A deformation of Robert-Wagner foam evaluation and link homology
.. A DEFORMATION OF ROBERT-WAGNER FOAM EVALUATION ANDLINK HOMOLOGY
MIKHAIL KHOVANOV AND NITU KITCHLOO
Abstract.
We consider a deformation of the Robert-Wagner foam evaluation formula, withan eye toward a relation to formal groups. Integrality of the deformed evaluation is es-tablished, giving rise to state spaces for planar GL ( N ) MOY graphs (Murakami-Ohtsuki-Yamada graphs). Skein relations for the deformation are worked out in details in the GL (2)case. These skein relations deform GL (2) foam relations of Beliakova, Hogancamp, Putyraand Wehrli. We establish the Reidemeister move invariance of the resulting chain complexesassigned to link diagrams, giving us a link homology theory. Contents
1. Introduction 11.1. MOY graphs and quantum invariants for level one representation 11.2. Foams and Robert-Wagner evaluation 31.3. Formal groups as a motivation 61.4. Plan of the paper 81.5. Acknowledgments 92. Deformed evaluation for GL ( N ) foams 93. Formal groups and generalized divided difference operators 144. Deformed GL (2) foam evaluation 204.1. GL (2) foams and their colorings 204.2. Deformed evaluation for GL (2) foams 224.3. Examples 244.4. Skein relations 314.5. Prefoams and ground ring reduction 404.6. GL (2) webs, their state spaces, and direct sum decompositions 435. Reidemeister moves invariance and link homology 47References 561. Introduction
MOY graphs and quantum invariants for level one representation.
Foams are2-dimensional combinatorial CW-complexes, often with extra decorations, embedded in R .They naturally appear [Kh2, KRo2, MV1, MSV, QR, RWd] in the study of link homologytheories that categorify quantum sl N or gl N link invariants for level one representations when N ≥ Date : April 30, 2020. a r X i v : . [ m a t h . QA ] A p r MIKHAIL KHOVANOV AND NITU KITCHLOO
Reshetikhin-Turaev-Witten invariants [RT, W] of oriented links L in the 3-sphere S dependon the choice of a simple Lie algebra g and an irreducible representation of g associated to eachcomponent of L . When g = sl N and the components are labelled by level one representationsof sl N , the Reshetikhin-Turaev-Witten invariant P ( L ) ∈ Z [ q, q − ] can be written [MOY] as alinear combinations of terms P (Γ) ∈ Z + [ q, q − ] over trivalent oriented planar graphs Γ withedges labelled by integers between 1 to N . P (Γ) is known as the Murakami-Ohtsuki-Yamadaor MOY invariant of Γ.An edge labelled a corresponds to the identity intertwiner of Λ aq V , the latter a quantumgroup representation which q -deforms the a -th exterior power of the fundamental representa-tion of the Lie algebra sl N . At this point it’s convenient to shift from sl N to gl N , and viewΛ aq V as a representation of U q ( gl N ) rather than that of sl N . This change will be more essen-tial at the categorified level of homological invariants rather than for uncategorified quantuminvariants, taking values in Z [ q, q − ].Oriented labelled graphs Γ are built out of trivalent vertices that correspond to suitablyscaled inclusion and projection of Λ a + bq V into and out of the tensor product Λ aq V ⊗ Λ bq V , seeFigure 1.1.1. a a b a ba + ba + b Figure 1.1.1.
Generating diagrams for GL ( N ) MOY graphs. They corre-spond to the identity intertwiner on Λ aq V and projection and inclusion (up toscaling) between Λ aq V ⊗ Λ bq V and Λ a + bq V .Quantum gl N (or MOY) invariant of Γ is given by a suitable convolution of these maps,which for closed graphs Γ results in a Laurent polynomial P (Γ) ∈ Z [ q, q − ] with nonnegativecoefficients, see [MOY] for integrality and [RW2, Appendix 2A] for nonnegativity via a suitablestate sum formula. Planar graph invariant P (Γ) can be computed either via a state sumformula or inductively via skein relations.As we mention earlier, Z [ q, q − ]-linear combinations of invariants P (Γ) give quantum linkinvariants P ( L ), when g = gl N and components of L are labelled by level one representations,that is, by Λ aq V , over different a ’s.The reason for the popularity of this specialization (from g to gl N and to level one repre-sentations), especially with an eye towards categorification, is the relative simplicity of theseformulas compared to the case of general g and its representations, where canonical choicesof intertwiners associated to graph’s vertices are harder to guess, spaces of these intertwinersmay be more than one-dimensional, decomposition of a crossing into a linear combinationsof planar graphs has more complicated coefficients or may be difficult to select, and evalu-ations of P (Γ) lose positivity, acquire denominators and live in Q ( q ) rather than Z + [ q, q − ].Any such complication makes categorical lifting noticeably harder. An approach to categori-fication of the Reshetikhin-Turaev-Witten link invariants for an arbitrary g and arbitrary DEFORMATION OF ROBERT-WAGNER FOAM EVALUATION AND LINK HOMOLOGY 3 representations has been developed by Webster [We]. It’s an open problem to find a foam-like interpretation of Webster link homology theories and refine them to achieve functorialityunder link cobordisms.1.2.
Foams and Robert-Wagner evaluation.
The key property of P (Γ) is it having non-negative coefficients, that is, taking values in Z + [ q, q − ], rather than just in Z [ q, q − ], wherelink invariants P ( L ) live. In the lifting of P ( L ) to homology groups, state spaces (cid:104) Γ (cid:105) will begraded, with graded rank (as a free module over the graded ring R N of symmetric functions,see below) having non-negative coefficients, thus lying in Z + [ q, q − ], Homology groups H ( L )come from complexes of state spaces (cid:104) Γ (cid:105) , built from various resolutions Γ of L .Louis-Hadrien Robert and Emmanuel Wagner discovered a remarkable evaluation formulafor GL ( N ) foams [RW1]. Their formula leads to a natural construction of homology groups(or state spaces) for each planar trivalent MOY graph Γ as above.At the categorified level of this story, Robert-Wagner foam evaluation leads to a statespace (cid:104) Γ (cid:105) , a graded module over the ring R N = Z [ x , . . . , x N ] S N of symmetric polynomials in x , . . . , x N with coefficients in Z . Robert and Wagner prove [RW1] that the graded R N -moduleis free and finitely-generated, of graded rank P (Γ).Thus, graded rank of R N -module (cid:104) Γ (cid:105) categorifies the quantum gl N invariant (the Murakami-Ohtsuki-Yamada invariant) of these planar graphs. Forming suitable complexes out of thesestate spaces and taking homology groups leads to bigraded homology theories of links thatcategorify the HOMFLYPT polynomial and its generalizations to other quantum exteriorpowers of the fundamental representation [ETW], see also earlier approaches [Y, Wu1, Wu2]to categorification of gl N link homology with components colored by arbitrary level one rep-resentations.We now recall the details of Robert-Wagner’s foam invariant. A GL ( N )-foam F is a two-dimensional piecewise-linear compact CW -complex F embedded in R . Its facets are orientedin a compatible way and labelled by numbers from 0 to N called the thickness of a facet (facetsof thickness 0 may be removed) with points of three types: • A regular point on a facet of thickness a . • A point on a singular edge, which has a neighbourhood homeomorphic to the productof a tripod T and an interval I . The three facets must have thickness a, b, a + b respectively. One can think of thickness a,b facets as merging into the thick facet orvice versa, of the facet of thickness a + b splitting into two thinner facets of thickness a and b . • A singular vertex where four singular edges meet. The six corners of the foam at thevertex have thickness a, b, c, a + b, b + c, a + b + c respectively.Neighbourhoods of these three types of points are depicted below.Orientations of facets are compatible at singular edges, see Figure 1.2.3 below.A singular vertex can be viewed, see Figure 1.2.2, as the singular point of the cobordismbetween two labelled trees that are the two splittings of an edge of thickness a + b + c intoedges of thickness a, b, c, respectively. This is a kind of ”associativity” cobordism, which isinvertible when viewed as an appropriate module map between state spaces associated toMOY planar graphs in the foam theory. We follow the orientation conventions from [ETW].They show compatible orientations on facets of thickness a and b attached along a singularedge to a facet of thickness a + b . The same diagram shows induced orientations on top and MIKHAIL KHOVANOV AND NITU KITCHLOO a a cab ba + b b + c a + b + c a + b Figure 1.2.1.
Three types of points on a foam a cba + b + ca cba + b + c a cba + b + c
Figure 1.2.2.
Cross-sections near a singular vertexbottom boundaries of foam F . This convention will be used once we pass from closed foamsto foams with boundary, viewed as cobordisms between GL ( N ) MOY graphs. a ab a + b Figure 1.2.3.
Orientation conventions from [ETW, Figure 1]. An orienta-tion of a facet induces an orientation of its top boundary (if non-empty, fornon-closed foams only) by sticking the first vector of the orientation basis upout of the foam. The remaining vector then induces an orientation of theboundary. For the bottom boundary the resulting orientation is reversed. Toinduce an orientation on a singular circle, approach it with an orientation basisfrom a thin facet and point the first vector into the thick facet. The secondvector then defines an orientation of the singular circle (or a singular arc, iffoam is not closed). This is the one convention we choose out of the four pos-sible conventions for inducing orientations on the boundary and on singularlines, given an orientation of a facet.
DEFORMATION OF ROBERT-WAGNER FOAM EVALUATION AND LINK HOMOLOGY 5
Facets f of a foam F are the connected components of the set F \ s ( F ), where s ( F ) isthe set of the singular points of F . Thickness of f is denoted (cid:96) ( f ). The set of facets of F isdenoted f ( F ). A coloring c of F is a map c : f ( F ) −→ I N from the set of facets to the set ofsubsets of I N = { , . . . , N } such that subset c ( f ) has cardinality (cid:96) ( f ) and for any three facets f , f , f attached to a singular edge with (cid:96) ( f ) = (cid:96) ( f ) + (cid:96) ( f ) equality c ( f ) = c ( f ) (cid:116) c ( f )holds. In other words, the subset for f is the union of subsets for f and f . A foam maycome with decorations (dots). A dot on a facet f of thickness a represents a homogeneoussymmetric polynomial P f in a variables.Any coloring c gives rise to closed surfaces F i ( c ), 1 ≤ i ≤ N , which are unions of facets f such that c ( f ) contains i . One also forms symmetric differences F ij ( c ) = F i ( c )∆ F j ( c ),which are the unions of facets f such that c ( f ) contains exactly one element of the set { i, j } .Surfaces F ij ( c ), i (cid:54) = j are closed orientable as well.Rogert-Wagner evaluation (cid:104) F, c (cid:105) RW of a foam on a coloring c is(1) (cid:104) F, c (cid:105) RW = ( − s ( F,c ) P ( F, c ) Q ( F, c ) , where s ( F, c ) = θ + ( c ) + N (cid:88) i =1 iχ ( F i ( c )) / ,θ + ( c ) = (cid:88) i In this paper we propose a deformation of theRobert-Wagner evaluation formula, motivated by algebraic topology and generalized coho-mology theories related to formal groups. Link homology theories in the SL (2) case havebeen lifted to spectra by Lipshitz and Sarkar [LS1, LS2] and Hu, Kriz and Kriz [HKK]. Morerecently, a lifting of bigraded GL ( N ) link homologies as well as the triply-graded homologyto equivariant spectra has been constructed by the second author [K1, K2, K3]. DEFORMATION OF ROBERT-WAGNER FOAM EVALUATION AND LINK HOMOLOGY 7 Application of generalized cohomology theories to these spectra results in new homologi-cal link invariants as well as cohomological operations on them. A purely combinatorial oralgebraic description of these homological invariants is clearly desirable, and modifying foamtheory and foam evaluation may be a natural first step in this direction. GL ( N ) foams are closely related to Grassmannians and partial and full complex flag va-rieties. A family of cohomology theories known as complex oriented cohomology theories isrelated to these varieties as well, and to deformations of the formula for the first Chern classin singular cohomology of the tensor product of line bundles c ( L ⊗ L ) = c ( L ) + c ( L )to formulas c ( L ⊗ L ) = F ( c ( L ) , c ( L ))that hold for the first Chern class invariant in these geralized cohomology theories, where F ( x, y ) = x + y + (cid:88) i + j> a i,j x i y j is, in general, a power series in x, y with coefficients in the ground ring. Such a powerseries admits rich internal structure, making it a Formal group law . In Section 3 we shallstudy formal group laws in detail, but let us briefly point out some relevant structure in thisintroduction. Among the relations satisfied by F ( x, y ) is the associativity relation, whichleads to polynomial relations on a i,j which admit a universal solution with one generator foreach k = i + j − k ≥ 1. This solution is hard to write down explicitly, and most manipulationswith general formal group laws are implicit [Ha, St] (see section 3 for examples).With formal group law F ( x, y ) at hand, one defines − F x or [ − x as power series − x + . . . which solves the equation F ( x, [ − x ) = 0, and forms the power series x − F y = F ( x, − F y ),also denoted x [ − y : x [ − y = x − F y := F ( x, − F y ) . This expression deforms x − y , so that x − F y = x − y + higher order terms . One can show x − F y = ( x − y ) q ( x, y ) for an invertible element q ( x, y ) of a suitable power series ring. Wewrite x − y = p ( x, y )( x − F y ) where p ( x, y ) q ( x, y ) = 1 , and use p ( x, y ) = q ( x, y ) − in ourcomputations.From the standpoint of algebraic topology, x − F y represents the Euler class (in the coho-mology theory corresponding to the formal group law F ( x, y )) of the line bundle L ⊗ L ∗ ,where L and L represent the tautological line bundles over the product space CP ∞ × CP ∞ .In other words, the expression q ( x, y ) should be interpreted as the relative Euler class for thebundle L ⊗ L ∗ , in the sense that one compares the Euler classes in the cohomology theorycorresponding to F ( x, y ), to the standard Euler class in singular cohomology. In this context,products of the form q ( x i , x j ) q ( x i , x j ) . . . q ( x i k , x j k ) (which we will come across often inthis paper) may be interpreted as the relative Euler class of the direct sum of the line bundlescorresponding to each factor.Robert-Wagner foam evaluation formulas contain powers of x i − x j in the denominator, anda natural idea would be to carefully replace them with x i − F x j . We pursue a variant of thisidea in this paper. Similar replacements have already been considered for various formulasin the theory of symmetric functions, including the Weyl formula for the Schur function,see [NN1, NN2, Na] and references therein. Foam evaluation specializes to the Weyl formulafor the Schur function in the case of the so-called theta-foam and its natural generalizations. MIKHAIL KHOVANOV AND NITU KITCHLOO On the algebraic topology side, the expressions x i − x j for 1 ≤ i (cid:54) = j ≤ N have a naturalmeaning as the Euler classes in singular cohomology for the roots α of GL ( N ) (we haveincluded both positive and negative roots). In particular, the deformation x i − F x j canbe interpreted as the Euler class of α in an exotic cohomology theory corresponding to theformal group law F ( x, y ). We may therefore speculate that the corresponding deformedfoam evaluation formula is obtained by applying an exotic cohomology theory to a (hithertoundefined) homotopy type. The existence of such a homotopy type for foam evaluations is verycompelling given the results by the second author [K1, K2, K3]. Since GL ( N ) has N ( N − α representing the weights x i − x j in the standard basis for 1 ≤ i (cid:54) = j ≤ N , we see thatour deformed evaluation formulas will be expressible in terms of N ( N − 1) parameters givenby the relative Euler classes q ( x i , x j ). However, these extra parameters will satisfy certainconstraints with coefficients in the algebra of symmetric power series in N -variables (whichis the GL ( N )-equivariant cohomology of a point). This suggests that the possible underlyinghomotopy type for foam evaluations is built from universal bundles using suitable subsets ofroots of GL ( N ).The discussion above motivates our deformation using the language of formal group lawsand related cohomology theories. We go into this further in section 3. Interestingly however,although motivated by it, our deformation setup will end up not requiring all the constraintson the power series q ( x, y ) imposed by a formal group law. For instance, we will not requireassociativity from our analogue of the power series F ( x, y ). We therefore take as Ansatz,the series p ( x, y ), the inverse of q ( x, y ), with arbitrary coefficients. Most of the informationin the coefficients of q ( x, y ) will turn out to be redundant in our framework, at least in the GL (2) case. However, it is conceivable that one may endow our constructions with the actionof cohomology operations which are sensitive to more coefficients in the power series q ( x, y ).1.4. Plan of the paper. In section 2, motivated by analogies with formal group laws, wewrite down a multi-parameter deformation of the Robert-Wagner evaluation of closed GL ( N )-foams and prove its integrality for any such foam. In section 3 we review formal group laws andcorresponding generalizations of the divided difference operators. In section 4 we specializeto N = 2 and study this deformation, which ultimately adds two more variables, of the GL (2) foam evaluation. Skein relations for the deformed GL (2) foam evaluation are derivedin section 4.4. In section 4.5 we work out the ground ring R for the deformed theory, which hasfour generators E , E , ρ , ρ of degrees 2 , , − , 0, respectively. For comparison, the groundring for the usual GL (2)-equivariant link homology has generators E , E (also denoted h, t , upto a minus sign). In section 4.6 we show, unsurprisingly, that the state spaces (or homology)of planar GL (2) webs are free modules over the graded ring R of rank ( q + q − ) k over theground ring R , where k is the number of thin circles in a web. In section 5 we extend thestate spaces to homology groups of planar link diagrams and show the invariance under theReidemeister moves.Specializing power series p ( x, y ) to p ( x, y ) = 1 recovers the GL (2) foam theory of Beliakova,Hogancamp, Putyra, and Wehrli [BHPW]. Simplifying computations in our section 4 to thiscase gives a foam evaluation approach to their theory. GL (2) foam theory that comes from this deformation seems very similar to the SL (2) the-ory as set up by Vogel [V] and extended by him to get a strong invariant of tangle cobordisms,without the sign indeterminacy. The relation is given by dropping double facets but remem-bering singular circles along which the facets attach to the thin surface of a GL (2) foam. DEFORMATION OF ROBERT-WAGNER FOAM EVALUATION AND LINK HOMOLOGY 9 GL (2) foam theory also has an overlap with Ehrig-Stroppel-Tubbenhauer’s generic GL (2)foams [EST1].We’ve already mentioned connections to Clark-Morrison-Walter [CMW] and Caprau [Ca1,Ca2] who have achieved full functoriality of the SL (2) link homology via diagrammaticalcalculi that employ singular circles on thin surfaces. These circles should be remnants ofattached double facets. Caprau’s theory is GL (2) equivariant, with variables h and a in placeof our E and E .Twisting of SL (2) theories in Vogel [V] is related to the deformation via power series p ( x, y )in this paper. We plan to elucidate connections to Vogel [V], Ehrig-Stroppel-Tubbenhauer [EST1],and to Turaev-Turner’s rank two Frobenius algebra structures [TT] in a follow-up paper andalso see whether the p ( x, y ) deformation corresponds to the twisting [Kh4, V] in the N = 2and the general case.1.5. Acknowledgments. M.K. was partially supported by NSF grants DMS-1664240 andDMS-1807425 while working on this paper. The authors are grateful to Yakov Kononov,Louis-Hadrien Robert and Lev Rozansky for valuable discussions and would like to thankElizaveta Babaeva for help with producing figures for the paper.2. Deformed evaluation for GL ( N ) foams The GL ( N ) Robert-Wagner formula has denominators of the form ( x i − x j ) χ ij ( c ) / , where χ ij ( c ) = χ ( F ij ( c )) is the Euler characteristic of the bicolored surface F ij ( c ). The expres-sion x i − x j can be generalized to x i − F x j = x i [ − x j , where F is a formal group law.Unlike the additive case, when x − y = − ( y − x ), most formal group laws do not satisfy x [ − y = − ( y [ − x ), while those that do are called symmetric . Converting ( x i − x j ) χ ij ( c ) / to ( x i [ − x j ) χ ij ( c ) / to modify the Robert-Wagner formula may be possible, but it wouldnot contain the opposite terms x j [ − x i , that perhaps should be present to maintain somesymmetry, despite us having fixed a set of positive roots { x i − x j } i F, c (cid:105) and the original one in [RW1] by (cid:104) F, c (cid:105) RW .Setting aside formal group laws at this point, let us now formally define x [ − y as follows.Choose a commutative graded ring k and homogeneous elements β k,(cid:96) ∈ k in degree − k + (cid:96) )for all k, (cid:96) ∈ Z + = { , , , . . . } such that ( k, (cid:96) ) (cid:54) = (0 , . The element(6) p ( x, y ) = 1 + (cid:88) ( k,(cid:96) ) (cid:54) =(0 , β k,(cid:96) x k y (cid:96) belongs to the power sum ring k (cid:74) x, y (cid:75) . In general, β k,(cid:96) (cid:54) = β (cid:96),k . The element p ( x, y ) has theinverse q ( x, y ) = p − ( x, y ) ∈ k (cid:74) x, y (cid:75) . Define(7) x [ − y = q ( x, y )( x − y ) = p ( x, y ) − ( x − y ) ∈ k (cid:74) x, y (cid:75) . Equivalently, x − y = p ( x, y )( x [ − y ) . Denote p i,j = p ( x i , x j ) or, interchangeably, p ij , and q ij = q ( x i , x j ) = p − ij . Then(8) x i [ − x j = ( x i − x j ) p − ij . Note that x [ − y = − ( y [ − x ) iff p ( x, y ) = p ( y, x ) iff β (cid:96),k = β k,(cid:96) for all k, (cid:96) . We refer tothis as the symmetric case.The universal case is that of the ring(9) k = Z [ β k,(cid:96) ]over all k, (cid:96) as above ( k, (cid:96) ∈ Z + , ( k, (cid:96) ) (cid:54) = (0 , , − , − , . . . . It’s a gradedpolynomial ring with k + 1 generators in degree − k over all k ≥ 1. The universal symmetriccase is when β (cid:96),k = β k,(cid:96) are formal variables over all 0 ≤ k ≤ (cid:96), ( k, (cid:96) ) (cid:54) = (0 , x i − x j in the numeratorto get ( x i − x j ) χ i ∩ j ( c ) ( x i [ − x j ) χ i ( c ) / ( x j [ − x i ) χ j ( c ) / = ( x i − x j ) χ i ∩ j ( c ) ( x i − x j ) χ i ( c ) / q χ i ( c ) / ij ( x j − x i ) χ j ( c ) / q χ j ( c ) / ji =( − χ j ( c ) / p χ i ( c ) / ij p χ j ( c ) / ji ( x i − x j ) χ ij ( c ) / Taking the product over all 1 ≤ i < j ≤ N , the minus signs will combine to ( − (cid:80) Nj =1 ( j − χ j ( c ) / and(10) (cid:104) F, c (cid:105) = ( − s (cid:48) ( F,c ) (cid:89) f ∈ f ( F ) P f ( c ) (cid:89) i 1, as in [RW1]. Adding this sign term is a matter of preference, while we hope thatthe deformation via p ( x, y ) will eventually prove significant.Let us now show that the formula (12) given by summing the expressions (10) over all coloringsgives rise to a symmetric power series that does not involve denominators. We begin with asimple lemma: Lemma 2.1. Given a coloring c , let p ( c ) denote the expression p ( c ) = (cid:89) ≤ i 1. Factoringthis expression, se see that ( x − x ) must divide p , ,s − (cid:3) Remark 2.2. Notice that the definition of p , ,s as a ratio of p ( c (cid:48) ) and p ( c ) can be extended tothe case when c (cid:48) is a coloring obtained from c by a (1 , -Kempe move along several connectedcomponents Σ s := Σ s (cid:116) . . . (cid:116) Σ s k . This expression, p , ,s satisfies a locality property that is crucial for the following theorem p , ,s = p , ,s . . . p , ,s k , where p , ,s i denotes the ratio of p ( c (cid:48) i ) and p ( c ) , with c (cid:48) i obtained from c by a (1 , -Kempemove on Σ s i . The above lemma allows us to prove Theorem 2.3. The GL ( N ) -foam evaluation (cid:104) F (cid:105) is a symmetric power series in the variables x , x , . . . , x N . In particular, (cid:104) F (cid:105) is free of denominators.Proof. The proof of the above theorem is essentially a simple variation on the argument givenin [RW1, Proposition 2.18]. Consider the expression (cid:104) F (cid:105) . It is clear that it is symmetric inthe variables x , x , . . . , x N with possible denominators of the form ( x i − x j ) k . By symmetry,the proof of the theorem will follow if we can show that the denominator ( x − x ) does notappear in (cid:104) F (cid:105) . DEFORMATION OF ROBERT-WAGNER FOAM EVALUATION AND LINK HOMOLOGY 13 Let us decompose the set of colorings of (cid:104) F (cid:105) into a collection of equivalence classes relativeto the colors 1 and 2. Given a coloring c of F , decompose F ( c ) into connected components, F ( c ) = Σ = Σ ∪ Σ ∪ . . . ∪ Σ r . The equivalence class of colorings C c that contains c consists of colorings of F that can beobtained from c by performing Kempe moves about various connected components Σ s ⊆ F ( c ), where we recall that a Kempe move about Σ s switches the colors 1 and 2 of the facetsin Σ s .As in [RW1, Proposition 2.18], consider the expressions P F/ Σ ( F, c ) = p ( c ) (cid:89) f not a facet in Σ P ( c ( f )) , (cid:101) Q ( F, c ) = Q ( F, c ) (cid:81) s,k> ( x − x k ) l Σ s ( c,k ) / ( x − x ) χ ( c ) / where p ( c ) is as defined in Lemma 2.1, and the integers l Σ s ( c, k ) are as defined in [RW1,Lemma 2.10]. Also, for 1 ≤ s ≤ r , define T s ( F, c ) = (cid:98) P Σ s ( F, c ) + ( − χ (Σ s ) / p , ,s σ ( (cid:98) P Σ s ( F, c )) , where σ is the transposition that swaps x and x , the term p , ,s is as defined in Lemma 2.1,and the expression (cid:98) P Σ s ( F, c ) is defined as (cid:98) P Σ s ( F, c ) = (cid:89) f a facet in Σ s P ( c ( f )) (cid:89) F, c (cid:105) on summingover the equivalence class C c as (cid:88) c (cid:48) ∈ C c (cid:104) F, c (cid:48) (cid:105) = ( − s (cid:48) ( F,c ) P F/ Σ ( F, c ) (cid:101) Q ( F, c ) r (cid:89) s =1 ( x − x ) − χ (Σ s ) / T s ( F, c ) . Since (cid:101) Q ( F, c ) is not divisible by x − x , it is enough for our purposes to show that theexpression ( x − x ) − χ (Σ s ) / T s ( F, c ) does not have a denominator given by a power of x − x .The only case that is relevant is when Σ s is a surface of genus zero. It is therefore sufficientto show that T s ( F, c ) is divisible by ( x − x ) when Σ s is a surface of genus zero. In this case,we have T s ( F, c ) = (cid:98) P Σ s ( F, c ) − p , ,s σ ( (cid:98) P Σ s ( F, c )) . By Lemma 2.1, recall that p , ,s is of the form 1 mod ( x − x ). We therefore have T s ( F, c ) = (cid:98) P Σ s ( F, c ) − σ ( (cid:98) P Σ s ( F, c )) mod ( x − x ) . However, the expression (cid:98) P Σ s ( F, c ) − σ ( (cid:98) P Σ s ( F, c )) is also divisible by x − x since it switchessign under σ . It follows that T s ( F, c ) is divisible by x − x whenever Σ s is a surface of genuszero. The proof of the theorem easily follows on summing (cid:104) F, c (cid:48) (cid:105) over all the equivalenceclasses C c . (cid:3) Formal groups and generalized divided difference operators In this section we study formal group laws and their relationship to topology in somedetail. Good references are [Ha, St] and the references therein. Due to standard conventionsthe choice of notation R in this section conflicts with its use in the next section.Let us begin by recalling the definition of a formal group law. A formal group law definedover a ring R is a power series F ( x, y ) with coefficients in R so that F ( x, y ) represents acommutative group structure on the formal affine line over R . In other words, one requires F ( x, y ) to satisfy the following three properties F ( x, y ) = F ( y, x ) commutativity F (0 , x ) = F ( x, 0) = x unitarity F ( x, F ( y, z )) = F ( F ( x, y ) , z ) associtivity . Remark 3.1. There is a universal ring known as the Lazard ring which is initial among allrings that support a formal group law. This ring L can be defined to be generated by symbols a i,j where the universal formal group law has the form F ( x, y ) = x + y + (cid:88) i,j> a i,j x i y j . We then impose relations on the generators a i,j that are forced by the relations of commu-tativity and associativity (the relation for unitarity is built into the form of F ( x, y ) ). Forinstance, commutativity implies that a i,j = a j,i . The relation for associtivity is clearly moreinvolved. In topology, formal group laws appear when one describes the E -cohomolgy of a spaceBU(1), where E is any complex oriented cohomology theory and BU(1) denotes the classifyingspace of the group U(1) (the space BU(1) is equivalent to the infinite projective plane CP ∞ ).More precisely, one starts with the observation that E ∗ (BU(1)) can be expressed as R (cid:74) x (cid:75) ,with R = E ∗ ( pt ) and x being the first Chern class in cohomological degree 2. The abeliangroup structure on U(1) induces a mapBU(1) × BU(1) −→ BU(1) . Evaluating this map in E -cohomology then gives rise to the underlying a formal group law F E ( x, y ) for the complex oriented cohomology theory E : R (cid:74) x (cid:75) = E ∗ (BU(1)) −→ E ∗ (BU(1) × BU(1)) = R (cid:74) x, y (cid:75) , x (cid:55)−→ F E ( x, y ) . In what follows therefore, we work in the graded setting. So F ( x, y ) will denote a formalgroup law over a graded power series ring R (cid:74) x, y (cid:75) , where R is a graded Z -algebra, and thevariables x and y are defined to have degree 2. We assume that F ( x, y ) is in homogeneousdegree 2, namely F ( x, y ) = x + y + (cid:88) i,j> a ij x i y j , a ij ∈ R − i + j ) . DEFORMATION OF ROBERT-WAGNER FOAM EVALUATION AND LINK HOMOLOGY 15 Definition 3.2. The formal negative of the variable x is defined to be the (unique) powerseries [ − x with the property F ( x, [ − x ) = F ([ − x, x ) = 0 . The formal difference x [ − y is defined as F ( x, [ − y ) . It is a power series in two variables x, y that has homogeneous degree 2. Example 3.3. Let R be the Z -algebra Z [ β ] , where β is in degree − . The multiplicativeformal group law F ( x, y ) and its formal difference is given by F ( x, y ) = x + y − βxy, x [ − y = x − y − βy , [ − x ) = − (cid:88) i ≥ β i x i +1 . Example 3.4. Let R be the Z -algebra Z [ β ] as before with β in degree − . The Lorentz orL-formal group law F ( x, y ) and its formal difference is given by F ( x, y ) = x + y β xy , x [ − y = x − y − β xy , [ − x ) = − x. Example 3.5. Let R be the Z -algebra Z [ , β ] with β in degree − . The ˆ A -formal group law F ( x, y ) and its formal difference is given by F ( x, y ) = x (cid:112) β ( y/ + y (cid:112) β ( x/ ,x [ − y = x (cid:112) β ( y/ − y (cid:112) β ( x/ , [ − x ) = − x. where the radicals are expressed as a power series (with coefficients in Z [ , β ] ) by the formalapplication of the binomial expansion. Example 3.6. Let R be the Z -algebra Z [ , (cid:15), δ ] with the degree of δ being − and that of (cid:15) being − . The Jacobi formal group law F ( x, y ) and its formal inverse is given by F ( x, y ) = x (cid:112) J ( y ) + y (cid:112) J ( x )1 − (cid:15)x y , where J ( z ) = 1 − δz + (cid:15)z .x [ − y = x (cid:112) J ( y ) − y (cid:112) J ( x )1 − (cid:15)x y , [ − x ) = − x. Examples 3.4 and 3.5 are specializations of 3.6 at the “cusps” described by (cid:15) = β , δ = β and (cid:15) = 0 , δ = − β / respectively. Let us return to the universal example. In other words, we consider the example of R beingthe Lazard ring introduced earlier. On introducing a grading on the variables x and y so thatthe universal formal group law belongs in homogeneous degree 2, the Lazard ring naturallyacquires a grading as described earlier. With this grading, the Lazard ring can be shown tobe isomorphic to the graded coefficient ring of a complex oriented cohomology theory knownas complex cobordism, MU. In other words R ∼ = MU ∗ ( pt ) as a graded ring. By the definition of complex cobordism, the elements of MU − k ( pt ) are cobordism classesof k -dimensional manifolds endowed with an almost complex structure on their stable normalbundle, and with the ring structure being induced by the cartesian product of manifolds. Thering MU ∗ ( pt ) can be shown to be a polynomial algebra over Z , with one generator in eachnegative even degree. Working rationally, the generator in degree − n may be chosen to bethe cobordism class of the complex projective space of dimension 2 n , denoted by [ CP n ].Any formal group law over a Q -algebra is isomorphic to the additive formal group law. Thisisomorphism is called the logarithm, written as log F ( x ), and is the unique power series withleading term being x , that interpolates the given formal group law F ( x, y ) with the additiveone G a ( x, y ) = x + y .On extending scalars from Z to Q , the logarithm in the universal case has an explicitdescriptionlog MU ( x ) = (cid:88) k ≥ [ CP k ] k + 1 x k +1 , so that log MU ( F MU ( x, y )) = log MU ( x ) + log MU ( y )An immediate corollary of the above description is the following example Example 3.7. Let R be the Z -algebra MU ∗ ( pt ) . The universal formal group law F MU ( x, y ) and its formal difference is given by F MU ( x, y ) = exp MU (log F ( x ) + log F ( y )) = exp MU ( (cid:88) k ≥ [ CP k ] k + 1 ( x k +1 + y k +1 )) ,x [ − y = exp MU ( (cid:88) k ≥ [ CP k ] k + 1 ( x k +1 − y k +1 )) = exp MU (( x − y ) (cid:88) k ≥ [ CP k ] k + 1 S k ( x, y )) , where exp MU ( z ) is the compositional inverse of log MU ( z ) , and S k ( x, y ) is the symmetric sumS k ( x, y ) = x k + x k − y + · · · + xy k − + y k . Notice that even though the expressions for F MU ( x, y ) and x [ − y above appear to have denom-inators, these denominators cancel away in the ring MU ∗ ( pt ) once one expands the expressionas a power series in x and y . One may notice that the formal difference x [ − y in each of the above examples appearsto be divisible by the expression ( x − y ). In fact, this is always true as we now show Claim 3.8. Given an arbitrary formal group law, there is a unique homogeneous degree 0element q ( x, y ) ∈ R (cid:74) x, y (cid:75) so that x [ − y = ( x − y ) · q ( x, y ) . Furthermore, q ( x, y ) is invertible and q ( x, y ) ≡ y ) .Proof. Consider the formal expansion of the expression ( y + z )[ − y := F ( y + z, [ − y ). Onsetting z as 0, we see that the expression vanishes. Therefore, it is divisible by z . Setting z as ( x − y ), we conclude that there is a power series q ( x, y ) that satisfies the relation( x − y ) q ( x, y ) := F ( y + x − y, [ − y ) = x [ − y. DEFORMATION OF ROBERT-WAGNER FOAM EVALUATION AND LINK HOMOLOGY 17 Since ( x − y ) is not a zero divisor in R (cid:74) x, y (cid:75) , we see that q ( x, y ) is unique. Next, by setting y as 0, we see that q ( x, 0) = 1. In particular, q ( x, y ) has the form 1 mod ( y ), and is thereforea unit. (cid:3) Remark 3.9. It is not hard to show using the definition of the power series q ( x, y ) that thecoefficients in its expansion generate the same sub algebra of R as the coefficients of the formalgroup law F ( x, y ) . To see this, first observe that the power series [ − y can be expressed interms of the coeffiients of q ( x, y ) using the fact that [ − y = − yq (0 , y ) . Next, observe that F ( x, y ) = x [ − − y ) = ( x − [ − y ) q ( x, [ − y ) . These two observations together establishwhat we seek to show. Remark 3.10. In example 3.6, one may verify that q ( x, y ) is the following (symmetric)expression q ( x, y ) = x + yx (cid:112) J ( y ) + y (cid:112) J ( x ) . In general however, q ( x, y ) need not be symmetric in x, y as is easily seen from example 3.3. Remark 3.11. The universal example 3.7 allows us to deduce some interesting propertiesabout q ( x, y ) . For instance, we see that q ( x, y ) has the form q ( x, y ) = (cid:88) k ≥ [ CP k ] k + 1 S k ( x, y ) + (cid:88) n ≥ q n ( x, y )( x − y ) n , where q n ( x, y ) are symmetric power series in x and y . Note that each individual series q n ( x, y ) involves denominators. However, on setting x = y , those terms vanish and we obtain theinteresting (universal) relation that does not involve denominators q ( x, x ) = ddx log F ( x ) . Claim 3.12. Assume that the Z -algebra R is torsion free. Then, given a formal group law F ( x, y ) over R , the power series q ( x, y ) is symmetric if and only if log F ( x ) is an odd powerseries. Equivalently, q ( x, y ) is symmetric if and only if q ( x, x ) is an even power series. Notethat these conditions are automatic if R has no nontrivial elements in degrees .Proof. The equivalence of the two conditions follows from remark 3.11 above. It remains toestablish the first condition. Now log F ( x ) is an odd power series if and only if its compositionalinverse exp F ( x ) is an odd power series. We will now proceed to show that symmetry of q ( x, y )is equivalent to exp F ( x ) being an odd power series. Using the universal example 3.7, we seethat the formal difference x [ − y for the formal group law F ( x, y ) has the form x [ − y = exp F ( z ) , z = ( x − y ) s ( x, y ) , with s ( x, y ) being a symmetric power series s ( x, y ) = (cid:88) k ≥ l k k + 1 S k ( x, y ) , where log F ( x ) = (cid:88) k ≥ l k k + 1 x k +1 . Note that s ( x, y ) is invertible in ( R ⊗ Q ) (cid:74) x, y (cid:75) , and so z can be chosen to be a power seriesgenerator. We therefore have an inclusion( R ⊗ Q ) (cid:74) z (cid:75) ⊂ ( R ⊗ Q ) (cid:74) x, y (cid:75) , z (cid:55)−→ ( x − y ) s ( x, y ) . It follows that q ( x, y ) s ( x, y ) − = exp F ( z ) /z is symmetric if and only if exp F ( z ) /z is even, orthat exp F ( z ) is odd. (cid:3) Let us now study the divided difference operators in the context of formal group laws. Definition 3.13. Consider the formal power series ring R (cid:74) x , x , . . . , x n (cid:75) . Let α denote anypair ( i, j ) for ≤ i < j ≤ n . We think of α as a positive root of U( n ) so that the pairs ( i, j ) are indexed by the set ∆ + of positive roots of U( n ) . Given a formal group law defined over R ,we define the generalized divided difference operator A α as the operator on R (cid:74) x , x , . . . , x n (cid:75) A α ( f ) := fx i [ − x j + r α ( f ) x j [ − x i , f ∈ R (cid:74) x , x , . . . , x n (cid:75) , where r α is the reflection on R (cid:74) x , x , . . . , x n (cid:75) given by switching x i and x j . Using elementaryalgebra, one can check that the operator A α is well defined and does not involve denominators. Remark 3.14. If E is a complex oriented cohomology theory with underlying formal grouplaw F E and the coefficients of a point being E ∗ ( pt ) = R , then the operators A α have a naturalmeaning in terms of push-pull oprators on the U( n ) -equivariant E -cohomology ring of the flagvariety U( n ) /T (see [BE] ). More precisely, recall that the U( n ) -equivariant E -cohomology ofa U( n ) -space X is defined as the E -cohomology of the space EU( n ) × U( n ) X with EU( n ) beingthe principal contractible U( n ) -space. For X = U( n ) /T , the U( n ) -equivariant E -cohomologyring is isomorphic to R (cid:74) x , x , . . . , x n (cid:75) , supporting the operator A α that is defined as thepushforward in equivariant E -cohomology followed by the pullback: π ∗ ◦ π ∗ , where π denotesthe U( n ) -equivariant fibration π : U( n ) /T −→ U( n ) / U α ( n ) , with U α ( n ) being the maximal compact subgroup in the parabolic subgroup corresponding tothe positive root α . Claim 3.15. Given a root α ∈ ∆ + defined by the pair ( i < j ) , let q ( α ) denote the unit q ( x i , x j ) as defined in 3.8. Then the intersection of the kernels of all the operators A α i , where α i = x i − x i +1 is a simple root, is a rank one free module over the ring of symmetric powerseries R (cid:74) x , x , . . . , x n (cid:75) Σ n generated by the unit q (∆ + ) , where q (∆ + ) := (cid:89) α ∈ ∆ + q ( α ) . Proof. Given a simple root α i ∈ ∆ + , let us rewrite the action of A α i on f as A α i ( f ) = fx i [ − x i +1 + r i ( fx i [ − x i +1 ) . Hence f is in the kernel of A α i if the expression fx i [ − x i +1 switches sign under r i . Notice thatthe ratio of any two such elements is invariant under r i . On the other hand, by claim 3.8, DEFORMATION OF ROBERT-WAGNER FOAM EVALUATION AND LINK HOMOLOGY 19 it follows that given an r i -invariant element g , the expression q ( α i ) g is in the kernel of A α i .In particular, we have shown that for a fixed α i , the kernel of A α i is precisely the rank onemodule of r i invariants generated by the unit q ( α i ). Now let us fix α i ∈ ∆ + , and considerˇ q ( α i ) := (cid:89) β ∈ ∆ + , β (cid:54) = α i q ( β ) , so that q ( α i ) ˇ q ( α i ) = q (∆ + ) . Since r i permutes all positive roots besides α i , we see that ˇ q ( α i ) is an r i -invariant unit. Hencethe kernel of A α i is a rank one free module of r i -invariants generated by the element q (∆ + ).Taking intersection over all simple roots α i ∈ ∆ + we get the required result. (cid:3) Example 3.16. For the multiplicative formal group law of example 3.3, the intersection ofthe kernels of all the generalized divided difference operators A α i is a rank one free moduleover symmetric power series R (cid:74) x , x , . . . , x n (cid:75) Σ n generated by the unit q (∆ + ) , where q (∆ + ) = 1(1 − βx )(1 − βx ) . . . (1 − βx n ) n − . Note that q (∆ + ) is not Σ n -invariant for any n > . Let D α denote the classical divided difference operator (i.e. the divided difference operatorfor the additive formal group law). Definition 3.13 and claim 3.8 imply that we have(17) A α ( f ) = D α ( q ( α ) − f ) , in other words A α = D α ◦ Q ( α ) − , where Q ( α ) denotes the operator given by multiplication with q ( α ). In particular, the q -twisted operators Q ( α i ) ◦ A α i satisfy the braid relations, and generate an algebra isomorphicto the nilHecke algebra, namely, the algebra generated by the operators D α i . We also have Theorem 3.17. Given a formal group law defined over R , let A ( n ) denote the algebra ofoperators on R (cid:74) x , x , . . . , x r (cid:75) generated by multiplication operators, and the generalized di-vided difference operators A α i for ≤ i < n . Then A ( n ) is identically the same as the(completed) affine nilHecke algebra over the ground ring R . In other words, A ( n ) agreeswith the algebra generated by the operators D α i and multiplication operators with respect to R (cid:74) x , x , . . . , x r (cid:75) . In particular, A ( n ) is a free (left or right) module of rank n ! over thesubalgebra R (cid:74) x , x , . . . , x n (cid:75) . Alternatively, A ( n ) is a matrix algebra of rank ( n !) over thesubalgebra R (cid:74) x , x , . . . , x n (cid:75) Σ n . It follows that R (cid:74) x , x , . . . , x n (cid:75) Σ n ⊂ A ( n ) is the center.Proof. By (17), we see that the operators A α i are of the form D α i ◦ Q ( α i ) − , where Q ( α i )is the invertible multiplication operator corresponding to q ( α i ). It follows that the operators A α i generate the same algebra as D α i when extended with multiplication operators, which isthe affine nilHecke algebra by definition (once we complete polynomials to power series). Therest of the claim follows from well-known results on the affine nilHecke algebra. (cid:3) Remark 3.18. The above theorem may come as a surprise to the reader, since it has beenknown for some time that for an arbitrary compact Lie group G , the push-pull operators A α i defined as in remark 3.14 and acting on the equivariant cohomology E ∗ G ( G/T ) , do not satisfythe braid relations, unless the formal group law F E underlying the cohomology theory E ishighly restrictive (see theorem 3.7 in [BE] , see also [HMSZ] ). It is possible that theorem3.17 only holds for the compact Lie group G = U( n ) , though we have not verified this. It isimportant to note that the classes q ( α i ) that allow for the proof of the above theorem have been studied before (see [C] , [Na] ), though the main observation of theorem 3.17 appears to benew. Deformed GL (2) foam evaluation GL (2) foams and their colorings. The original formulation [Kh1] of SL (2) link ho-mology did not use foams. Hints at foams appeared in the work of Clark, Morrison, andWalker [CMW] and Caprau [Ca1, Ca2], who used disorientation lines on surfaces involved inthe construction of SL (2) homology to control minus signs that appear throughout the theory.This allowed them to establish full functoriality of the theory under cobordisms rather thanthe functoriality up to an overall minus sign, as shown in the earlier work [J, Kh3, BN]. Onecan think of disorientation lines as remnants of the 2-facets of GL (2) foams along which theywere attached to the 1-facets.Earliest constructions of SL (3) and SL ( N ) link homology for N > SL (2) (more precisely, GL (2)) homologytheory. A detailed exploration of various flavours of GL (2) foams and applications can befound in [EST1, EST2].We find its useful to follow the GL (2) foam calculus of Beliakova, Hogancamp, Putyra,Wehrli [BHPW]. That’s the calculus deformed in this section.We consider GL (2) foams (or, simply, foams) in this paper. A closed GL (2) foam F isa combinatorial compact two-dimensional CW-complex embedded in R (or S ). The onlyallowed singularities of the CW-complex are singular circles , such that any point on the circlehas a neighbourhood homeomorphic to the product of the tripod and the interval.The set of points of F on its singular circles is denoted s ( F ), and connected components of F \ s ( F ) are called facets . Facets of F are subdivided into 1-facets and 2-facets. One-facetsare also called thin facets, two-facets are also called double or thick facets. We require thatalong each singular circle two 1-facets and one 2-facet meet, see Figure 4.1.1. thinfacets doublefacetsingular circle F IGURE Figure 4.1.1. Part of a singular circle and its neighbourhood, with two thinand one double facet.This implies, in particular, that no ’monodromy’ is possible along any singular circle, so ithas a neighbourhood in F homeomorphic to the product of S and a tripod with ’two thinlegs and a double leg’. DEFORMATION OF ROBERT-WAGNER FOAM EVALUATION AND LINK HOMOLOGY 21 Each facet is oriented in such a way that all three facets along any singular circle induce acompatible orientation on this circle, see Figure 4.1.2. In most diagrams that follow, it’s clearwhether a facet is thin or double, and we usually omit the corresponding label 1 or 2. Figure 4.1.2. Left: oriented think and thick facets, with induced orienta-tions on the top and bottom boundary. Right: compatibility between orienta-tions of a singular circle and adjacent thin facets and the double facet. Alsoshown induced orientations of the top and bottom foam’s boundary. They’llbe needed when we pass from closed foams to foams with boundary in Sec-tion 4.6Thin facets may carry dots, which can move freely along a facet, but cannot jump to anadjacent facet. If a facet carries n dots, we may record them as a single dot with label n .It’s possible to allow similar decorations on 2-facets, namely symmetric polynomials in twovariables, but we avoid doing so in the paper, instead moving any such decoration from a2-facet to the coefficient of the foam. Remark: Unlike GL ( N ) foams for N ≥ SL ( N ) foams for N ≥ GL (2) foams can’thave singular vertices.A coloring (or admissible coloring ) of a foam F is a map c from the set of its thin facets tothe set { , } such that along any singular circle, the two thin facets are mapped to differentnumbers. It’s convenient to extend c to double facets, coloring each double facet by theset { , } . This produces the flow condition, that the union of colors of 1-facets along eachsingular circle is the color of the double facet, that is, the entire set { , } .Notice that F ( c ) does not depend on the coloring c and is a closed surface which is theunion of closures of 1-facets of F . We denote it by F and call the thin surface of F . Likewise, F ∩ ( c ) does not depend on c and is the union of closures of 2-facets of F . We denote it by F ∩ and call the double surface of F . The boundary of F ∩ is exactly the set of singularcircles of F .Often it’s convenient to identify a facet f with its closure f in F . In particular, the Eulercharacteristics of f and f are equal, since the two spaces differ only by a union of circles,which is the boundary of f . From now on, unless otherwise specified, by a facet we mean aclosed facet.Surface F has finitely many connected components Σ , . . . , Σ n . Each component maycontain one or more singular circles. The union of these singular circles is zero when viewedas an element of H (Σ k , Z / k , due to our orientation requirements on F . Inparticular, each Σ k admits exactly two checkerboard colorings of its regions, where along eachsingular circle in Σ k the coloring is reversed. A choice of such coloring for each Σ k is equivalent to a coloring of F . Hence, F has 2 n colorings, where n is the number of connected components of F .Quantity θ + ( c ) = θ +12 ( c ) counts the number of positive circles for a coloring c , see Fig-ure 4.1.3. positive { 1 , 2 } { 1 , 2 } { 1 , 2 } { 1 , 2 } negative Figure 4.1.3. θ +12 ( c ) or simply θ + ( c ) counts the number of positive circles.4.2. Deformed evaluation for GL (2) foams. Modified Robert-Wagner evaluation formula,in the GL (2) case, specializes to(18) (cid:104) F, c (cid:105) = ( − θ +12 ( c ) ( x − x ) χ ( F ∩ ( c )) x d ( c )1 x d ( c )2 ( x [ − x ) χ ( c ) / ( x [ − x ) χ ( c ) / Since(19) x [ − x = ( x − x ) p − , x [ − x = ( x − x ) p − , we have(20) ( x − x ) χ ∩ ( c ) ( x [ − x ) χ ( c ) / ( x [ − x ) χ ( c ) / = ( − χ ( c ) / p χ ( c ) / p χ ( c ) / ( x − x ) χ ( c ) / In the 2-color case, F ∩ ( c ) = F ∩ is the union of facets of thickness two and does not dependon c . Likewise, F ( c ) = F does not depend on c either. Its Euler characteristic is denoted χ ( F ) = χ ( F ) . Equation (18) can be rewritten (cid:104) F, c (cid:105) = ( − θ +12 ( c )+ χ ( c ) / x d ( c )1 x d ( c )2 ( x − x ) χ ( F ) / p − χ ( c ) / p − χ ( c ) / = ( − θ +12 ( c )+ χ ( c ) / x d ( c )1 x d ( c )2 p χ ( c ) / p χ ( c ) / ( x − x ) χ ( F ) / = ( − s (cid:48) ( F,c ) P ( F, c ) Q ( F, c ) p χ ( c ) / p χ ( c ) / = ( − χ ( F ) / (cid:104) F, c (cid:105) RW p χ ( c ) / p χ ( c ) / . Above, s (cid:48) ( F, c ) = θ +12 ( c ) + χ ( c ) / d ( c ), d ( c ) is the number of dots on thin facets coloredby 1, resp. 2 by c . We see that the original evaluation (cid:104) F, c (cid:105) RW is scaled by an invertibleelement, which is a product of powers of p and p and a sign. Also, the power of x − x in the denominator depends on F only. Let us write down the formula again. DEFORMATION OF ROBERT-WAGNER FOAM EVALUATION AND LINK HOMOLOGY 23 (21) (cid:104) F, c (cid:105) = ( − θ +12 ( c )+ χ ( c ) / x d ( c )1 x d ( c )2 p χ ( c ) / p χ ( c ) / ( x − x ) χ ( F ) / We now define(22) (cid:104) F (cid:105) = (cid:88) c (cid:104) F, c (cid:105) , the sum over all colorings of F . Let E = x + x , E = x x . The symmetric group S actson k (cid:74) x , x (cid:75) by permuting x , x . Theorem 4.1. (cid:104) F (cid:105) ∈ k (cid:74) x , x (cid:75) S ∼ = k (cid:74) E , E (cid:75) for any GL (2) foam F . In other words, (cid:104) F (cid:105) is a power series in x , x that’s symmetric under the permutationaction of S on x , x . Equivalently, it’s a power series in elementary symmetric functions E , E . Consider the chain of inclusions(23) k ⊂ k (cid:74) E , E (cid:75) ⊂ k (cid:74) x , x (cid:75) ⊂ k (cid:74) x , x (cid:75) (cid:20) x − x (cid:21) . Denote these rings by (cid:101) R = k (cid:74) E , E (cid:75) , (24) R (cid:48) = k (cid:74) x , x (cid:75) , (25) R (cid:48)(cid:48) = k (cid:74) x , x (cid:75) (cid:20) x − x (cid:21) , (26)resulting in the chain of ring inclusions(27) k ⊂ (cid:101) R ⊂ R (cid:48) ⊂ R (cid:48)(cid:48) . The theorem above has already been proved in Section 2 for general N , see Theorem 2.3. Weinclude a more detailed proof for the special case N = 2 to make this section independentfrom Section 2. Proof. The evaluation (cid:104) F, c (cid:105) can be written, via (18), as a power series in x , x with coeffi-cients in k divided by a power of x − x , either positive or negative, thus it belongs to thering R (cid:48)(cid:48) , see above.Group S acts on 2-colorings of F by transposing the colors 1 and 2. This action iscompatible with the evaluation in the sense that σ ( (cid:104) F, c (cid:105) ) = (cid:104) F, σ ( c ) (cid:105) , where σ = (12) is thenontrivial element of S . Therefore, (cid:104) F (cid:105) is in the subring ( R (cid:48)(cid:48) ) S of S -invariants of R (cid:48)(cid:48) . (cid:104) F (cid:105) potentially has a denominator ( x − x ) χ ( F ( c )) / . Surface F ( c ) = F is a unionof connected components Σ , . . . , Σ m , each one contributing ( x − x ) χ (Σ k ) / to the product.Only connected components of genus 0 have positive Euler characteristic, χ (Σ k ) = 2, andcontribute x − x to the denominator.Consider one such component Σ and a coloring c . The Kempe move on Σ replaces c witha coloring c = ( c, Σ) which is identical to c outside Σ and swaps colors 1 , c on Σ. Wecompare (cid:104) F, c (cid:105) and (cid:104) F, c (cid:105) in formula (21).If there are t i dots on color i facets of Σ under c , i = 1 , 2, then x d ( c )1 x d ( c )2 = x t x t u, x d ( c )1 x d (1)2 = x t x t u, for a monomial u in x , x counting dots on facets not in Σ.If Σ has r singular circles, let θ + ( c, Σ) be the number of positive circles on Σ under c and θ − ( c, Σ) be the number of negative circles. Under the swap c ↔ c , positive circles on Σbecome negative circles on Σ and vice versa, so that θ + ( c, Σ) + θ + ( c , Σ) = r .Let (cid:101) χ i (Σ) = χ (Σ ∩ F i ( c )), i = 1 , , be the Euler characteristic of the union of color i facetsof Σ, for coloring c . We have χ ( c ) = χ ( c ) + (cid:101) χ (Σ) − (cid:101) χ (Σ) = χ ( c ) + χ (Σ) − (cid:101) χ (Σ) , since χ (Σ) = (cid:101) χ (Σ) + (cid:101) χ (Σ) . Defining integer (cid:96) by 2 (cid:96) = (cid:101) χ (Σ) − (cid:101) χ (Σ) . , we have χ ( c ) = χ ( c ) + 2 (cid:96), χ ( c ) = χ ( c ) − (cid:96). Consequently, one can write p χ ( c ) / p χ ( c ) / = p (cid:96) u (cid:48) , p χ ( c ) / p χ ( c ) / = p (cid:96) u (cid:48) , for a suitable monomial u (cid:48) in p , p , possibly with negative exponents.Also, ( − s (cid:48) ( F,c ) ( − s (cid:48) ( F,c ) = ( − θ + ( c )+ χ ( c ) / ( − θ + ( c )+ χ ( c ) / = ( − r + χ (Σ) / − (cid:101) χ (Σ) . When Σ ∼ = S , r + χ (Σ) / − (cid:101) χ (Σ) ≡ r − (cid:101) χ (Σ) ≡ , since r ≡ χ (Σ)(mod 2). The last comparison modulo 2 can be proved by induction on r , byremoving an innermost singular circle of Σ. This operation reduces r by 1 and changes (cid:101) χ (Σ)by ± 1. We see that ( − s (cid:48) ( F,c ) = − ( − s (cid:48) ( F,c ) .Putting these relations together, (cid:104) F, c (cid:105) + (cid:104) F, c (cid:105) = s (cid:48) ( F, c ) · (cid:0) x t x t p (cid:96) − x t x t p (cid:96) (cid:1) uu (cid:48) ( x − x ) − χ ( F ) / . The expression ( x t x t p (cid:96) − x t x t p (cid:96) ) is divisible by x − x and allows to cancel out thatterm from the denominator.Repeating this argument simultaneously for all S -components of F shows that (cid:104) F (cid:105) ∈ k (cid:74) x , x (cid:75) . Permutation action of S on k (cid:74) x , x (cid:75) and on colorings shows that (cid:104) F (cid:105) ∈ k (cid:74) x , x (cid:75) S = k (cid:74) E , E (cid:75) . (cid:3) The sum χ ( F ) = χ ( c ) + χ ( c ) does not depend on the coloring c and is the Euler char-acteristic of the surface F . In particular, in the symmetric case (when p = p ), one hasthat(28) (cid:104) F (cid:105) = (cid:104) F (cid:105) RW · ( − p ) χ ( F ) / so that the new evaluation is proportional to the original one with the coefficient that dependsonly on χ ( F ). We expect that non-symmetric case will prove more interesting.4.3. Examples. Example 1: Let F = S ,n be the two-sphere of thickness one with n dots (or, equivalently,with a single dot labelled n ). Here the lower index (1 , n ) lists thickness followed by the numberof dots. S ,n has two colorings c and c , where in the coloring c i the 2-sphere carries color i . Forthe coloring c F ( c ) ∼ = S , F ( c ) = ∅ , F ∩ ( c ) = ∅ , DEFORMATION OF ROBERT-WAGNER FOAM EVALUATION AND LINK HOMOLOGY 25 and likewise for c , so that (cid:104) S ,n , c (cid:105) = x n x [ − x = x n p x − x , (cid:104) S ,n , c (cid:105) = x n x [ − x = x n p x − x , and (cid:104) S ,n (cid:105) = x n p − x n p x − x . To explicitly cancel x − x in the denominator, expand p and p into power series and thencancel. The result is a power series symmetric in x , x with coefficients which are polynomialsin β i,j .We denote(29) ρ n = (cid:104) S ,n (cid:105) = x n p − x n p x − x . Note that the following relation holds:(30) ρ n +2 − E ρ n +1 + E ρ n = 0 , where, recall, E = x + x , E = x x . It follows from the relation( x n +21 p − x n +22 p ) − ( x + x )( x n +11 p − x n +12 p ) + x x ( x n p − x n p ) = 0 . Relation (30) allows to inductively write ρ n = (cid:104) S ,n (cid:105) as a linear combination of ρ = (cid:104) S , (cid:105) and ρ = (cid:104) S , (cid:105) with coefficients in Z [ E , E ]. The latter are(31) ρ = (cid:104) S , (cid:105) = p − p x − x , ρ = (cid:104) S , (cid:105) = x p − x p x − x . Example 2: Let the foam F be a thin two-torus T with n dots and standardly embeddedin R (embedding of a surface does not influence its evaluation). As in the previous example,there are two colorings, c and c , with F ( c ) ∼ = T , F ( c ) = ∅ , F ∩ ( c ) = ∅ , and (cid:10) T (cid:11) (cid:48) = x n + x n . Example 3: Closed surface M of genus g ≥ n dots.(32) (cid:104) M (cid:105) = x n p − g ( x − x ) − g + x n p − g ( x − x ) − g = ( x n p − g + ( − g − x n p − g )( x − x ) g − . Example 4: F is 2-sphere S of thickness two, also denoted S . It has a unique coloring c ,with the facet labelled by { , } and F ( c ) = F ( c ) = F ∩ ( c ) ∼ = S so that (cid:104) S (cid:105) = (cid:104) S , c (cid:105) = ( x − x ) ( x [ − x )( x [ − x ) = ( x − x ) p p ( x − x )( x − x )= − p p Denote the value of this foam by ρ , so that(33) ρ = (cid:104) S (cid:105) = − p p . Note that ρ is an invertible element of the ground ring. In the original case, when p ( x, y ) = 1,the double sphere S evaluates to − Example 5: An oriented closed surface M of genus g ≥ (cid:104) M (cid:105) = ( x − x ) − g p − g q − g ( x − x ) − g ( x − x ) − g = ( − p p ) − g = ρ − g . In the special case, when g = 1 so that M = T is a two-torus, (cid:104) T (cid:105) = 1 . Example 6: The theta-foam Θ with n and n dots on thin facets, suitably oriented. (cid:399) – foam (cid:399) ´ reversed orientation n n n n F IGURE Figure 4.3.1. On the left: Θ-foam. On the right: same foam Θ (cid:48) with thereversed orientation.Let c be the coloring of Θ with its top facet colored 1. Then the bottom facet is colored2. Surfaces F ( c ) , F ( c ) , and F ( c ) = F are all 2-spheres, with Euler characteristics 2 . The sign θ +12 ( c ) = 1. coloring c (cid:423) + (c ) = 1F (c ) =F (c ) =F (c ) = = S = S = S n n 12 12 F IGURE Figure 4.3.2. Computing (cid:104) Θ , c (cid:105) .For the sign, s (cid:48) (Θ , c ) = θ + ( c ) + χ ( c ) / , ( − s (cid:48) (Θ ,c ) = 1 . We get (cid:104) Θ , c (cid:105) = x n x n p p x − x , (cid:104) Θ , c (cid:105) = − x n x n p p x − x , DEFORMATION OF ROBERT-WAGNER FOAM EVALUATION AND LINK HOMOLOGY 27 where c is the other coloring (with the opposite sign in the evaluation and transposed expo-nents of x , x ). Assuming n ≥ n ,(35) (cid:104) Θ (cid:105) = ( x x ) n x n − n − x n − n x − x p p = ( x x ) n h n − n − ( x , x ) p p , where h k ( x , x ) = x k + x k − x + · · · + x k is the k -the complete symmetric function in x , x .Note that we can write(36) (cid:104) Θ (cid:105) = − E n h n − n − ( x , x ) ρ, and that h k ( x , x ) is a polynomial in E , E , the latter elementary symmetric functions in x , x . Also, (cid:104) Θ (cid:105) is the product of a Schur function for GL (2) and − ρ .Note that if the two thin facets of the theta-foam carry the same number of dots, n = n ,then it evaluates to zero, (cid:104) Θ (cid:105) = 0. If we reverse the orientation of Θ to get a foam Θ (cid:48) , then (cid:104) Θ (cid:48) (cid:105) = −(cid:104) Θ (cid:105) . In general, if foam F contains k singular circles and F is given by reversing theorientation of F , then (cid:104) F (cid:105) = ( − k (cid:104) F (cid:105) .Recall that our ground ring (cid:101) R is the power series k (cid:74) E , E (cid:75) , where k = Z [ β i,j ] is polynomialsin various negative degree generators with integer coefficients, see formulas (9), (24). Let R be the subring of (cid:101) R = k (cid:74) E , E (cid:75) generated by E , E , ρ , ρ , ρ ± over Z :(37) R = (cid:104) E , E , ρ , ρ , ρ ± (cid:105) ⊂ (cid:101) R = k (cid:74) E , E (cid:75) . In all the examples above, the foam evaluates to an element of this subring. We’ll see soonthat this is true for any closed foam and that the ground ring of the theory can be reducedfrom the rather large power series ring k (cid:74) E , E (cid:75) to the subring R , which is finitely generatedover the image of Z in k .Let us summarize that ρ = (cid:104) S , (cid:105) = p − p x − x ,ρ = (cid:104) S , (cid:105) = x p − x p x − x , (38) ρ = (cid:104) S , (cid:105) = − p p are the evaluations of the thin 2-sphere with zero dots, with one dot, and the double 2-sphere,respectively. The subring R is graded, with homogeneous generators in degreesgenerator ρ ρ ρ E E degree -2 0 0 2 4Notice that only ρ has a negative degree. Using that E − E = ( x − x ) , it’s easy tocompute(39) ρ − E ρ ρ + E ρ = E − E ( x − x ) p p = − ρ. Define the ring(40) R = Z [ E , E , ρ , ρ , ( ρ − E ρ ρ + E ρ ) − ]as the localization of the polynomial ring with generators E , E , ρ , ρ at the element(41) ρ = − ( ρ − E ρ ρ + E ρ ) . There is an obvious homomorphism R −→ R ⊂ (cid:101) R , and we now prove that it’s an isomorphismbetween R and R .Consequently, we can think of R as the localization,(42) R ∼ = Z [ E , E , ρ , ρ , ( ρ − E ρ ρ + E ρ ) − ] . We will show that this localization has a basis over Z [ E , E ]:(43) B := { ρ n ρ n ρ n , n ∈ { , } , n ∈ Z + , n ∈ Z } To establish isomorphism R ∼ = R of rings, denote by(44) R − := Z [ ρ , ρ ± ] (cid:74) E , E (cid:75) the graded ring of power series in E , E with coefficients in the ring Z [ ρ , ρ ± ]. In thisdefinition, we view ρ , ρ as additional generators and not as power series. Lemma 4.2. Ring R − is naturally a subring of (cid:101) R , via power series expansions (38) for ρ and ρ .Proof. We can write the power series p = 1 + Ax + Bx + (cid:88) i + j> β ij x i x j , where A = β , and B = β , , also see formula (6). The power series for ρ is invertible, sincethe expansion starts with 1 + A ( x + x ) = 1 + AE followed by higher degree terms in x and x with coefficients in the variables β ij for i + j > ρ (cid:55)−→ x p − x p x − x = 1 + A ( x + x ) + h.o.t., where h.o.t. stands for ’higher order terms’. Furthermore, the series expansion for ρ doesnot involve the coefficient B . The series ρ (cid:55)−→ p − p x − x = A − B + h.o.t. begins with the element A − B followed by higher degree terms in x and x with coefficientsonly involving the parameters β ij for i + j > 1. Consider the homomorphism τ : R − = Z [ ρ , ρ ± ] (cid:74) E , E (cid:75) −→ (cid:101) R, given by expanding ρ and ρ as power series, so that τ ( ρ ) = A − B + h.o.t., τ ( ρ ) = 1 + AE + h.o.t. To show that τ is injective, compose τ with the involution of (cid:101) R that sends the generator B = β , to τ ( ρ ), and fixes all other generators (generators E , E and β i,j for ( i, j ) (cid:54) = (0 , π : R − −→ (cid:101) R, π ( ρ ) = B, π ( ρ ) = τ ( ρ ) = 1 + AE + h.o.t., π ( E i ) = E i , i = 1 , . Now consider any homogeneous element of degree 2 n in the kernel of π (cid:88) i,j,k, i +2 j = n + k ρ k f ijk ( ρ ) E i E j , DEFORMATION OF ROBERT-WAGNER FOAM EVALUATION AND LINK HOMOLOGY 29 where f ijk ( ρ ) is a Laurent polynomial in ρ . Mapping to (cid:101) R under π and observing that theelements B k are linearly independent over the subring of ˜ R given by power series in E and E with values in the polynomial algebra Z [ β i,j , ( i, j ) (cid:54) = (0 , k ≥ (cid:88) i +2 j =2 n + k f ijk ( τ ( ρ )) E i E j . Notice that for any k , the above expression is a finite sum. So by multiplying by a suitablepower of τ ( ρ ), we may assume that each Laurent polynomial f ijk ( ρ ) is in fact a polynomialin ρ . The algebraic independence of the classes τ ( ρ ) , E , E easily implies that each f ijk ( ρ )must be trivial. In other words, the map π is injective, which is what we wanted to prove. (cid:3) Corollary 4.3. The power series homomorphism R −→ (cid:101) R takes R isomorphically onto thesubring R of (cid:101) R . Moreover, the ring R has a basis over Z [ E , E ] given by B := { ρ n ρ n ρ n , n ∈ { , } , n ∈ Z + , n ∈ Z } . Proof. By definition, the image of R in (cid:101) R is equal to the ring R . Now both rings R and R aregenerated as modules over Z [ E , E ] by the set of elements of B . To be more precise, bothrings R and R have a collection of generators B ( R ) and B ( R ), respectively, as defined abovethat are compatible under the map from R to R . Hence, to demonstrate the isomorphismbetween R and R , it is sufficient to show that the elements B ( R ) are linearly independentover Z [ E , E ] when seen as elements in R , thereby showing that the elements B ( R ) forma Z [ E , E ]-module basis of R . It follows from this that the collection B ( R ) also forms a Z [ E , E ]-module basis of R , and consequently, that the map from R to R is an isomorphism.In what follows, we will actually show that the elements B ( R ) are linearly independentover Z (cid:74) E , E (cid:75) in the larger ring R − , once we observe that the ring R is contained in theimage of R − ⊂ (cid:101) R . For this it suffices to show that ρ − is in R − , which follows from formula(39) that expresses ρ − as a power series in E and E with polynomial coefficients in ρ , ρ ± : ρ − = − ( ρ − E ρ ρ + E ρ ) − = − ρ − (1 − E ρ ρ − + E ρ ρ − ) − , and then formally expanding the inverse as power series. This shows that the inclusion R ⊂ (cid:101) R factors through the subring R − .It remains to show linear independence of the elements B ( R ) over Z (cid:74) E , E (cid:75) inside R − .Since the set of elements { ρ n } are linearly independent over Z (cid:74) E , E (cid:75) , it is sufficient to showthat the sub-collection of B ( R ) given by the elements { ρ n ρ n } is linearly independent over Z [ ρ ] (cid:74) E , E (cid:75) . Let us consider a homogeneous relation(45) 0 = (cid:88) n :=( n ,n ) A n ( ρ , E , E ) ρ n ρ n , where the indexing set is some finite subset of distinct pairs n := ( n , n ) as above with A n ( ρ , E , E ) being a homogeneous element of Z [ ρ ] (cid:74) E , E (cid:75) . Reducing relation (45) mod ρ and using equation (39), we obtain the relation in Z [ ρ ± ] (cid:74) E , E (cid:75) (cid:88) n :=( n ,n ) ( − n A n (0 , E , E ) ρ n +2 n , which is clearly true only if A n (0 , E , E ) = 0 for all n . This condition implies that each A n ( ρ , E , E ) is divisible by ρ . We may therefore factor ρ out of the entire relation (45), and repeat the argument (note that ρ is not a zero divisor). This shows that A n ( ρ , E , E )must be trivial for all n , which is what we needed to establish. (cid:3) Remark 4.4. The inclusion R ⊂ R − is dense in the power series ring topology. In order toshow this, it is sufficient to show that ρ − can be described in terms of a power series in E and E , with coefficients that are polynomials in ρ , ρ , ρ ± . This follows from formula (39)which implies that ρ − = − ρ − (1 + ρ ρ − E ) − ( ρ − ρ E ) . Notice that in addition to the chain of ring inclusions in formulas (23)-(27), there is also achain of inclusions(46) R ⊂ (cid:101) R ⊂ R (cid:48) ⊂ R (cid:48)(cid:48) . The example 6 above for the evaluation of the Θ-foam is straightforward to generalize to GL ( N ), where Θ-foam has a disk of thickness N with N disks of thickness one attached toit, carrying n , . . . , n N dots, respectively, where we can assume n ≥ n ≥ · · · ≥ n N , seeFigure 4.3.3. F IGURE n n n N N Figure 4.3.3. Θ-foam for GL ( N )Let λ i = n i − N + i, so that λ = ( λ , . . . , λ N ) is a partition iff n i > n i +1 for all i . Denotethis foam by Θ λ . One can compute the foam evaluation(47) (cid:104) Θ λ (cid:105) = ± (cid:88) σ ∈ S N ( − (cid:96) ( σ ) (cid:81) Ni =1 x n i σ ( i ) (cid:81) i Figure 4.4.1. Singular neck-cutting relation4.4. Skein relations.Proposition 4.5. The skein relation ( singular neck-cutting relation ) in Figure 4.4.1 holds.Proof: Coloring c of F induces a coloring c (cid:48) of F , F (the latter two foams differ only bydot placement, and we use c (cid:48) to denote corresponding coloring of both foams). Coloring c (cid:48) has opposite colors on the two disks of F (and F ). If a coloring c of F and F has thesame color on the two disks, (cid:104) F , c (cid:105) = (cid:104) F , c (cid:105) , since dots will contribute with the same x i , i ∈ { , } , to the evaluations, and this coloring will not contribute to the difference (cid:104) F (cid:105)−(cid:104) F (cid:105) .Thus, we can restrict to colorings c (cid:48) as above, in bijection with colorings c of F . c c ´ c ´ 121 1 12 2 2 positiveF F F F IGURE Figure 4.4.2. When top facet is colored 1If the top facet of c is colored 1, see Figure 4.4.2, then the circle of F in the figure ispositive and θ + ( c ) = θ + ( c (cid:48) )+1. Also, χ ( F, c ) = χ ( F , c (cid:48) ) = χ ( F , c (cid:48) ), so that − ( − s (cid:48) ( F,c ) =( − (cid:48) ( F ,c (cid:48) ) = ( − (cid:48) ( F ,c (cid:48) ) . We have χ ( F, c ) = χ ( F , c (cid:48) ) − 2, so that (cid:104) F, c (cid:105) has an additional ( x − x ) in thenumerator, compared to (cid:104) F , c (cid:48) (cid:105) and (cid:104) F , c (cid:48) (cid:105) . Due to a dot on facet colored 1 there’s an extra x in (cid:104) F , c (cid:48) (cid:105) and an extra x in (cid:104) F , c (cid:48) (cid:105) . More accurately, we can write (cid:104) F, c (cid:105) = − ( x − x ) y, (cid:104) F , c (cid:48) (cid:105) = x y, (cid:104) F , c (cid:48) (cid:105) = x y for some y , so that (cid:104) F, c (cid:105) = −(cid:104) F , c (cid:48) (cid:105) + (cid:104) F , c (cid:48) (cid:105) . c c ´ 212 2 21 1 1 negativeF F F F IGURE Figure 4.4.3. When top facet is colored 2The other case is when the top facet of F is colored 2 by c , see Figure 4.4.3. In this casethe singular circle of F in the figure is negative for the coloring c , so that θ + ( c ) = θ + ( c (cid:48) ) and s (cid:48) ( F, c ) = s (cid:48) ( F , c (cid:48) ) = s (cid:48) ( F , c (cid:48) ) . This change of sign is balanced by the opposite coloring ofthe two disks in F , F , so that (cid:104) F, c (cid:105) = ( x − x ) y, (cid:104) F , c (cid:48) (cid:105) = x y, (cid:104) F , c (cid:48) (cid:105) = x y for some y , and we still have (cid:104) F, c (cid:105) = −(cid:104) F , c (cid:48) (cid:105) + (cid:104) F , c (cid:48) (cid:105) . Summing over all c implies theproposition. (cid:3) Reversing the orientation of the singular circle (and hence of the entire connected compo-nent of F ) changes the signs in the relation, see Figure 4.4.4 and Proposition 4.9 below. = – F IGURE Figure 4.4.4. Relation for the other orientation Proposition 4.6. The skein relation ( canceling double disks ) in Figure 4.4.5 holds.Proof: There is a bijection between colorings c of F and colorings c of F , see Figure 4.4.6.One checks that χ k ( F , c ) = χ k ( F, c ) + 2, k = 1 , 2, and χ ( F, c ) = χ ( F , c ). For anycoloring, θ + ( c (cid:48) ) ≡ θ + ( c )(mod 2) , since the two singular circles in F have the same parity,and ( − s (cid:48) ( F,c ) = − ( − s (cid:48) ( F ,c ) . Comparing the contributions, (cid:104) F, c (cid:105) = − p p (cid:104) F , c (cid:105) = ρ (cid:104) F , c (cid:105) . Summing over all c , the result follows. (cid:3) Since ρ is invertible, this relation shows that either of the two foams in Figure 4.4.5 can bewritten as the other foam times ρ ± . DEFORMATION OF ROBERT-WAGNER FOAM EVALUATION AND LINK HOMOLOGY 33 F F = F IGURE Figure 4.4.5. Canceling parallel double disks F, c F , c i iij F IGURE Figure 4.4.6. A coloring c of F and the corresponding coloring c of F Reversing orientation of the two singular circles on the left hand side of Figure 4.4.5 givesa similar skein relation, with no sign added since the parity of the number of singular circlesis the same on both sides of the relation. Proposition 4.7. The skein relation ( neck-cutting relation ) in Figure 4.4.7 holds. = – F F F F IGURE Figure 4.4.7. Neck-cutting relation Again, that ρ is invertible, and the relation allows us to do a surgery on an annulus whichis part of a thin facet of F . Proof: Apply Figure 4.4.5 relation to pass to a tube with two double disks and then useFigure 4.4.4 relation to do surgery on the top double disk. (cid:3) Doing the surgery on the bottom double disk using Figure 4.4.1 results in a similar relation,depicted in Figure 4.4.8 where singular disks now appear at the top rather than the bottomon the right hand side. = – F IGURE Figure 4.4.8. Neck-cutting relation with double disks at the top Proposition 4.8. If a double disk D bounding a singular circle in a foam F can be completedto a 2-sphere without additional interections with F , denote by F the foam given by removingthe 2-disk from F and adding its complement in S , see Figure 4.4.9. Then (cid:104) F (cid:105) = −(cid:104) F (cid:105) . double D F F F IGURE Figure 4.4.9. Double disk flipping Proof: There is a bijection between colorings c of F and colorings c of F , with theonly difference in evaluations coming from the type of the singular circle, so that s (cid:48) ( F, c ) = s (cid:48) ( F , c ) ± (cid:104) F, c (cid:105) = −(cid:104) F , c (cid:105) . (cid:3) Proposition 4.9. If F is a foam F with the reversed orientation of all facets, then (cid:104) F (cid:105) =( − k (cid:104) F (cid:105) , where k is the number of singular circles of F . DEFORMATION OF ROBERT-WAGNER FOAM EVALUATION AND LINK HOMOLOGY 35 Proof: Each coloring c of F is a coloring of F as well, and (cid:104) F , c (cid:105) = ( − k (cid:104) F, c (cid:105) , since s (cid:48) ( F , c ) = ( − k s (cid:48) ( F, c ) as the type (positive or negative) of each singular circle of F isreversed in F . Summing over c implies the proposition. (cid:3) This proposition can be applied, for instance, to the neck-cutting relation in Figure 4.4.7.Reversing the orientation of singular circles in F , F reverses the orientation of all facets aswell. Since F has one less singular circle than F , F , there’ll be an additional overall minussign, which can go either to the left or right hand side. Proposition 4.10. For a foam F with a facet with two dots, the relation (cid:104) F (cid:105) = E (cid:104) F (cid:105) − E (cid:104) F (cid:105) hold, where F and F are the foams with one fewer and two fewer dots on the same facet,see Figure 4.4.10. = E – E F F F Figure 4.4.10. Dot reduction relation Proof: Follows, since x i = E x i − E for i = 1 , (cid:3) Proposition 4.11. (Double facet neck-cutting relation) Evaluations of foams F and F inFigure 4.4.11 satisfy (cid:104) F (cid:105) = ρ (cid:104) F (cid:105) . = F F F IGURE Figure 4.4.11. Neck cutting on a double facet Proof: Again, there’s a bijection between colorings c of F and colorings c of F . Differ-ence in the Euler characteristics χ i ( F, c ) = χ i ( F , c ) + 2 , for i = 1 , 2, contributes the term − p p = ρ to the evaluation of (cid:104) F, c (cid:105) compared to that of (cid:104) F , c (cid:105) . Summing over c impliesthe proposition. (cid:3) Proposition 4.12. (Dot migration relations) = E + FF F F IGURE Figure 4.4.12. Dot migration relation = E F IGURE Figure 4.4.13. Second dot migration relation(1) Evaluations of foams F , F and F in Figure 4.4.12 satisfy (cid:104) F (cid:105) + (cid:104) F (cid:105) = E (cid:104) F (cid:105) . (2) Figure 4.4.13 relation holds.Proof: Follows, since these foams differ only by dot placement, any for any coloring c thetwo facets with dots carry opposite colors. These dots contribute x and x to the evaluation.Consequently, (cid:104) F , c (cid:105) + (cid:104) F , c (cid:105) = ( x + x ) (cid:104) F , c (cid:105) = E (cid:104) F , c (cid:105) The same argument implies the second relation. (cid:3) Proposition 4.13. Skein relation in Figure 4.4.14 holds.Proof: Colorings c (cid:48) of F , F that don’t come from colorings of F have the property thatthe front thin bottom and back thin top facets are colored by the same color, see Figure 4.4.15left.The dots on F , F will have the same color and these terms will cancel out from thedifference, with (cid:104) F , c (cid:48) (cid:105) − (cid:104) F , c (cid:48) (cid:105) = 0.The remaining colorings c of F , F are in bijection with colorings c of F , see Figure 4.4.15right. For these colorings we have χ ( F, c ) = χ ( F k , c ) − , χ (cid:96) ( F, c ) = χ (cid:96) ( F k , c ) , (cid:96), k = 1 , . The rest of the computation is similar to that in the proof of Proposition 4.7. If i = 1 , j = 2,one checks that θ + ( c ) = θ + ( c ) and the signs ( − s (cid:48) are the same in the three evaluations. DEFORMATION OF ROBERT-WAGNER FOAM EVALUATION AND LINK HOMOLOGY 37 = – F F F F IGURE Figure 4.4.14. Cutting a tube with two singular edges ii ijij F , F , c F, c F IGURE Figure 4.4.15. Left: a coloring of F , F that does not come from a coloringof F . Right: a coloring c (cid:48) of F and induced coloring c of F , F .Due to difference in χ , the evaluation (cid:104) F, c (cid:105) will acquire ( x − x ) in the numerator comparedto the other two foams. This will be matched by the dots, contributing x to (cid:104) F , c (cid:105) and x to (cid:104) F , c (cid:105) , correspondingly.If i = 2 , j = 1, there will be sign difference ( − s (cid:48) ( F,c ) = − ( − s (cid:48) ( F k ,c ) , k = 1 , 2. Dots willnow contribute x ’s with the opposite indices to the evaluations of F , F , canceling the signdifference, so that again (cid:104) F, c (cid:105) = (cid:104) F , c (cid:105) − (cid:104) F , c (cid:105) . (cid:3) This relation with the opposite singular circles orientation, see Figure 4.4.16, can be ob-tained from that in Figure 4.4.14 by looking at foams there from the opposite side of the plane.Furthermore, rotating foams in Figure 4.4.14 by 180 ◦ (or using dot migration relation in Fig-ure 4.4.12 twice) yields a similar to Figure 4.4.14 relation but with a different distribution ofdots across thin facets. = – F IGURE Figure 4.4.16. Tube-cutting with the other orientation. In a foam F , let γ be a curve that connects two points on singular lines and lies in asingle facet of F , see Figure 4.4.17 left. Let us call such a curve a proper curve. The foam F = m ( γ, F ) on that figure on the right is called the modification of F along γ . In theundeformed case, when p ( x, y ) = 1, GL (2) foam evaluation satisfies (cid:104) F (cid:105) = ± (cid:104) F (cid:105) , with thesign depending on orientation of singular edges of F , see [BHPW, Equation (2.10)]. F (cid:534) F = m ( (cid:534) , F) F IGURE (cid:534) Figure 4.4.17. Modifying foam F along curve γ in a thin facet. γ connectstwo points on the singular set of F .The relation is more subtle in our case. We start with orientations of singular edges asshown on Figure 4.4.18; note that choosing orientation of one edge forces the orientation ofthe other edge of F shown. Choose a coloring c of F and denote by c (cid:48) the correspondingcoloring of F (there’s a bijection between colorings of F and F ), see Figure 4.4.18. ii ij ij c F c F F IGURE i Figure 4.4.18. Notice orientation of singular edges.We have χ ( F , c ) = χ ( F, c ) , χ i ( c ) = χ i ( c ) − , χ j ( c ) = χ j ( c ) . The number of singular circles of F is one less or more of that of F , depending on whetherthe two singular edges shown in F belong to different or the same singular circle.If i = 1 , j = 2 then these circles are negative, they make no contribution to θ + ( c ) and θ + ( c ), and s (cid:48) ( F, c ) = s (cid:48) ( F , c ) , since χ ( c ) = χ ( c ) and θ + ( c ) = θ + ( c ). If i = 2 , j = 1, thecircles are positive and ( − θ + ( c ) = − ( − θ + ( c ) . Also, the Euler characteristics χ ( c ) , χ ( c )differ by two and contribute a sign to the differece as well. We again get s (cid:48) ( F, c ) = s (cid:48) ( F , c ) , so that for any coloring s (cid:48) ( F, c ) = s (cid:48) ( F , c ) . Combining these computations,(48) (cid:104) F, c (cid:105) = p ij (cid:104) F (cid:48) , c (cid:48) (cid:105) . Note that p ij is not an element of our ground ring k (cid:74) E , E (cid:75) and summing this equality overall colorings c of F will not get an immediate relation between evaluations of F and F (cid:48) . DEFORMATION OF ROBERT-WAGNER FOAM EVALUATION AND LINK HOMOLOGY 39 We now look at the oppositely oriented case, see Figure 4.4.19. Circles now carry oppositesigns from that of the previous case, and one can check that s (cid:48) ( F, c ) = − s (cid:48) ( F , c ) in each ofthe cases ( i, j ) = (1 , 2) and ( i, j ) = (2 , . A similar computation now gives ii ij ij c F c F F IGURE i Figure 4.4.19. Singular edges have the opposite orientation from that inthe previous figure(49) (cid:104) F, c (cid:105) = − p ij (cid:104) F , c (cid:105) , which is similar to (48) but with an additional sign.Consider the thin surface F of F and choose a connected component Σ in it. Recallthat we are looking at modifications of F along proper curves γ and now restrict to γ on acomponent Σ. Notice that the double facets at the endpoints of γ are pointing in the samedirection relative to Σ, either both outward or both inward. Also, if we were to redraw F in Figure 4.4.18 keeping orientations of the singular edges but drawing double facets on theopposite side of Σ (’below’, rather than ’above’), the type of the diagram would change tothe one in Figure 4.4.19, and vice versa. Proper disjoint curves or arcs γ , γ ∈ Σ are called complementary if for a coloring c of F they lie in differently colored regions. Thi propertydoes not depend on the choice of c .To a pair ( γ , γ ) of complementary arcs we assign a sign s ( γ , γ ) . Namely, consider thefour double facets of F at the endpoints of γ and γ . If these four facets all point into thesame connected component of R \ Σ, we set s ( γ , γ ) = 1. Otherwise we define s ( γ , γ ) = − s ( γ , γ ) = 1 is shown in Figure 4.4.20. In general, γ , γ don’t have tohave an endpoint on the same singular circle. (cid:534) (cid:534) F IGURE (cid:534) (cid:534) Figure 4.4.20. Complementary proper arcs γ , γ with s ( γ , γ ) = 1 Given complementary proper arcs γ , γ in Σ, we can do commuting modifications along γ , γ to get from F to the foam F = m ( γ , m ( γ , F )) = m ( γ , m ( γ , F )) . Proposition 4.14. For F and F as above, (cid:104) F (cid:105) = s ( γ , γ ) ρ · (cid:104) F (cid:105) . Proof: For a coloring c of F curves γ and γ lie in differently colored regions of Σ, say i and j -colored regions, { i, j } = { , } . When s ( γ , γ ) = 1, orientations on singular edgeswill make one of curves γ the type in Figure 4.4.18 and the other in Figure 4.4.19, with( i, j ) replaced by ( j, i ) in one of these two cases. Using equations (48) and (49), we obtain (cid:104) F, c (cid:105) = − p p (cid:104) F , c (cid:105) = ρ (cid:104) F , c (cid:105) for the corresponding coloring c of F .When s ( γ , γ ) = − 1, orientations on singular edges will make both γ , γ either the typein Figure 4.4.18 or the type in Figure 4.4.19, with ( j, i ) in place for ( i, j ) for one of γ , γ .This will introduce minus sign, with (cid:104) F, c (cid:105) = − ρ (cid:104) F , c (cid:105) . (cid:3) This proposition may be generalized in some cases when one of γ , γ is not a proper arc.One would need γ to be a proper arc in m ( γ , F ), in the region of color opposite to that of γ , with a coloring of F naturally converted to a coloring of m ( γ , F ). We provide an exampleof such pair of arcs in Figure 4.4.21 and leave the details to the reader. (cid:534) (cid:534) F IGURE (cid:534) (cid:534) Figure 4.4.21. Arc γ is not proper but becomes proper in m ( γ , F ) Corollary 4.15. (cid:104) F (cid:105) = ρ (cid:104) F (cid:105) for foams F, F in Figure 4.4.22. This follows from Proposition 4.14 using the pair of arcs in Figure 4.4.23 with s ( γ , γ ) = 1. (cid:3) Corollary 4.16. Figure 4.4.24 relation on foam evaluations holds. The corollary follows at one from the previous one. (cid:3) Prefoams and ground ring reduction. To prove Proposition 4.19 below, it’s convenient to introduce the notion of GL (2) prefoamand its evaluation. An (oriented) GL (2) prefoam (or pre-foam) F has the same local structureas a GL (2) foam, but without an embedding into R . It has oriented thin and double facets,with facets orientations compatible along singular edges as in Figure 4.1.2. In particular, DEFORMATION OF ROBERT-WAGNER FOAM EVALUATION AND LINK HOMOLOGY 41 F F = F IGURE Figure 4.4.22. Notice additional double cap on the foam F , used to createa pair of complementary proper arcs on it. (cid:534) (cid:534) F F IGURE (cid:534) Figure 4.4.23. Arcs γ , γ are complementary proper with s ( γ , γ ) = 1 = F IGURE Figure 4.4.24. The bubble on the thin plane on the LHS foam points itsdouble facet toward us, being on the same side of the thin plane as portionsof the other double facets shown on the LHS.orientations of facets induce orientations of singular circles. Vice versa, an orientation of asingular circle in a connected component of a prefoam will induce orientation on all facets ofthat component.Along each singular edge a preferred facet out of two adjacent thin facets is specified. Onecan encode this choice by an arrow (a normal direction) out of the singular edge and into the thin surface of the pre-foam (the union of its thin facets). A pre-foam may carry dots on itsthin facets.A GL (2) foam F gives rise to a GL (2) prefoam, also denoted F . Embedding of foam F in R together with orientation of singular circles induces an order on the two thin facetsattached to a given singular circle. Namely, look in the direction of the orientation on thecircle and choose the first thin facet counterclockwise starting from the double facet attachedto the circle. This is then the preferred facet for the singular circle in the underlying pre-foam F .Coloring c of a pre-foam is defined in the same way as for foams. For each coloring c surfaces F ( c ) and F ( c ) inherit orientations from the facets of F they contain. Surface F ( c )is orientable as well, say with orientation matching that of thin facets of F ( c ) colored 1 andopposite to that of thin facets colored 2.Orientation requirements for facets ensure that each connected component of the thinsurface F of a prefoam F will admit two checkerboard colorings, so that a prefoam F willadmit 2 k colorings, where k is the number of connected components of F .Given a coloring c of F , the preferred thin facet at a singular circle u allows to label thecircle positive or negative , as in Figure 4.1.3. Namely, if the preferred facet is colored 1, thecircle is positive . If the preferred facet is colored 2, the circle is negative.Define θ + ( c ) = θ +12 ( c ) as the number of positive singular circles for the coloring c .Thus, in a pre-foam F , each singular circle u comes with both an orientation (induced fromthe orientation of attached facets and, vice versa, determining them) and a choice of preferredthin facet (normal direction to the thin surface F ) along u . The evaluation of F , though,will only depend on the choice of preferred facet at each singular circle, not on its orientation.Unlike the foam case, in a GL (2) pre-foam we can reverse the thin normal direction (reversethe choice of preferred thin facet) at any subset of its singular circles without making any otherchanges, such as reversing orientations of facets or singular circles, changing the embeddinginto R , etc. In a foam, the analogous operation of reversing the cyclic order of facets ata single circle via a simple modification of the embedding is possible only sometimes, seeFigure 4.4.9 for an example.Recall the chain of inclusions of rings R ⊂ (cid:101) R ⊂ R (cid:48) ⊂ R (cid:48)(cid:48) defined in formulas (23)-(27) and(37).Now, to a coloring c of a prefoam F we assign an element (cid:104) F, c (cid:105) ∈ R (cid:48)(cid:48) using the formula(21). Furthermore, define (cid:104) F (cid:105) via the formula (22). Proposition 4.17. Evaluation (cid:104) F (cid:105) of any GL (2) prefoam F belongs to the subring k (cid:74) E , E (cid:75) of R (cid:48)(cid:48) .Proof: Our proof of this result for foams, Theorem 4.1, extends to prefoams without change. (cid:3) Proposition 4.18. Evaluation (cid:104) F (cid:105) of any GL (2) prefoam F belongs to the subring R of (cid:101) R = k (cid:74) E , E (cid:75) . Proof: Evaluations of surfaces and theta-foams, with dots, in Section 4.3 depend only onthe pre-foam structure, not on an embedding in R . Skein relations described in Section 4.4extend, with suitable care, to pre-foams. In Figure 4.4.1 relation, a pre-foam on the LHSinduces pre-foam structures on terms on the right, with orientations of facets in the RHS DEFORMATION OF ROBERT-WAGNER FOAM EVALUATION AND LINK HOMOLOGY 43 coming from those of the LHS. With this convention, Figure 4.4.1 relation holds for pre-foams, where in the pre-foam F on the LHS one also remembers the cyclic order of the facetsRelations in Figures 4.4.4, 4.4.5 extend likewise. In Figure 4.4.7 choice of orientations ofall foams (respectively, pre-foams) is encoded in the orientation of the singular circle on theRHS (equivalently, of the cyclic order of the 3 facets at the circle).Analogue of Proposition 4.9 for prefoams is that (cid:104) F (cid:48) (cid:105) = ( − k (cid:104) F (cid:105) , where F (cid:48) is obtainedfrom F by reversing the cyclic order of facets at some k singular circles of F .Figure 4.4.10 relation obviously extends to prefoams. In the double facet neck-cuttingrelation in Figure 4.4.11 relation prefoam F on the right induces an orientation on theprefoam F on the left. With this convention, Figure 4.4.11 relation extends to pr-foams.Dot migration relations in Figures 4.4.12 and 4.4.13 as well as the tube-cutting relation inFigure 4.4.14 extend to prefoams.Modification m ( γ, F ) in Figure 4.4.18 can be done to a prefoam F , assuming compatibleorientations and cyclic orders along the two singular edges of F . Proposition 4.14 will holdfor prefoams as well, again assuming compatibility of the orientations and cyclic orders alongthe three singular edges shown in Figure 4.4.20.Starting with a prefoam F , look at the thin surface F . It may have several connected com-ponents, some of which are connected in F by double facets. Applying the double neck-cuttingrelation in Figure 4.4.11, using multiplicativity of (cid:104) F (cid:105) on the disjoint union of prefoams, andthe evaluation of closed double surfaces (Examples 4, 5 in Section 4.3), we can reduce theevaluation to the case when F is connected and each double facet is a disk. Applying thesingular neck-cutting relation in Figure 4.4.1 along each singular circle of F , the evaluationreduces to that of a closed thin surface, possibly with dots, see Examples 1-3 in Section 4.3.All coefficients in the skein relations and in the evaluation of closed surfaces belong to thering R , implying the proposition. (cid:3) The proposition implies the next result. Proposition 4.19. Evaluation (cid:104) F (cid:105) of any closed foam F coincides with the evaluation of theassociated prefoam. In particular, it belongs to the subring R of (cid:101) R = k (cid:74) E , E (cid:75) . Proof: Foam F lives in R , but to evaluate it using the formulas (21) and (22) we can passto the associated prefoam and evaluate it instead. (cid:3) Consequently, evaluations of all closed foams belong to the subring R of (cid:101) R . It can then bechosen as the ground ring of the theory instead of (cid:101) R , in the GL (2) case.4.6. GL (2) webs, their state spaces, and direct sum decompositions. We define GL (2) closed webs Γ as generic intersections of GL (2) foams with planes R in R . A GL (2) web Γ is a plane trivalent oriented graph with thin and thick (or double ) edgesand vertices as in Figure 4.6.1.Vertices of GL (2) foams may be of two types. In one type, a pair of oriented thin edgesflows into the vertex and a double edge flows out. In the other type, a double edge flows inand a pair of oriented thin edges flows out of the vertex. The web in Figure 4.6.1 has twovertices of each type.Single and double closed loops are allowed, as well as the empty web. The union of thinedges of Γ is called the thin one-manifold of Γ, or the thin cycles of Γ and denoted Γ (1) . ForΓ in Figure 4.6.1, the thin one-manifold Γ (1) has three connected components. Figure 4.6.1. A GL (2) web Γ with two thick edges, four thin edges, onethin and one thick circle.One defines GL (2) foams with boundary a GL (2) web Γ in the usual way. We use Fig-ure 4.1.2 as the convention for the induced orientation of the web that’s the boundary of a GL (2) foam. Note that GL (2) foams F in R × [0 , 1] with the boundary ( − Γ ) (cid:116) Γ , whereΓ i = F ∩ ( R × { i } ), i = 0 , 1, may be viewed as cobordisms between Γ and Γ .Define Foam as the category where objects are GL (2) webs Γ and morphisms from Γ to Γ are isotopy classes (rel boundary) of GL (2) foams with the boundary ( − Γ ) (cid:116) Γ . Compositionis the concatenation of foams.Define the degree of a foam F , not necessarily closed, as(50) deg( F ) = − χ ( F ) + 2 | d ( F ) | , where d ( F ) is the number of dots of F . Thin surface F of F is well-defined for foams withboundary. The boundary of F is the union of thin circles on the boundary of F .For closed foams F , deg( F ) equals the degree of (cid:104) F (cid:105) , viewed as a homogeneous element ofeither (cid:101) R or its subring R . Degree of a foam is additive under composition of foams.We define the state space (cid:104) Γ (cid:105) of a GL (2) web Γ using the universal construction asin [BHMV, Kh2].First, let Fr(Γ) be the free graded R -module with a basis { [ F ] } F , over all foams F from theempty web to Γ. The degree of the generator [ F ] is defined to be deg( F ). Define a bilinearform on Fr(Γ) by(51) ([ F ] , [ G ]) = (cid:104) w ( G ) F (cid:105) , where w ( G ) is the reflection of G in the horizontal plane together with the orientation reversalof all facets of G to make F and w ( G ) composable along F . The foam w ( G ) F is closed andcan be evaluated to an element of R . Given a closed foam H , reflecting it about a planeinto a foam H (cid:48) may add sign to the evaluation, (cid:104) H (cid:48) (cid:105) = ( − k (cid:104) H (cid:105) , where k is the number ofsingular circles of H . To get rid of the sign, reverse orientation of all facets of H (cid:48) to get a foam w ( H ) with (cid:104) w ( H ) (cid:105) = (cid:104) H (cid:105) . A similar argument works for non-closed foams. Consequently,the bilinear form (51) is symmetric.Define the state space (cid:104) Γ (cid:105) as the quotient of Fr(Γ) by the kernel of the bilinear form ( , ). Thestate space (cid:104) Γ (cid:105) is a graded R -module, via the degree formula (50). As usual in the universal DEFORMATION OF ROBERT-WAGNER FOAM EVALUATION AND LINK HOMOLOGY 45 construction, a foam F with boundary ( − Γ ) (cid:116) Γ induces a homogeneous R -module map (cid:104) F (cid:105) : (cid:104) Γ (cid:105) −→ (cid:104) Γ (cid:105) of degree deg( F ) taking an element (cid:104) G (cid:105) ∈ (cid:104) Γ (cid:105) associated to a foam G with boundary Γ to the element (cid:104) F G (cid:105) associated to the foam F G with boundary Γ . These maps assembleinto a functor from the category of GL (2) foams to the category of graded R -modules andhomogeneous R -module maps. The results below imply that the functor is monoidal.The state space of the empty web is naturally isomorphic to the free rank one module over R with a generator in degree zero, (cid:104)∅(cid:105) ∼ = R .Let Γ (cid:48) denote the web Γ with an innermost thin circle (with one of the two orientations)added in a region of Γ. Thus, Γ (cid:48) depends on the choice of a region of Γ and the orientationof the circle. Proposition 4.20. There are natural isomorphisms of graded R -modules (cid:104) Γ (cid:48) (cid:105) ∼ = (cid:104) Γ (cid:105){ } ⊕ (cid:104) Γ (cid:105){− } , for Γ , Γ (cid:48) as above and { m } the grading shift up by m . , – –1 { 1 }{ –1 }F IGURE Figure 4.6.2. Direct sum decomposition for an innermost thin circle withthe clockwise orientation. For the opposite orientation of the circle, reversethe orientation of the singular circle as well and add an overall minus sign toone of the two maps. Proof: Foam cobordisms that deliver this direct sum decomposition are shown in Fig-ure 4.6.2. The composition of the maps in either order is the identity, as follows from Θ-foamevaluations in section 4.3 and neck-cutting relation in Figure 4.4.7. (cid:3) Proposition 4.21. The saddle cobordism in a thick facet induces a grading-preserving iso-morphism between the state spaces of its two boundary webs, see Figure 4.6.3. The inverseisomorphism is given by the adjoint saddle cobordism scaled by ρ − .Proof: This follows from the thick neck-cutting relation in Figure 4.4.11. (cid:3) –1 F IGURE Figure 4.6.3. Saddle isomorphism on a double facet. Double edges on theleft and right must carry compatible orientation (that is, extendable to theorientation of the surface). Proposition 4.22. Let Γ be a web and Γ (cid:48) be Γ with added innermost thick circle. There is acanonical degree zero isomorphism of state spaces (cid:104) Γ (cid:48) (cid:105) ∼ = (cid:104) Γ (cid:105) given by the cobordisms in Figure 4.6.4. –1 F IGURE Figure 4.6.4. An isomorphism between a diagram with an innermost doublecircle and the diagram without it, via double cup and cap cobordisms. Proposition 4.23. Let web Γ have a thin edge and denote by Γ (cid:48) the web Γ with an attacheddouble edge along the thin edge. The state spaces of Γ and Γ (cid:48) are naturally isomorphic asgraded R -modules via the maps given in Figure 4.6.5.Proof: This follows from relations in Figures 4.4.24 and 4.4.5. (cid:3) Theorem 4.24. (cid:104) Γ (cid:105) is a free graded R -module of graded rank [2] m , where m is the numberof components (circles) of the thin one-manifold Γ (1) and [2] = q + q − .Proof: This can be proved by induction on m . An innermost thin or double circle of Γ, seeFigure 4.6.6, can be removed using isomorphisms in Figures 4.6.2 and 4.6.4, respectively. DEFORMATION OF ROBERT-WAGNER FOAM EVALUATION AND LINK HOMOLOGY 47 F IGURE Figure 4.6.5. Mutually-inverse isomorphisms between a thin edge and athin edge with an attached double edge. Figure 4.6.6. Thin and double innermost circles.Now look at Γ (1) and choose an innermost circle α in it. We distinguish between innermostcircles of Γ and those of Γ (1) . The latter correspond to thin circles in Γ which may containvertices and thus have attached double edges. α bounds a disk D in R . Double edgesemanating out of α split into those inside and outside of D . Repeatedly applying the doublesaddle isomorphism in Figure 4.6.3, we can reduce to the case when each of these double edgeshas both endpoints on α . Going along α one encounters 2 n vertices (an even number dueto orientation reversal along α at each vertex). If at two consecutive vertices double edgesboth point in or out of D , one can apply an isomorphism in Figure 4.6.3 followed by anisomorphism in Figure 4.6.5 to reduce from 2 n to 2 n − α . A configurationwhere such a pair of vertices does not exist is impossible for n > 0, for then the n ends ofdouble edges pointing into D from α would all have the same orientations and there would beno room for the other n ends of these edges to land. This concludes the inductive argument. (cid:3) Corollary 4.25. Associating the state space (cid:104) Γ (cid:105) to a GL (2) web Γ and the map (cid:104) F (cid:105) of statespaces to a foam F with boundary is a monoidal functor from the category of GL (2) foams tothe category of free graded R -modules of finite rank. Reidemeister moves invariance and link homology With the state spaces (cid:104) Γ (cid:105) of GL (2) webs Γ defined, we can associate homology groups toa generic projection D of an oriented link L ⊂ R , as follows. Let D has n crossings. Weresolve each crossing into two resolutions, 0- and 1-resolutions, as in Figure 5.0.1. positive0 – resolution 1 – resolution 0 – resolution 1 – resolutionnegative Figure 5.0.1. Resolutions of a positive and a negative crossings.One of the resolutions consists of two disjoint thin edges, the other contains a double edgeand four adjoint thin edges. All the edges are oriented. Choose a total order on crossingsof D . Doing this procedure over all crossings results in 2 n resolutions of D into GL (2) webs D ( µ ), for µ = ( µ , . . . , µ n ), with µ i ∈ { , } . In a web D ( µ ) the i -th crossing is resolvedaccording to µ i .To a crossing now associate a complex of two webs with boundaries and the differentialinduced by the ”singular saddle” cobordism between them, see Figure 5.0.2 which sets us theterms in the complex, and Figure 5.0.3 which depics ”singular saddle” foams inducing thedifferential. These complexes make sense whenever the two webs are closed on the outside positivenegative { 2 }0 0{ 1 }{ – – – Figure 5.0.2. Complexes associated to positive and negative crossings.Numbers at the top show homological gradings of the terms. Resolution intotwo edges is always in homological degree 0.into two closed GL (2) webs. Grading shifts are inserted to make the map induced by the”singular saddle” cobordism grading-preserving (and, later, to have full invariance under theReidemeister I move, rather than an invariance up to an overall grading shift).In this way, one can form a commutative n -dimensional cube which has the graded R -module (cid:104) D ( µ ) (cid:105) in its vertex labelled by the sequence µ and maps induced by ”singular saddle”foams associated to oriented edges of the cube. The maps commute for every square of thecube. DEFORMATION OF ROBERT-WAGNER FOAM EVALUATION AND LINK HOMOLOGY 49 d d Figure 5.0.3. Foams that induce the differential in the complexes for apositive and negative crossings. Upward-pointing arrows next to the foamsindicate the ’direction’ of the differential.This setup with ”singular saddle” cobordisms goes back to Blanchet [B], and is also visiblein the earlier papers of Clark-Morrison-Walker [CMW] and Caprau [Ca1, Ca2], where thedouble facet is not there, but its boundary, a singular edge along the foam, together with achoice of normal direction, is present.The commuting cube of graded R -modules (cid:104) Γ( µ ) (cid:105) and grading-preserving homomorphismsbetween them collapses, in the standard way upon adding minus signs, to a complex of graded R -modules with a degree-preserving differential. This complex starts in the homological degree– minus the number of negative crossings of D and ends in the homological degree which isthe number of positive crossings of D .Denote this complex by F ( D ). Theorem 5.1. For two diagrams D and D of an oriented link L , complexes F ( D ) and F ( D ) are chain homotopy equivalent as complexes of graded R -modules.Proof: Consider the Reidemeister move R1, undoing a positive curl in Figure 5.0.4. D ˜ D Figure 5.0.4. Reidemeister move R1, for a positive twist. Proposition 5.2. The following relations hold on maps f , g , h and d in Figures 5.0.5, 5.0.6: dh = id , (52) df = 0 , (53) g f = id F ( D ) , (54) id = f g + hd. (55) The map id in the first equation is the identity of the complex F ( D (1)) , associated to thediagram in the top right corner of Figure 5.0.5, while id in the last equation is the identity ofthe complex F ( D (0)) associated to the diagram the top left corner of the figure. F ( D ) :F ( D ) : ƒ d = h = g = – – Figure 5.0.5. Top row, together with the right-pointing arrow d , encodesthe complex F ( D ). Top left-pointing arrow h is a self-homotopy of F ( D ).Down and up arrows h and g are maps of complexes F ( D ) and F ( D ). Map f is given in the next figure. – E ƒ = + Figure 5.0.6. Map f : F ( D ) −→ F ( D ) of complexes. Proof is a direct computation using skein relations derived in Section 4.4. This proof isvery similar to the proof of the invariance under the Reidemeister move in [MSV], that doesit in the non-equivariant GL ( N ) case, in particular see Figure 8 there. (cid:3) Corollary 5.3. Complexes F ( D ) and F ( D ) , for diagrams in Figure 5.0.4, are chain ho-motopy equivalent as complexes of graded R -modules. Proposition 5.4. For each pair of the diagrams D , D in Figure 5.0.7, which shows Rei-demeister moves R2 and R3, complexes F ( D ) and F ( D ) are chain homotopy equivalent ascomplexes of graded R -modules. DEFORMATION OF ROBERT-WAGNER FOAM EVALUATION AND LINK HOMOLOGY 51 ˜ ˜ Figure 5.0.7. Reidemeister moves R2 and R3. Proof: For the Reidemeister R2 move, relation (4.4.16) used in the direct sum of decompo-sition of a web Γ with a digon facet into the sum of two copies of the simpler web Γ is nodifferent from the corresponding decomposition in the usual SL ( N ) graphical calculus, for anarbitrary N (see [Kh2, Proposition 8] for the analogous decomposition in the non-equivariant SL (3) case). As one of the relations for this decomposition, the relation of removing a bubbleon a double facet with at most one dot on one of the two thin facets is identical with thecorresponding relation in the usual SL ( N ) foam calculus, whether for the standard calculusor the equivariant one. Bubble removal relation follows from the combination of theta foamevaluation in Example 6 in Section 4.3 for n , n ≤ SL (3) case see, for instance, the top two relations in [Kh2, Figure18].For this reason, the usual proof of the Reidemester R2 relation, when both strands areoriented in the same direction, as in Figure 5.0.7 left, repeats without any changes in ourcase, see for instance [Kh2, Section 5.2], [MSV, Theorem 7.1], and many other sources. D D D' D' Figure 5.0.8. Two partial resolutions of each of D and D (cid:48) . Note that D and D (cid:48) are identical diagrams.Consider the Reidemeister R3 move in Figure 5.0.7. Denote by D and D (cid:48) the diagrams onthe left and right of this move.We start by resolving a single crossing in each of D and D (cid:48) , see Figure 5.0.8. Complexes C ( D ) and C ( D (cid:48) ) are isomorphic to cones of maps C ( D ) −→ C ( D ) and C ( D (cid:48) ) −→ C ( D (cid:48) )built out of foams between complete resolutions of these diagrams.Tangle diagrams D and D (cid:48) are canonically isomorphic, and their resolutions result in thetotal complex of the square shown in Figure 5.0.9 with the differential coming from the fourfoams associated to the arrows of the diagram, with each foam a standard singular saddle inthe appropriate position.Consider now the diagram D and its resolution in Figure 5.0.10. Maps ψ k , k = 1 , . . . , D (00) D (10) D (01) D (11) Figure 5.0.9. Resolution of the diagram D ∼ = D (cid:48) .foams). Summing over all possible resolutions of crossings of D not shown in the diagramgives homomorphisms, also denoted ψ , . . . , ψ , of corresponding complexes.The four terms C ( D ( k(cid:96) )), k, (cid:96) ∈ { , } , will also map to the corresponding four terms C ( D ( k(cid:96) )) in C ( D ) in Figure 5.0.9 to constitute a 3-dimensional cube diagram (not shown).The complex C ( D (00)) of the diagram in the upper left of Figure 5.0.10 is isomorphic (andnot just homotopy equivalent) to the complex C ( D ) of the diagram D shown in Figure 5.0.11left.Foam F going from D (00) to D ’straightens out’ the long thin arc u of D (00) bycanceling in pairs the four vertices on this arc where double edges meet u . Arc u becomesthe rightmost arc u of D . Seam edges that cancel the four vertices in pairs are shown inFigure 5.0.12 as two arcs in the upper half of the diagram. The upper half shows the thinfacet of F where singular vertices along u are cancelled in pairs. These cancellations aredone via singular arcs, shown in Figure 5.0.12 top, along which double facets are attached tothe thin facet.Foam F goes back from D to D (00) and is given by reflecting F in the horizontal plane.The thin facet of F is shown as the lower half of Figure 5.0.12. Semicircles depict singularedges along the thin facet.Denote the maps F , F induce on state spaces and on complexes built out of the statespaces of all resolutions of D (00) and D by(56) τ : C ( D (00)) −→ C ( D ) , τ : C ( D ) −→ C ( D (00)) . DEFORMATION OF ROBERT-WAGNER FOAM EVALUATION AND LINK HOMOLOGY 53 D (00) φ φ φ – φ D (10) δ D (01) D (11) Figure 5.0.10. Resolution of the diagram D . D (00) D F F F F u u u u Figure 5.0.11. Diagrams D (00) and D have isomorphic state spaces forany resolution of these diagrams. Complexes C ( D (00)) and C ( D ), with thedifferentials induced by various foams between their resolutions, are canonicallyisomorphic, C ( D (00)) ∼ = C ( D ) in the abelian category of complexes (beforefactoring by homotopies).We know that both τ and τ are isomorphisms of the state spaces and corresponding com-plexes, since annihilating a digon facet with a thick edge is an isomorphism, see Proposi-tion 4.23. F F F F u u u u u u Figure 5.0.12. Flattened thin facets of F and F containing arcs u , u .Composition F F contains thin surface S (shown on the right) given by gluingthe two thin surfaces along the common arc u . This surface has two singularcircles where double facets attach.More precisely, τ τ = − ρ − Id. Indeed, the composition τ τ is an endomorphism of thestate space (cid:104) Γ (cid:105) for each web resolution Γ of D and the induced endomorphism of the complex C ( D ). The map τ τ : (cid:104) Γ (cid:105) −→ (cid:104) Γ (cid:105) transforms arc u of the diagram D to the arc u of D (00) and back, via the composition of foams F F .Consider the thin surface S bounded by u at the top and bottom of the cobordism F F .It can be visualized by gluing the two thin surfaces for F and F shown in Figure 5.0.12along the common arc u , shown in red. Surface S contains two nested singular circles, wheredouble facets of F F meet S . Double facets at these two circles attach to S from oppositesides, as one can glean from Figure 5.0.11. This corresponds to having two double edgesattached to arc u on one side and the other double edge attached to u on the other side ofthe plane, at both endpoints, see the leftmost diagram in Figure 5.0.11.Apply Proposition 4.11 at each of these attached double facets to simplify the non-trivialpart of the foam F F to the surface S with two double disks attached to it from the oppositesides along the two singular circles, with an additional factor ρ − . We then apply Propo-sition 4.8 to flip one of the disks to the opposite side, gaining a minus sign, and then useProposition 4.11 to reduce to the identity foam times − ρ − .Consequently, maps τ and − ρτ are mutually-inverse isomorphisms.Note that diagrams D (11) and D are isotopic and their complexes are canonically iso-morphic. Complex C ( D (01)) decomposes into direct sum of two copies of D (11) in theusual way. The composition ψ τ : C ( D ) −→ C ( D (01)) is a split inclusion into one of thesecopies.Since τ is an isomorphism, this composition allows to split off contractible summand0 −→ C ( D (00)) ∼ = −→ im( ψ ) −→ D , also see Figure 5.0.10. The map ψ induces an isomorphism fromthe complementary direct summand of C ( D (01)), also isomorphic to C ( D ), to C ( D (11)),allowing to split the second contractible summand from C ( D ). After removing these con-tractible summands, the entire complex C ( D ) in Figure 5.0.10 is downsized to C ( D (10)). DEFORMATION OF ROBERT-WAGNER FOAM EVALUATION AND LINK HOMOLOGY 55 The inclusion C ( D (10)) ⊂ C ( D ) realizing this chain homotopy equivalence is given in coor-dinates by (id , δ ) , see Figure 5.0.10 with δ the diagonal map induced by the simplest cobordismfrom D (10) to D (01), with the property ψ = ψ δ . Figure 5.0.13. Common reduction of C ( D ) and C ( D (cid:48) ).Reducing the map of complexes C ( D ) −→ C ( D ) to the map C ( D (10)) −→ C ( D ) viathe above inclusion of complexes results in the complex shown in Figure 5.0.13, with all arrowsgiven by maps induced by the elementary foams between these webs. Signs need to be addedto make each square anticommute, but the isomorphism class of the complex does not dependon the distribution of signs. This complex has an obvious symmetry given by reflecting alldiagrams and foams about the vertical axis (or plane, in case of foams) and permuting topand bottom terms in the complex.The cone of the map C ( D (cid:48) ) −→ C ( D (cid:48) ) in Figure 5.0.8 right reduces to the isomorphiccomplex, by removing contractible summands of C ( D (cid:48) ) in the same fashion as for C ( D ). (cid:3) This completes the proof of Theorem 5.1. (cid:3) Our proofs of the Reidemeister R2 and R3 relations, for upwards orientations and in N = 2case, are essentially identical to those in the usual equivariant case, when p ij = 1. Thisobservation mirrors our earlier Theorem 3.17 and Remark 3.18 that our deformation doesnot change the nilHecke algebra relation. This makes it likely that our p ( x, y ) deformationdoes not modify the Soergel category and that the Soergel category will act in the deformedsituation as well, with the proofs of Reidemeister R2 and R3 moves for upward orientationsidentical to that in the p ( x, y ) = 1 case. Then p ( x, y ) deformation would only modify the firstReidemeister move and Reidemeister moves R2 and R3 for non-braid orientations of strands.This expectation mirrors our observation that p ( x, y ) may only contribute to the deformationof the Frobenius structure, not of multiplication. In the N = 2 case, similar deformations canbe hidden at the level of link homology, see [V, Kh4]. 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