shows that [ ] R + = R + holds. The proof of ths lemma follows, snce = R + R + and g + = R +. Before we contnue wth square decompostons we defne the followng bmodule maps: Defnton.. We defne the R k -bmodule maps μ k : o k r k and Δ k : r k o k by μ k (a b) =ab and Δ k () = k k ( ) e ( x k x ). The elements e are the elementary symmetrc polynomals n k +varables. =0 The formula for Δ k canalsobewrttenas k Δ k () = ( ) x k e, =0 where e s the -th elementary symmetrc polynomal n the last k varables x,...,x k+.notethatμ k has degree 0 and Δ k has degree k. It s not hard to see that μ and Δ are ndeed R k bmodule maps. One can check ths by drect computaton or, as above, note that R k s somorphc to HU(k+) (G(,k)) as an R k+-module. The maps above are well known (t s an mmedate consequence of exercse. of lecture 4 n [], for example) to be the multplcaton and comultplcaton n ths commutatve Frobenus extenson wth respect to the trace defned by tr(x k ) =. The observaton now follows from the fact that the multplcaton and comultplcaton n a commutatve Frobenus extenson A are always A-bmodule maps. When t s not mmedately clear to whch varables one apples a multplcaton or comultplcaton, we wll ndcate them n a superscrpt. When there s no confuson possble, we wll wrte ab for μ,k (a b). Defnton.. We defne the R -bmodule maps =0 by μ : o r and Δ : r o μ (a b) =ab

9 098 MARCO MACKAAY, MARKO STOŠIĆ, AND PEDRO VAZ and Δ () = π π π 0 + π 0 π + π π 0 π 0 π + π +. The π and π are the Schur polynomals n x,x and x,x 4, respectvely. The ndcates the terms whch are obtaned from all the prevous terms by nterchangng the π and the π so that Δ becomes cocommutatve. One easly checks that μ and Δ are R -bmodule maps by drect computaton. They have degree 0 and 8, respectvely. The map Δ s the comultplcaton n HU(4) (G(, 4)) wth respect to the trace defned by tr(π ) =. Sometmes t wll be useful to rewrte the formula of Δ entrely n terms of the π and elements of R 4. To do so, use the followng relatons: π 0 = π 000 π 0, π = π 00 π 0π π 0, π 0 = π 000 π 0e + π, π = π 00 π 0(π π 00 )+π 0π π π 000 π, π = π 00 π 0π 00 + π π π 0π 00 π π π. We can now prove some square decompostons. Lemma.4. Let s bethesquaren(a5). The followng decomposton holds: t = o r {}. Proof. Note that t = R R R and o = R R. Let Γ be the webs n Fgure, for =,. Then we have Γ Γ Fgure. Intermedate webs Γ and Γ Γ = R R R R = R R R R = Γ. The map f : t o s the composte of f t Γ { 4} = Γ { 4} f o. We defne f by f (a b c) =a Δ (b) c, where Δ s defned as n Defnton.. The map f s defned by applyng to both dgons the map R R {} whch corresponds to the proecton onto the second summand n the decomposton R = R x R ; e.g. we have x and x =(x +x ) x and x =(x +x )x x x x +x.

10 THE,-COLOURED HOMFLY-PT LINK HOMOLOGY 099 Smlarly defne g : o t as the composte of g o Γ g = Γ t. We defne g by twce applyng the ncluson map R R to create the dgons. The map g s defned by g (a b c d) =a bc d, where c s mapped to R by the ncluson map before applyng μ. One easly verfes by drect computaton that fg =d,soo s a drect summand of t. To show that r s a drect summand as well, we also use an ntermedate web, denoted Γ and shown n Fgure 4. Note that Γ = R R. Defne h: t r as the composte of Fgure 4. Intermedate web Γ h t h Γ r {}. In ths case h s gven by h (a b c) =ab c and h by applyng the same proecton R R {} as above. Inversely, we defne : r t as the composte of r {} Γ {} t. The frst map s defned by the ncluson R R. The second map s defned by (a b) =Δ x,x (a) b. Agan, by drect computaton, t s straghtforward to check that h = d, so r {} s also a drect summand of t. Also by drect computaton one easly checks that hg =0andf = 0. Fnally we have to show that (g, ) s surectve. Snce all maps nvolved are bmodule maps and R = R x R and R = R x R, we only have to show that and x are n ts mage. We have g( ) = and() + g(x ) = x. Ths fnshes the proof of the lemma. Lemma.5. Let t bethesquaren(a6). Then we have t = o r {} r {4}. Proof. The arguments are analogous to the ones used n the proof of Lemma.4. To show that o s a drect summand of t,usethentermedatewebsnfgure5. Wth the same notaton as before, let f (a b c) =a Δ (b) c,

11 00 MARCO MACKAAY, MARKO STOŠIĆ, AND PEDRO VAZ 4 4 Fgure 5. Intermedate webs for s and for f use for both dgons the map R R {4} whch corresponds to the proecton onto the thrd drect summand n the decomposton R = R x R x R. For g we use the ncluson R R twce to create the dgons and g (a b c d) =a bc d. To show that r {} r {4} s a drect summand of t,usethentermedate webnfgure6. Fgure 6. Intermedate web Defne h by h (a b c) =ab c and h by applyng the map R R {} R {4} correspondng to the proecton on the last two drect summands n the decomposton of R above. For use the map R {} R {4} R defned by to create the dgon. Defne by An easy calculaton shows that (, 0) and (0, ) x (a b) =Δ x,x 4 (a) b. h = ( ) x4, 0 whch s nvertble. Ths shows that R {} R {4} s a drect summand of t. Usng the rewrtng rules for Δ, whch were gven below ts defnton, one easly checks that gf = d. Therefore o s a drect summand of t,too. Another easy calculaton shows that hg = 0 and a slghtly harder one that f =0.

12 THE,-COLOURED HOMFLY-PT LINK HOMOLOGY 0 It remans to show that (g, ) s surectve. It suffces to show that, x and x are n ts mage. We have =g( ), x =(, 0) + g(x 4 ), x =(0, ) + (x 4, 0) + g(x 4 ). Lemma.6. Let t bethesquaren(a7). We have t = o u {}. Proof. We frst defne the bmodule map φ : o = R 4 R R R R = t. We use the two ntermedate bmodules Γ = R R 4 R R and Γ = R R 4 R R (see Fgure 7). Then φ s the composte 4 4 Γ Γ Fgure 7. Intermedate webs Γ and Γ φ Γ φ Γ t wth φ (a b) = a b, φ (a b c d) =ab c d, φ (a b c d) =a bc d. Conversely, we defne ψ as the composte wth o φ ψ t ψ Γ ψ Γ o ψ(a b c) =a Δ (b) c, ψ(a b c d) =ab c d and wth ψ defned by applyng the maps R R {4} and R R {} whch are the proectons onto the last drect summands n the decompostons R = R x 4 R x 4R and R = R x R. A short calculaton shows that ψ φ = d, whch proves that o s a drect summand of t. Let us now defne φ : u t. Note that u = R R. Agan we use certan ntermedate bmodules: Λ = R R R, Λ = R R R, Λ = R R R R and Λ 4 = R R R R (see Fgure 8).

13 0 MARCO MACKAAY, MARKO STOŠIĆ, AND PEDRO VAZ Λ Λ Λ Λ 4 We defne φ as the composte u Fgure 8. Intermedate webs Λ,..., Λ 4 φ φ Λ φ Λ φ Λ 4 φ Λ 5 4 t wth φ (a b) = a b, φ (a b c) =ab c, φ (a b c) =a Δ x x x (b) c, φ 4 (a b c d) =ab c d and wth φ 5 defned by the map R R {}, whch s the proecton onto the second drect summand n the decomposton R = R x R. Conversely, we defne ψ as the composte ψ t ψ Λ ψ 4 Λ ψ Λ 4 ψ Λ 5 u wth ψ(a b c) = a b c, ψ(a b c d) =ab c d, ψ(a b c d) =a bc d, ψ(a b c) 4 =ab c and wth ψ 5 defned by the map R R {}, whch s the proecton onto the second drect summand n the decomposton R = R x R.Asmple calculaton shows that ψ φ =d. One can also easly check that ψ φ =0andψ φ = 0. Ths shows that (φ,φ ) s nectve. To show that (φ,φ )ssurectvenotethatbothmapsareleftr - module maps and that the source and the target are both free R -modules of rank 9 wth the same gradngs. 4. The lnk homology Let us defne the coloured HOMFLY-PT homology for lnks wth components labelled and. We use a smlar setup to the one n [6]. To each brad dagram we assocate a complex of bmodules (defned below) obtaned from the categorfed MOY calculus. Ths complex s nvarant up to homotopy under the bradlke Redemester II and III moves. Then we take the Hochschld homology of each bmodule n the complex, whch corresponds to the categorfcaton of the Markov trace. Ths nduces a complex of vector spaces whose homology s the one we are lookng for. The latter s stll nvarant under the second and thrd Redemester

14 THE,-COLOURED HOMFLY-PT LINK HOMOLOGY 0 moves, because the Hochschld homology s a covarant functor, and also under the Markov moves, as we wll show. Therefore we obtan a trply graded lnk homology. By takng the graded dmensons of the homology groups we get a trply graded lnk polynomal. To defne the complex of bmodules assocated to a brad, t suffces to defne t for a postve and for a negatve crossng only. For an arbtrary brad one then tensors these complexes over all crossngs. To each crossng wth both strands labelled by, we assocate a complex wth three terms. For a postve, resp. negatve, crossng between strands labelled, the terms n the complex are r {6} t {} o,resp.o { 8} t { 8} r { 6} (see Fgures 9-0). = q q 6 d + d + 4 Fgure 9. The complex of a postve crossng q 8 d q 8 d = 4 q 6 Fgure 0. The complex of a negatve crossng In both cases, the cohomologcal degree s fxed by puttng the bmodule o n cohomologcal degree 0. To defne the dfferentals we need the ntermedate webs Φ, ΨandΩ(seeFgure ). Note that Ψ = Ω. 4 4 The dfferental d + Φ Ψ Ω Fgure. Intermedate webs Φ, Ψ and Ω s the composte of d + r Φ d+ t { 4}, where d + s defned by the ncluson R R to create the dgon and d + (a b) =Δx,x,x (a) b. The dfferental d + s the composte of t { 4} d+ Ψ{ 0} = Ω{ 0} d+ o { 6},

15 04 MARCO MACKAAY, MARKO STOŠIĆ, AND PEDRO VAZ d + (a b c) =a Δ (b) c and d + by twce applyng the map R R {} gven by the proecton onto the second drect summand n the decomposton R = R x R. Drect computaton shows that d + d+ =0. The frst dfferental n the complex assocated to a negatve crossng s the composte of d o Ω = Ψ d t, where d s defned by twce applyng the ncluson R dgons and d (a b c d) =a bc d. The dfferental d s the composte of R to create the d t Φ d r {}, where d s defned by d (a b c) =ab c and d by applyng the same proecton R R {} as above. Agan, drect computaton shows that d d = 0. Note that ths calculaton s much easer than the one whch shows d + d+ = 0. The latter s a drect consequence of the former by the observaton that all maps nvolved n the defnton of d ± d± are unts and counts of the two badont functors gven by nducton and restrcton. As a matter of fact, whenever we use a unt n the defnton of one, we use the correspondng count n the defnton of the other. Therefore the fact that d d = 0 mples d + d+ =0. Next we defne a complex of bmodules assocated to a crossng of a strand labelled and a strand labelled. To a postve crossng we assocate the complex = q d +, and to a negatve one d = q 4 q 4. Agan, n both cases, we put the bmodule o n the cohomologcal degree 0. Note that u = R R and o = R R.Weusethentermedate bmodules Λ = R R R = R R R = Λ (see Fgure ). Then d + s the composte of u d + Λ { 4} = Λ { 4} d+ o { } We thank Mkhal Khovanov for ths observaton.

16 THE,-COLOURED HOMFLY-PT LINK HOMOLOGY 05 Λ Λ Fgure. Intermedate webs Λ and Λ wth d + (a b) =a Δ (b), a b c = ab c and wth d + defned by the map R R {} correspondng to the proecton onto the second drect summand n the decomposton R = R x R. It s easy to compute the mage of d + on generators of the R R bmodule R R : d + ( ) = x x and d + (x ) = x x x x. Smlarly we defne d as the composte wth o d = d Λ Λ u d d (a b) = a b, a b c = ab c, (a b c) =a bc. Note that ths yelds d (a b) =a b. We get smlar complexes for the crossngs wth and swapped. The pctures can be obtaned from the ones above by rotaton around the y-axs, and the shfts are the same. For a crossng wth both strands labelled we use the same complex of bmodules as Khovanov n [6]: = q, = q q wth v n cohomologcal degree 0.

17 06 MARCO MACKAAY, MARKO STOŠIĆ, AND PEDRO VAZ Let HHH(D) denote the trply graded homology we defned above. Then to obtan the,-coloured HOMFLY-PT homology H, (D) wehavetoapplysome overall shfts. We defne Defnton 4.. H, (D) = HHH(D) n + n s (D)+n + n s (D) { n + +n +s (D) n + +n +s (D), n + +n +s (D) n + +n +s (D) The defntons of n +,n and s (D) were gven n Secton, s an upward shft by n the homologcal degree and {k, l} denotes an upward shft by k n the Hochschld degree and by l n the q-degree. Fnally n the next two sectons we prove the followng. Theorem 4.. For a gven lnk L, H, (D) s ndependent of the chosen brad dagram D whch represents t. Hence, H, (L) s a lnk nvarant. 5. Invarance under the R and R moves The next thng to do s to prove nvarance of the,-coloured HOMFLY-PT homology under the second and thrd Redemester moves. If the strands nvolved are all labelled, we already have nvarance by Khovanov s [6] and Khovanov and Rozansky s [8] results. If there are strands nvolved whch are labelled, we wll use a trck, nspred by the analogous trck n [9], whch reduces to the case wth all lnk components labelled. The argument s slghtly trcky, so let us explan the general dea frst. Suppose we have a brad B wth n strands, all labelled. Create a dgon U on top of each strand. Our lnk homology complex correspondng to ths brad wth dgons s ( + q ) n tmes our lnk homology complex correspondng to B, whch means that the latter complex s the drect sum of n copes of the former wth gradng shfts accordng to the powers of q n the polynomal ( + q ) n. We wll prove n Lemma 5. that sldng the lower parts of the dgons past the crossngs s a gradng preservng homotopy equvalence. After sldng ths way the lower parts of all dgons past all crossngs (see Fgure ), we obtan a braded dagram B whch s the -cable of B wth the two top endponts, and the two bottom endponts respectvely, of each cable zpped together. The complex of bmodules }. Fgure. Creatng and sldng dgons assocated to B s the tensor product of the HOMFLY-PT complex assocated to the -cable of B wth two complexes, one assocated to the top endponts of the cables and one to the bottom endponts. Performng a Redemester II or III move on B corresponds to performng a seres of Redemester II and III moves on ts

18 THE,-COLOURED HOMFLY-PT LINK HOMOLOGY 07 -cable. By Khovanov and Rozansky s [8] results we know that the complex of bmodules assocated to the -cable s nvarant up to homotopy equvalence under Redemester II and III moves. Therefore the complex of bmodules assocated to B s nvarant under the Redemester II and III moves up to homotopy equvalence. In the next paragraph we wll show that the category C whose obects are complexes of graded bmodules, such that ts homogeneous summands are fnte-dmensonal for each degree and whose morphsms are graded maps of complexes modulo homotopy, s Krull-Schmdt,.e. all obects have a unque decomposton nto ndecomposables. Therefore, the fact that B s nvarant under the bradlke Redemester II and III moves mples that B s nvarant under the same moves, because the former s ( + q ) n tmes the latter. Let us now prove that C s Krull-Schmdt. Frst of all, ths category s addtve wth fnte-dmensonal hom-spaces, so t suffces to show that t s Karouban [],.e. each dempotent splts. Note that the category C whose obects are complexes of graded bmodules, such that ts homogeneous summands are fnte-dmensonal for each degree and whose morphsms are graded maps of complexes, s an Abelan category wth fnte-dmensonal hom-spaces. Therefore t s Krull-Schmdt (Theorem.8 n []). Note also that any complex decomposes unquely nto a maxmal contractble subcomplex and ts complement. Now, let M be an obect n C and e A =End C (M) be an dempotent. By the above, we can assume that M has no contractble summands n C. Note that A =End C (M) s Artnan, because t s fnte-dmensonal, so ts Jacobson radcal J s nlpotent by Theorem..7 of []. Note also that the deal N A of null-homotopc maps s contaned n J. If M s ndecomposable, ths s true because A s local by Lemma.4.5 n []. If M s a fnte drect sum of ndecomposables, then A s the drect sum of Hom-spaces between the ndecomposable summands. Wrte J n terms of the Jacobson radcals of the endomorphsm rngs of the ndecomposables usng Proposton..5 of []. Agan we see that N J. Therefore, the nlpotency of J mples that of N. By Theorem.7. n [] ths mples that we can lft e from A = A/N to A. SnceC s Krull-Schmdt, we can splt e n C. Of course that splttng descends to a splttng n C, whch s what we had to prove. Alternatvely, one could note that the Poncaré polynomal of the,-coloured HOMFLY-PT homology of B s ( + q ) n tmes that of B and that therefore the nvarance of the former under the bradlke Redemester II and III mples the nvarance of the latter. Note that ths alternatve argument s really dfferent from the one n the prevous two paragraphs. The Poncaré polynomals of the,- coloured HOMFLY-PT homology of B and B can only be obtaned after takng the Hochschld homology of B and B, whereas the argument of the prevous two paragraphs apples to B and B drectly, before takng ther Hochschld homology. Ether way, note that ths dgon trck does not gve a specfc homotopy equvalence between the complexes before and after a Redemester move. It only shows that the correspondng Poncaré polynomals are equal. Ths would be a problem f we wanted to show functoralty under lnk cobordsms of the whole constructon. However, even the ordnary HOMFLY-PT-homology by Khovanov and Rozansky has not been proven to be functoral and probably s not. We thank Mkhal Khovanov for explanng the results n ths paragraph to us.

19 08 MARCO MACKAAY, MARKO STOŠIĆ, AND PEDRO VAZ Before we prove the crucal lemma we have to prove an auxlary result. In the followng lemma the top and the bottom of the dagram are complexes. The reader can easly check the followng auxlary result. Lemma 5.. The dagram below gves a homotopy equvalence between the top and the bottom complex. C : A B (d,0) ) ( rs 0 ( ) 00 0 (0, g) (f,g) B B C ( ) u 0 v ( ) ( ) h0 ( h ) B D (0,d) C : A f d C D Usng the lemma above we can now prove the followng crucal lemma. Lemma 5.. We have the homotopy equvalence (5.) =. of Equa- Proof. Note that the complex of bmodules C assocated to the r.h.s. ton(5.)sgvenby C : q 4 q q. The complex C assocated to the l.h.s. s gven by C : q 6 q 4 4.

20 THE,-COLOURED HOMFLY-PT LINK HOMOLOGY 09 By Lemma.4 we have By Lemma. we have = q A B. = = q B and by Lemma.6 we have B, Note that = q B 4 = 4 4. Fnally apply Lemma 5., whch s ustfed because () d: A B s zero, () d: B D s zero, () d: B B s the dentty, (4) d: B B s mnus the dentty, D. (5) q d q q and = q d q, (6) q q d q 4 q 4 4 = q d q 4 4.

21 0 MARCO MACKAAY, MARKO STOŠIĆ, AND PEDRO VAZ All assertons follow from straghtforward computatons and can easly be checked. For the frst asserton one only has to compute the mage of, whch s ndeed zero. For the second, thrd and fourth t suffces to compute the mages of and x. For the ffth t suffces to compare the mages of. For the sxth, one has to compare the mages of, x andx. Of course there are also homotopy equvalences analogous to the one n Lemma 5. for a negatve crossng or an -splttng of the rght top strand. In the followng lemma the top and bottom parts are complexes agan. Lemma 5.. If sf =0, the dagram below defnes a homotopy equvalence between the top and the bottom complex. ( A A g ) ( r s ) u ( ) (,0) D : A ( 0f ) C A ( h,0) h ( ) f 0 B (d,0) (d, s) D : 0 B d Proof. All maps are ndcated n the fgure, so the reader can check everythng easly. Two denttes are helpful: snce the top of the fgure s a complex, we have h = r + sg and fh = g u. Lemma 5.4. We have the homotopy equvalence (5.) = Proof. The complex assocated to the l.h.s., denoted D, s gven n Fgure 4. The. C D : q Fgure 4. Complex of the l.h.s. of Equaton (5.) complex assocated to the r.h.s., denoted D, s gven n Fgure 5. Ths tme we have = q A A

22 THE,-COLOURED HOMFLY-PT LINK HOMOLOGY q D : q 4 q Fgure 5. Complex of the r.h.s. of Equaton (5.) and = q We can apply Lemma 5. because C A () f s equal to the dentty, () u s equal to mnus the dentty, () sf =0and. (4) q d q 4 = q d q 4 q 4 q 4 Snce we have gven all the maps, the reader can check the clams by straghtforward computatons. Note that for the frst two clams t suffces to compute the mage of. For the last two clams one has to compute the mages of and x. Agan, there are analogous homotopy equvalences for a negatve crossng or f one swaps the - and -strands n the lemma above. 6. Invarance under the Markov moves 6.. Hochschld homology of bmodules as the homology of a Koszul complex of polynomal rngs. The Hochschld homology of a bmodule over the polynomal rng can be obtaned as the homology of a correspondng Koszul complex of certan polynomal rngs n many varables. Ths dea was explaned and used by Khovanov n [6]. Here we shall brefly descrbe how to extend t to our case..

23 MARCO MACKAAY, MARKO STOŠIĆ, AND PEDRO VAZ Frst of all, we change our polynomal notaton n ths secton. Namely, the polynomal rng R... k can be represented as the polynomal rng n the varables whch are the elementary symmetrc polynomals n the frst varables, the elementary symmetrc polynomals n the followng varables, etc., and the k elementary symmetrc polynomals n the last k varables. In ths secton we wll always work wth these new varables,.e. the elementary symmetrc polynomals, because t s more convenent for our purposes here. Thus to each strand labelled by k we assocate the k varables x,...,x k, such that the degree of x s equal to for =,...,k. To begn wth, we descrbe whch polynomal rng to assocate to a web. Take a web and choose a heght functon to separate the vertces accordng to heght. Ths way chop up the web nto several layers wth only one vertex. To each layer we assocate a new set of varables. To each vertex wth ncdent edges labelled, and +, weassocatethe + polynomals whch are the dfferences of the k-th symmetrc polynomals n the outgong varables and the ncomng varables for every k =,..., +. Moreover, f there are two dfferent sets of varables x,...,x k and x,...,x k assocated to a gven edge labelled k, we also assocate the k polynomals x x for all =,...,k. Fnally, to the whole graph we assocate the polynomal rng n all the varables modded out by the deal generated by all the polynomals assocated to the vertces and edges. There s an somorphsm between these polynomal rngs and the correspondng bmodules assocated to the graph. Indeed, to the tensor product p(x) q(x) n varables x there corresponds the polynomal p(x )q(x), where the varables x are of the bottom layer and x are of the top layer. Loosely speakng, the poston of the factor n the tensor product corresponds to the same polynomal n the varables correspondng to the layer. Letusdoanexample: Example 6.. Consder the web l : x,x y x,x y. The polynomal rng assocated to ths web s the rng P l := C[x,x,y,x,x,y ]. The deal I l by whch we have to quotent s generated by the dfferences of the symmetrc polynomals n all top and bottom varables (snce the mddle edge s labelled by ). There are three elementary symmetrc polynomal n ths case: Σ = x + y,σ = x + x y and Σ = x y,andso I l = x + y x y,x + x y x x y,x y x y. Hence, the polynomal rng we assocate to t s gven by R l = P l /I l.

24 THE,-COLOURED HOMFLY-PT LINK HOMOLOGY On the other hand, the bmodule o assocated to the web l s R R, and ts elements are a lnear combnaton of elements of the form p(x,x,y ) q(x,x,y ) for some polynomals p and q. Fnally, the somorphsm between o and R l s gven by p(x,x,y ) q(x,x,y ) p(x,x,y )q(x,x,y ). In such a way we have obtaned the bectve correspondence between the bmodules Γ and the polynomal rngs R Γ that are assocated to the open trvalent graph Γ. The closure of Γ n the bmodule pcture corresponds to takng the Hochschld homology of Γ. In the polynomal rng pcture ths s somorphc to the homology of the Koszul complex over R Γ whch s the tensor product of the complexes (6.) 0 R Γ {, } x x R Γ 0, where the x s are the bottom layer varables and the x s are the top layer varables. We put the rght-hand sde term n (co)homologcal degree zero. The frst shft s n the Hochschld (homologcal) degree, and the second one s n the q-degree, so that the maps have b-degree (, ). 6.. Invarance under the frst Markov move. Essentally we have to show that the Hochschld homology of the tensor product B B of the bmodules B and B s somorphc to the Hochschld homology of B B,.e. (6.) HH(B B )=HH(B B ). We shall prove ths by passng to the polynomal rng and Koszul complex descrpton from above. Recall that to the open trvalent graph Γ we have assocated the polynomal algebra P Γ (the rng of polynomals n all varables) quotented by the deal I Γ generated by certan polynomals. Snce for each layer we ntroduced new varables, the sequence of these polynomals s regular, and so we have that R Γ = P Γ /I Γ s the homology of the Koszul complex obtaned by tensorng together all complexes of the form f 0 P Γ P Γ 0 for f I Γ. We call ths the Koszul complex generated by I Γ. Fnally, the obect assocated to the closure of the graph Γ s gven by the homology of the Koszul complex defned by (6.), so t s somorphc to the homology of the Koszul complex generated by the polynomals whch defne I Γ together wth the polynomals x x whch come from the closure. If Γ s the vertcal glueng of Γ and Γ,thennI Γ we have the polynomals y y whch correspond to the edges whch are glued together. The Hochschld homology of Γ Γ s somorphc to the homology generated by I Γ,I Γ and the polynomals x x and y y. Clearly the same holds for Γ Γ, wth the role of the x x and y y nterchanged. Thus we have proved (6.). 6.. Invarance under the second Markov move. Invarance under the second Markov move corresponds to nvarance under the Redemester move I. If the strand nvolved s labelled, the result was proved by Khovanov and Khovanov and Rozansky [6, 8] (see below).

25 4 MARCO MACKAAY, MARKO STOŠIĆ, AND PEDRO VAZ It remans to show nvarance f the strand s labelled. For both the postve and the negatve crossng, we have the same three resolutons: 4 For each of these, we shall gve ts descrpton n a polynomal language and compute the homology of the correspondng Koszul complex. The resoluton q: Before closng the rght strand, we have an open graph. To the bottom layer we assocate varables x and x to the left strand, and y and y to the rght strand, whle to the top layer we assocate varables x and x to the left strand, and y and y to the rght strand. Then the rng R q,whchwe assocate to t, s the rng of polynomals n all these varables modded out by the deal generated by the polynomals x x, x x, y y, y y,.e. t s somorphc to the rng B := C[x,x,y,y ]. The resoluton s: The varables that we assocate to the bottom and the top layers are the same as above. To the bottom mddle strand we assocate the varable z, to the top mddle strand we assocate z and to the rght strand we assocate the varable t. Then the correspondng rng R s s the rng of polynomals n all these varables, modded out by the deal generated by the polynomals y z t, y z t, y z t, y z t, x + z x z, x + x z x x z, x z x z. From the frst four relatons, we can exclude t and obtan the quadratc relatons for z and z : z = y z y and z = y z y. Hence, every element from R s canbewrttenasa + bz + cz + dz z,wherea, b, c and d are polynomals only n x s and y s (wth or wthout prmes). The resoluton l: The varables we assocate to the bottom and top layers are agan the same. Ths tme we obtan the rng R l by quotentng by the deal generated by the followng four polynomals: x + y x y, x + y + x y x y x y, x y + x y x y x y, x y x y. To the closure of the rght strands of each of these graphs there corresponds the homology of the tensor product of the followng two Koszul complexes: 0 R Γ {, } y y R Γ 0, 0 R Γ {, } y y R Γ 0.

26 THE,-COLOURED HOMFLY-PT LINK HOMOLOGY 5 Hence, n all three cases we can have homology n three homologcal gradngs, 0, and. We denote ths homology by HH R (Γ). Inthecaseofq we have that both dfferentals are 0. Hence HH0 R (q) = B = C[x,x,y,y ], HH (q) R =B{} B{}, HH (q) R =B{4}. For the other two cases the computatons are a bt more nvolved, and we want to explan the general dea frst. The Hochschld homology s the homology of a complex whch s the tensor product of complexes of the form 0 P/I p P/I 0, where P s a polynomal rng, I s an deal and p R s a polynomal. Let us explan how to compute the homology of one such complex. The man part s the computaton of the kernel and the cokernel of the map above. The cokernel s easly computed and s equal to the quotent rng P/ I,p. Now, let us pass to the kernel. For any polynomal q P to be n the kernel,.e. to be a cocycle, we must have pq I. In other words, we have to compute the colon deal Q =(I : pp ). In the cases we are nterested n, Q s always a prncpal deal,.e. Q = qp for some q P. Then the kernel s gven by Q/I = qp/i. However, ths s somorphc to Q/ I,p, sncepq I, whch s the form n whch we present the results below. It s not hard to extend these calculatons to a tensor product of complexes of the above form. In partcular, all homologes are certan deals, modded out by the deal generated by I Γ and the polynomals y y, =,, whch n partcular ncludes the polynomals x x, =,, and also z z n the case of the web s. Hence, n the homology, all varables wth prmes are equal to the correspondng varables wthout the prmes. Inthecaseofs we have that y y = t (z z )=(y z )(y y ), and straghtforward computatons along the lnes sketched above gve HH0 R (s) =A = C[x,x,y,y,z ]/ z y z + y, HH (s) R ={((y z )g +(x y z (x y ))h, g) g, h A} A{} A{}, HH (s) R =(x y z (x y ))A{4}. Note that we have A = B z B. Fnally, n the case of l, weobtan (6.) (6.4) (6.5) HH0 R (l) = B = C[x,x,y,y ], HH (l) R ={( c(x y )+(cy + dy )(x y ),c(x y ) +d(x y )) c, d B} B{} B{}, HH (l) R =pb{4}, where p =(x y ) +(x y )(x y x y ). Now, we pass to the dfferentals. In our polynomal notaton, n the case of the postve crossng, the maps are (up to a non-zero scalar) R q R s { 4} : (x + x y y )+(y x )z +(y x )z.

27 6 MARCO MACKAAY, MARKO STOŠIĆ, AND PEDRO VAZ In the case of the second map (s q l), t s gven by the coeffcent of z z after multplcaton wth ( y +y y + x (y y ) x y y +y y + x (y y ) x y ) +(z + z )(x x y + y y + x x y + y y ) +z z (x + x y y ). Recall that the elements of R s are of the form a+bz +cz +dz z,wherea, b, c and d are the polynomals only n x s and y s, whle z = y z y and z = y z y. For the negatve crossng, the maps are the followng: R l R s : and R s R q {} : a + bz + cz + dz z b + c + dy. Snce we are nterested n the dfferentals n the case when the rght strands are closed, n the homology all varables wth prmes are equal to the ones wthout the prmes (as we explaned above), and the dfferentals reduce to the followng (agan up to a non-zero scalar): q 4 : (x y ) (x y )z, q : a + bz a(x y )+b(x y ), :, q : a + bz b, where a, b B. Now, by straghtforward computaton of the homology for both postve and negatve crossng n each Hochschld degree, we obtan the followng smple behavour of HHH under the Redemester I move: ( ) ( ) HHH = HHH, HHH ( ) ( = HHH ) {, }.

28 THE,-COLOURED HOMFLY-PT LINK HOMOLOGY 7 We recall that denotes the upward shft by n the homologcal degree, whle {k, l} denotes the upward shft by k n the Hochschld degree and by l n the q- degree. For the Redemester I move nvolvng a strand labelled, we get a smlar shft (see [6, 8], or apply the same methods as above): ( ) ( ) HHH = HHH, HHH ( ) ( = HHH ) {, }. The overall shft n the defnton of H, (D) compensates ths behavour under the Redemester I moves, and we get a genune lnk nvarant Categorfcaton of the axoms (A) and (A). In ths subsecton we shall show that our constructon categorfes the axoms (A) and (A) of the MOY calculus, whch nclude the closures of the strands. The powers of q n (A) and (A) wll correspond to the nternal q-gradng (gradng of the polynomal rng), whle the powers of t wll correspond to the Hochschld gradng. Frst we focus on the (A) axom for arbtrary k. The crcle s the closure of the sngle strand labelled wth k. Denote the graph that conssts of ths strand by w k and the varables that we assocate to t by x,...,x k (remember that n ths secton we assume that deg x =). Then to ths w k we assocate the rng R wk = C[x,...,x k ], and to ts closure the Hochschld homology HH(w k )whchs the homology of the Koszul complex obtaned by tensorng 0 R wk {, } R 0 wk 0 for all =,...,k. Hence we have HH(w k )= k (C[x,...,x k ] C[x,...,x k ]{, }), = whch categorfes axom (A). Now we consder axom (A). The left-hand sde of the axom s the closure of the rght strand of the graph, whch we denoted by l. As we prevously sad, we are only nterested n the cases when the ndces and are from the set {, }, andso we have four cases. Each of these cases we treat n the same way as the case of l, whch we dealt wth n the prevous subsecton. We assocate the varables x,...,x and x,...,x to the strands on the left-hand sde and the varables y,...,y and y,...,y to the strands on the rght-hand sde, and compute the homology (that we denote by HH R (l )) of the tensor product of the Koszul complexes 0 R l {, l } y l y l R l 0 for l =,...,. In the remanng part of the graph, the varables y don t appear agan, whle as before, n the HH R (l )wehavethatx l = x l, for all l =,...,. Fnally, the rght-hand sde of axom (A) s w, to whch we assocate the varables x,...,x, and consequently the rng R w = C[x,...,x ].

29 8 MARCO MACKAAY, MARKO STOŠIĆ, AND PEDRO VAZ Now, n the four cases we are nterested n, the homologes are the followng: = = : The correspondng rng n ths case s R l = C[x,x,y,y ]/ x + y x y,x y x y, whle HH R l s the homology of the complex 0 R l {, } y y R l 0. Drect computaton of ths homology gves HH0 R (l ) = C[x,y ], HH (l R )=(x y )C[x,y ]{}, whch categorfes (A) n ths case. =, = : The correspondng rng n ths case s R l = C[x,x,x,x,y,y ]/ x +y x y,x y +x x y x,x y x y, whle HH R l s the homology of the complex 0 R l {, } y y R l 0. Then we have HH0 R (l ) = C[x,x,y ], HH (l R )=(x x y + y)c[x,x,y ]{}, as wanted. =, = : The correspondng rng n ths case s R l = C[x,x,y,y,y,y ]/ x + y x y,x y + y x y y,x y x y, whle HH R l s the homology of the complex obtaned by tensorng 0 R l {, } y y R l 0 and 0 R l {, } y y R l 0. Thus, we obtan HH0 R (l ) = C[x,y,y ], HH (l R )={((c(x y )+dy, c + dx ) c, d C[x,y,y ]} C[x,y,y ]{} C[x,y,y ]{}, HH (l R )=(x x y + y )C[x,y,y ]{4}, whch categorfes (A) n ths case. = = : We have already computed ths case n the prevous subsecton n the formulas (6.)-(6.5), whch gves the categorfcaton of (A) n ths case. The categorfcaton of axom (A) for general labellngs s obtaned n [].

30 THE,-COLOURED HOMFLY-PT LINK HOMOLOGY 9 7. Computaton of the,-coloured Hopf Lnk Consder the followng brad dagram labelled and : K, =. The complex of bmodules assocated to K, s q 4 d + q d + d + d +, q where d + and d+ are as gven n Secton 4 and the rghtmost term s n cohomologcal degree zero. The bmodules n cohomologcal degree are both somorphc to o o {} shfted up by. Usng the drect sum decompostons (A) and (A5) of Lemmas. and.4, one fnds that the complex assocated to K s homotopyequvalent to (7.) q 6 Δ q x x or, n the language of bmodules, q 6 Δ R q x R R x R R. Closng the strands of K, gves the,-coloured Hopf lnk ),. To obtan the HOMFLY-PT homology we have to take the Hochschld homology of the complex above. We use the notaton and conventons of Secton 6. More precsely, the bmodules R and R R become the polynomal rngs C[x,x,y ]andr l, respectvely (see Example 6.). Then the complex becomes the followng complex of polynomal rngs: q 6 C[x,x,y ] y x y +x q R l y y R R. The Hochschld homology groups can be computed as at the end of Secton 6.

31 0 MARCO MACKAAY, MARKO STOŠIĆ, AND PEDRO VAZ The complex n Hochschld degree s q R q 0 R q 9 R and has homology q 9 R n cohomologcal degree 0. The complex n Hochschld degree s Δ 0 0 q0 0 0 R 0 q 8 R q6 R q 8 R 0 q4 R q 6 R q 0 R q 0 R q 8 R and has homology q 6 R /Δ () = q 6 R n cohomologcal degree, and q 4 R q 6 R q 8 R n cohomologcal degree 0. The complex n Hochschld degree s Δ 0 0 q7 0 Δ 0 R 0 q 9 R q R q 5 R 0 qr q R q 7 R q 7 R q 5 R and has homology q R q 5 R n cohomologcal degree, and qr q R q 5 R n cohomologcal degree 0. The complex n Hochschld degree 0 s q 6 Δ R q 0 R R and has homology q R n cohomologcal degree, and R n cohomologcal degree 0. Therefore the Poncaré polynomal of HHH(), )s P(), )= u / t / q / ( q ) ( u ( t q 6 + t (q + q 5 )+q ) + t q 9 + t (q 4 + q 6 + q 8 )+t (q + q + q 5 )+ q 4 where the varables t and u correspond to the Hochschld and the cohomologcal degrees, respectvely, and we have ncluded the normalzaton defned at the end of Secton 4. In [4] Gukov, Iqbal, Kozçaz and Vafa (GIKV) conectured the exstence of a trply graded lnk homology for lnks wth components coloured by arbtrary representatons of sl(n). Ths conecture s motvated by the physcs of topologcal strngs. They establshed a map between the refned topologcal vertex and the sl(n) homologcal nvarants of the Hopf lnk. When the components of the Hopf lnk are coloured wth the frst and second fundamental representatons of sl(n), the GIKV s superpolynomal s equal to P (, ) () )(u, q, a) = ( q ) ( a a (q +)+aq where u corresponds to the cohomologcal degree. ), + u a q 6 a (q + q 4 + q 6 )+a( + q + q 4 ) a q 4 ),