aa r X i v : . [ m a t h . G T ] F e b ON TRANSVERSE INVARIANTS FROM KHOVANOV-TYPEHOMOLOGIES
CARLO COLLARIA bstract . In this article we introduce a family of transverse invariants arisingfrom the deformations of Khovanov homology. This family includes the invari-ants introduced by Plamenevskaya and by Lipshitz, Ng, and Sarkar. Then, we in-vestigate the invariants arising from Bar-Natan’s deformation. These invariants,called β -invariants, are essentially equivalent to Lipshitz, Ng, and Sarkar’s invari-ants ψ ± . From the β -invariants we extract two non-negative integers which aretransverse invariants (the c -invariants). Finally, we give several conditions whichimply the non-effectiveness of the c -invariants, and use them to prove severalvanishing criteria for the Plamenevskaya invariant [ ψ ] , and the non-effectivenessof the vanishing of [ ψ ] , for all prime knots with less than 12 crossings.
1. I ntroduction
Motivations and background. A contact -manifold is a 3-dimensional (smooth)manifold M endowed with a totally non-integrable plane distribution ξ . A link in M is a smooth embedding of a number of copies of S into M . A link iscalled transverse with respect to ξ if it is nowhere tangent to ξ . Two transverselinks in ( M , ξ ) are equivalent if they are ambient isotopic through a one-parameterfamily of transverse links. Transverse links are a central object of study in low-dimensional contact topology. In this paper we are concerned with the study oftransverse links in a special setting. Fix a system of coordinates ( x , y , z ) on R .The symmetric contact structure is the plane field ξ sym = ker ( dz + xdy − ydx ) . Sincewe are interested only in transverse links in ( R , ξ sym ) , henceforth we shall restrictourselves to this setting. Once an orientation of R is fixed, each transverselink inherits a natural orientation from the contact structure ([8]). All transverseinvariants defined in this paper are defined with respect to this orientation. Remark . With an abuse of language, the words “link”,“knot”,“transverse link”and “transverse knot” shall denote both embeddings and equivalence classes ofembeddings.A natural way to tackle the problem of classifying transverse links is to pro-duce invariants capable of distinguishing them. It follows immediately from thedefinitions that the underlying link (i.e. the ambient isotopy class of the ori-ented link) is a transverse invariant. Another well-known invariant for transverselinks is the self-linking number , which is denoted by sl . The underlying link andthe self-linking number are called classical invariants . These are by no meanscomplete invariants for transverse links. Nonetheless, there are links, which arecalled simple , whose transverse representatives are classified by their self-linkingnumber (e.g. unknot, the positive torus knots). Among the earliest examples of For transverse links there is a refinement of the self-linking number which is called the self-linking matrix. Strictly speaking, the classical invariants for transverse links are the underlying linkand the self-linking matrix. However, for consticency with the literature ([16, 20]) we are going to usethe self-linking number instead of the self-linking matrix. non-simple links there are those given by Etnyre and Honda ([9]), and by Birmanand Menasco ([4]). More precisely, Birman and Menasco described a construc-tion which yelds pairs of non-equivalent transverse links with the same classicalinvariants: the negative flype . We call a pair of (non-equivalent) transverse linkswith the same classical invariants a non-simple pair . Any transverse invariant cap-able of distinguishing the elements of a non-simple pair is called effective . Remark . There is a paucity of non-simple pairs, whose crossing number issufficiently small, which are not obtained via negative flype (cf. [16, Section 1and Subsection 4.5]). Therefore, there is a lack of sufficiently simple exampleswhere to test the effectiveness of transverse invariants which are not capable ofdistinguishing negative flypes.In classical link theory, the well-know theorems of Alexander and Markovassert, respectively, that any oriented link in R can be represented as the closureof a braid, and that all these representations are related by a finite sequence ofcombinatorial moves (called Markov moves). In the theory of transverse linksthere are similar results; the first, due to Bennequin ([3]) states that all transverselinks can be represented as closed braids. The second result, due to Orevkovand Shevchishin ([19]) and, independently, to Wrinkle ([27]), provides a completeset of combinatorial moves relating all braids whose closure represent the sametransverse link. We summarise these results into the following theorem, whichshall be referred as the transverse Markov theorem in the rest of the paper. Theorem 1.1 (Bennequin [3], Orevkov and Shevchishin [19], Wrinkle [27]) . Anytransverse link is transversely isotopic to the closure of a braid (with axis the z-axis).Moreover, two braids represent the same transverse link if and only if they are relatedby a finite sequence of braid relations, conjugations, positive stabilizations, and positivedestabilizations . These moves are called transverse Markov moves . Remark . Braids are naturally oriented, and their orientation coincide with theorientation of the corresponding transverse link.
Remark . By adding the negative stabilization to the set of transverse Markovmoves one recovers the full set of
Markov moves .Any sequence of Markov moves between braids, naturally translates into asequence of oriented Reidemeister moves between their closures. In particular,conjugation in the braid group can be seen as a sequence of Reidemeister movesof the second type followed by a planar isotopy, while the braid relations canbe seen as either second or third Reidemeister moves. We remark that not allthe oriented versions of second and third Reidemeister moves arise in this way;those which can be obtained as composition of Markov moves and braid relationsare called braid-like or coherent . Finally, a positive (resp. negative) stabilizationtranslates into a positive (resp. negative) first Reidemeister move, as shown inFigure 1.Let B be a braid. The self-linking number of B is defined as follows: sl ( B ) = w ( B ) − b ( B ) ,where w denotes the writhe , and b denotes the braid index (i.e. the number ofstrands). In the light of the transverse Markov theorem, it is immediate that the For coerence, we use the word “flype” with the same meaning as in [16]. Let B ∈ B m − , the positive (resp. negative ) stabilization of B ∈ B m − is the braid B σ m ∈ B m (resp. B σ − m ∈ B m ). The destabilization is just the inverse process: if one considers a braid of the form A σ m B (resp. A σ − m B ), where A , B ∈ B m − , then its positive (resp. negative) destabilization is the braid AB . N TRANSVERSE INVARIANTS FROM KHOVANOV-TYPE HOMOLOGIES 3 B Stabilization ⇄ Destabilization B F igure
1. Stabilization and destabilization as Reidemeistermoves between closures.self-linking number is a transverse invariant. (The integer sl ( B ) is indeed theself-linking number of the transverse link represented by B .) Moreover, from thedefinition of sl follows easily that a negative stabilization (resp. destabilization)does not preserve the transverse link.Now, let us turn to the known results concerning transverse invariants inKhovanov-type homologies. By Khovanov-type homologies we mean all link ho-mology theories obtained from a given Frobenius algebra using the constructionoriginally due to Khovanov ([1, 12]). This construction shall be reviewed more indetail in Section 3. For the moment the reader should keep in mind that, given aFrobenius algebra F (over a ring R ), there is a way to associate to each orientedlink diagram D a chain complex C •F ( D , R ) . Moreover, this complex is combinat-orially defined and can be endowed with either a second grading or a filtration.Furthermore, it is possible to associate to each sequence of Reidemeister movesbetween two diagrams a chain homotopy equivalence between the correspondingcomplexes.To each braid we associate the chain complex corresponding to its closure.Similarly, the map associated to a sequence of Markov moves between two braidsis the map associated to the corresponding sequence of Reidemeister movesbetween the braid closures.The first invariant for transverse links in a Khovanov-type homology is due toPlamenevskaya ([20]). Given a braid B , its Plamenevskaya invariant is a homologyclass [ ψ ( B )] , of bi-degree ( sl ( B )) , in the Khovanov homology of b B . This homo-logy class is invariant in the following sense; given two braids B and B ′ relatedby a sequence of transverse Markov moves Σ , we have ( Φ Σ ) ∗ [ ψ ( B )] = [ ψ ( B ′ )] ,where Φ Σ denotes the map associated to Σ . After Plamenevskaya’s groundbreak-ing work, invariants of similar flavour have been introduced in Heegard-Floerhomologies. For example, the Ng-Oszváth-Thurston b θ -invariant in [ HFK ([18]).The vanishing of this invariant is an effective invariant; there exist a non-simplepair of transverse knots, say T and T ′ , such that: b θ ( T ) = = b θ ( T ′ ) .Since the b θ -invariant can be interpreted as the Heegaard-Floer analogue of thePlamenevskaya invariant, it is natural to ask whether or not the vanishing of [ ψ ] is effective. This is an open question at the time of writing.More recently, Lipshitz, Ng and Sarkar introduced (in [16]) two invariants inthe Khovanov-type homology associated to the Frobenius algebra TLee (see Sub-section 2.2). The chain complex associated to
TLee is naturally filtered, denote by F i C • TLee this filtration. The
Lipshitz-Ng-Sarakar ( LNS ) invariants are two elements CARLO COLLARI ψ + ( B ) and ψ − ( B ) in F sl ( B ) C TLee ( b B , R ) , and are invariant in a sense which is madeprecise in the following proposition. Proposition 1.2 (Theorem 4.2 and Proposition 4.7 of [16]) . Let B and B ′ be twobraids. If Φ is the map induced by a sequence of transverse Markov moves from B to B ′ ,then Φ ( ψ ∗ ( B )) = ± ψ ∗ ( B ′ ) + d TLee θ , where θ ∈ F sl ( B ) C − TLee ( b B ′ , R ) and ∗ ∈ { + , −} . Moreover, if Φ is the map induced by anegative stabilization (resp. destabilization), then Φ ( ψ ∗ ( B )) = ± ψ ∗ ( B ′ ) + d TLee θ ′ ∈ F sl ( B ′ ) C − TLee ( b B ′ , R ) ( resp. F sl ( B ) C − TLee ( b B ′ , R ) ) , where θ ′ ∈ F sl ( B ′ ) C − TLee ( b B ′ , R ) (resp. θ ′ ∈ F sl ( B ) C − TLee ( b B ′ , R ) ) and ∗ ∈ { + , −} . Remark . The maps associated to the Reidemeister moves (and also the set ofReidemiester moves) used by Lishitz, Ng and Sarkar, are different from the onesused in the present paper.As the reader may notice, the filtration plays a key role. Lipshitz, Ng andSarkar defined also a family of auxiliary invariants ψ ± p , q , for p ≤ < q , as theimages of ψ ± under the composition F sl ( B ) C TLee ( b B , R ) ֒ → F sl ( B )+ p C TLee ( b B , R ) ։ F sl ( B )+ p C TLee ( b B , R ) F sl ( B )+ q C TLee ( b B , R ) In particular, one obtains that ψ + ( B ) = ψ − ( B ) can be identified with ψ ( B ) . Asa consequence, the non-effectiveness of the Lipshitz-Ng-Sarkar invariants impliesthe non-effectiveness of (the homology class of) the Plamenevskaya invariant. Tothis end, it has been proved by Lipshitz, Ng and Sarkar ([16]) that ψ ± cannot dis-tinguish non-simple pairs obtained via negative flypes, and all non simple pairswhose underlying knot is a prime knot with crossing number <
12 with the ex-ception of 29 cases (see [16, Subsection 4.5]). In particular, the Plamenevskayainvariant (and its vanishing) is non-effective for the transverse links describedabove. Moreover, Lipshitz, Ng and Sarkar proved that the Plamenvskaya in-variant cannot distinguish transverse links which become transversely isotopicafter a single negative stabilization. However, it is still unknown (to the best ofthe author’s knowledge) whether there are non-simple pairs representing the re-maining 29 knots with less than 12 crossing, and if the Plamenevskaya invariant,its vanishing or the LNS invariants can be used to distinguish them.
Statement of results.
The aim of this paper is to investigate the transverse in-formation contained in Khovanov-type homologies. We start by providing a ma-chinery to produce transverse invariants in Khovanov-type homologies, general-ising the Plamenevskaya and LNS invariants. One of the interesting features ofour construction is that it can be generalised to equivariant Khovanov-Rozanskyhomologies ([14]). This will be the subject of forthcoming work.Going back to the present paper, in Section 4 we show how to associate toeach oriented link diagram D , and each Frobenius algebra F (over the ring R ) as inSubsection 2.2, two cycles denoted by β F ( D , R ) and β F ( D , R ) . One of the mainresults in this paper is the following Theorem 1.3.
Let F be a Frobenius algebra as in Subsection 2.2. Given a braid B, thereexist two (possibly equal) cycles β F ( B ) = β F ( b B , R ) , β F ( B ) = β F ( b B , R ) ∈ C F ( b B , R ) such that: if the braids B and B ′ are related by a sequence Σ of transverse Markov movesthen the map Φ Σ : C •F ( b B , R ) −→ C •F ( b B ′ , R ) , N TRANSVERSE INVARIANTS FROM KHOVANOV-TYPE HOMOLOGIES 5 induced by Σ is such that Φ Σ ( β F ( B )) = β F ( B ′ ) Φ Σ ( β F ( B )) = β F ( B ′ ) . Moreover, if Σ consists of a single negative stabilization then ( x − x ) Φ Σ ( β F ( B )) = ± β F ( B ) + d F x ( x − x ) Φ Σ ( β F ( B ′ )) = ± β F ( B ′ ) + d F y , for some x , y ∈ C •F ( B ′ ) , where x and x are defined as in Subsection 2.2. Remark . In this paper, the maps associated to a generating set of Reidemeistermoves are those commonly used through the literature (see for example [1, 12,25]). The maps associated to the other Reidemeister moves are obtained by com-position.The cycles β F ( B ) and β F ( B ) are called β F -invariants. The cycle ψ ( B ) underly-ing the Plamenevskaya invariant belongs to this family since β Kh ( B ) = β Kh ( B ) = ψ ( B ) . With an abuse of notation we refer to ψ ( B ) as the Plamenevskaya invari-ant, and denote by [ ψ ] the original Plamenevskaya invariant when confusion canarise. Similarly, the LNS invariants ψ + ( B ) and ψ − ( B ) are β TLee ( B ) and β TLee ( B ) ,respectively. Motivated by the common properties between the Plamenevskayaand the LNS invariants, we give the following definition. Definition 1.1.
Given a Frobenius algebra F , a Plamenevskaya-type invariant x F isa function associating to each braid B a chain x F ( B ) ∈ C •F ( b B , R ) , such that: ⊲ x F ( B ) is non trivial; ⊲ x F ( B ) is an enhanced state, and its underlying resolution is the orientedresolution (see Subsection 3.1 for the definitions); ⊲ x F ( B ) is invariant under transverse Markov moves in the sense of The-orem 1.3.It turns out that the β F -invariants are essentially all the Plamenevskaya-typeinvariants. The second main result in this paper is the following Theorem 1.4.
Let F be a Frobenius algebra (as in Subsection 2.2) over R, and let Rbe an UFD. Given a Plamenevskaya-type invariant x F , for each braid B there existsr = r ( B ) ∈ R such that: either x F ( B ) = r β F ( B ) or x F ( B ) = r β F ( B ) . In particular,there is a bijection between Plamenevskaya-type invariants and R-valued invariants fortransverse braids. The usage of the β F -invariants to distinguish transverse links is quite far frombeing practical. To distinguish two transverse links, one must verify that allthe chain homotopy equivalences induced by sequences of either Markov or Re-idemeister moves do not preserve the β F -invariants (see Subsection 4.5). In orderto obtain a computable invariant, in Section 5, we set F = BN . The correspond-ing β ( = β BN )-invariants are equivalent to the LNS invariants. More precisely, weprove the following proposition. Proposition 1.5.
Let Σ be a sequence of Markov (resp. Reidemeister) moves from a braidB to a braid B ′ (resp. from b B to b B ′ ), and suppose that sl ( B ) = sl ( B ′ ) . Denoted by Φ Σ : C • , • BN ( b B ) −→ C • , • BN ( b B ′ ) and φ Σ : C • TLee ( b B ) −→ C • TLee ( b B ′ ) the chain maps induced by Σ . Then, Φ Σ ( β ( B )) = β ( B ′ ) ⇔ φ Σ ( ψ + ( B )) = ψ + ( B ′ ) and Φ Σ ( β ( B )) = β ( B ′ ) ⇔ φ Σ ( ψ − ( B )) = ψ − ( B ′ ) CARLO COLLARI
To extract new information from the β -invariants we make use of the F [ U ] -module structure of Bar-Natan homology. In particular, we define two (non-negative) integer invariants, called c -invariants. The c -invariants determine thevanishing of the Plamenevskaya invariant [ ψ ( B )] (Proposition 5.2). However, apriori the c -invariants contain more information than the vanishing of ψ . Remark . There is a class of knots (non-right-veering knots, see [21]) for whichboth [ ψ ] and b θ vanish on all braid representatives. In particular, both [ ψ ] and b θ do not provide any information on transverse representatives of these knots.It will be interesting to understand which kind of information the c -invariantscan provide in this case. As of now it is complicated to find examples of non-equivalent transverse braids representing non-right-veering knots, whose Khovanovhomology (and c -invariants) can be computed.Building on the result of Lipshitz, Ng and Sarkar concerning the flype invari-ance (in the weak sense of Proposition 1.2) of ψ ± , we prove our third main result. Proposition 1.6.
The c-invariants are invariant under negative flypes.
Remark . We prove a slightly stronger statement: given two braids related bya negative flype, there exists an isomorphism of bi-graded modules between theirBar-Natan homologies which preserves the homology classes of the β -invariants(see Proposition 5.4 for a precise statement).Moreover, we provide some sufficient conditions for the c -invariants to be noneffective. We say that a link λ is c-simple if all transverse links representing λ withthe same self-linking number have the same c -invariants. Proposition 1.7.
Let κ be an oriented knot. If κ satisfies one of the following conditions(1) H jKh ( κ , F ) ≡ for each j < s ( κ ) − ;(2) H − jKh ( κ , F ) ≡ for each j < s ( κ ) − ;(3) the torsion sub-module of H • BN ( κ , F [ U ]) is isomorphic to the F [ U ] -moduleM = m M i = F [ U ]( U k i ) , for some m, k , ..., k m ∈ N \ { } , and H − jKh ( κ , F ) ≡ for each j < s ( κ ) − ;where s ( κ ) = s ( κ , F ) is the Rasmussen invariant ( [23] ), then κ is c-simple. In particular,all Kh-thin and Kh-pseudo-thin (i.e. H • Kh ( κ , F ) is concentrated in two quantum degrees)knots are c-simple. Moreover, in the cases (1)-(3) we haves ( κ , F ) − = sl ( B ) + c F ( B ) , for each braid representative B of κ . Since [ ψ ( B )] = c F ( B ) > [ ψ ] . The corresponding result in the specialcase of Kh -thin knots is due to Plamenevskaya ([21, Theorem 1.2]). Corollary 1.8.
Let κ be a knot satisfying one among the conditions (1), (2) or (3) inProposition 1.7. For each braid B representing κ , we have [ ψ ( B )] = if, and only if,s ( κ , F ) − = sl ( B ) . Finally, from the analysis of the Khovanov homology of all small knots (i.e.prime knots with less that 12 crossings, up to mirror), we shall prove that theysatisfy the hypotheses of Proposition 1.7 (see Corollary 5.6). As a consequence,we obtain their c -simplicity and the following result. N TRANSVERSE INVARIANTS FROM KHOVANOV-TYPE HOMOLOGIES 7
Corollary 1.9.
Let F be a field with char ( F ) = , and let κ be (up to mirror) a primeknot with less than crossings. Given a braid B representing κ , the class [ ψ ( B )] istrivial (over F ) if, and only if, s ( κ , F ) − = sl ( B ) . In particular, the vanishing of [ ψ ] (over F ) is a non-effective invariant for all such κ ’s. Acknowledgments.
The author wishes to thank Prof. Paolo Lisca the helpfulconversations and his continuous support, Alberto Cavallo for his careful read-ing of a draft of this paper, and André Belotto for his advice. Many thanks go toDaniele Angella for his patience and support. The author is grateful to the an-onymous referees for their helpful suggestions. The content of the present paperis a part of the author’s PhD thesis, during the writing of which he was supportedby a PhD scholarship “Firenze-Perugia-Indam”.2. F robenius A lgebras This section contains some basic material concerning Frobenius algebras, andis divided into two subsections. The first subsection concerns the definition ofgraded and filtered Frobenius algebras. The second subsection is devoted to thedescription of some examples which play a central role in this paper.2.1.
Definitions. A Frobenius algebra F , over the ring R F , is a commutative unit-ary R F -algebra A F , together with two maps ∆ F : A F → A F ⊗ R F A F , ǫ F : A F → R F ,satisfying the following requirements(a) A F is a finitely generated and projective R F -module;(b) ∆ F is an A F -bi-module isomorphism (i.e. commutes with the left and rightaction of A F over A F ⊗ A F );(c) ǫ F is R F -linear;(d) ∆ F is co-associative, that is ( id ⊗ ∆ F ) ◦ ∆ F = ( ∆ F ⊗ id ) ◦ ∆ F ;(e) ∆ F is co-commutative, that is τ ◦ ∆ F = ∆ F , where τ ( a ⊗ b ) = b ⊗ a ;(f) ( id A F ⊗ ǫ F ) ◦ ∆ F = id A F = ( ǫ F ⊗ id A F ) ◦ ∆ F .The map ∆ F is called co-multiplication , while ǫ F is the co-unit relative to ∆ F .Whenever we deal with more than one Frobenius algebra at once, we willusually keep the subscript indicating to which Frobenius algebra the maps ∆ , ǫ , the algebra A , and the ring R belong to. Sometimes, it will be necessary tospecify the multiplicative structure on A F , so we will denote by m F the ( R F -linear) multiplication map from A F ⊗ R F A F to A F . Finally, we will denote by ι F the unit map , that is the R F -linear map from R F to A F sending 1 R F to 1 A F .We require the graded and filtered versions of Frobenius algebras. Definition 2.1. A graded Frobenius algebra is a Frobenius algebra F , satisfying thefollowing properties(a) R F = L k R k is a graded ring;(b) A F = L i A i is a graded R -module;(c) ǫ F , ι F are graded maps;(d) ∆ F , m F are graded maps (where A F ⊗ A F is given the usual tensor grading); Definition 2.2. A filtered Frobenius algebra is a Frobenius algebra F over a (possiblytrivially) graded ring R together with a filtration F ◦ of A = A F as an R -module,for which there exists an integer d such that: F i F j ⊆ F j + i + d for each i and each j , and ∆ F ( F n ) ⊆ ∑ k F k ⊗ F n − d − k ⊆ A ⊗ A , CARLO COLLARI for each n . Definition 2.3.
Let F = ( R , A , m , ι , ∆ , ǫ ) , and G = ( R ′ , B , n , , Γ , η ) be two (graded)Frobenius Algebras. A Frobenius algebra morphism is a pair of ring homomorph-isms f : R → R ′ , F : A → B ,such that F ◦ ι = ◦ f , η ◦ F = f ◦ ǫ .and F ⊗ F ◦ ∆ = Γ ◦ F .Given two Frobenius algebra morphisms, say ( f , F ) and ( g , G ) , their composition is defined as ( f ◦ g , F ◦ G ) . An isomorphism of Frobenius algebras is a morphism ( f , F ) such that both f and F are ring isomorphisms.2.2. Examples.
A large family of Frobenius algebras, which plays a central rolein this paper, is defined as follows A F = R F [ X ]( X + PX + Q ) R F = F [ U , V ]( p ( U , V ) , q ( U , V )) with p and q such that ( p , q ) is a (possibly trivial) prime ideal in F [ U , V ] .Up to twist equivalence ([11, Theorem 1.6], see also [13]) we may assume ǫ F ( X ) = R F , ǫ F ( A F ) = A F . This implies that X + PX + Q factorsover R F . Thus, we may write ( X + PX + Q ) = ( X − x )( X − x ) ,where x , x ∈ R F . Remark . Notice that the case x = x is not excluded.Set x ◦ = ( X − x ) and x • = ( X − x ) .In the rest of the paper we will be using the properties of x ◦ and x • . In particular,we need to understand the behaviour of x ◦ and x • with respect to the operations ∆ F , m F and ǫ F . First, notice that(1) m F ( x ◦ , x ◦ ) = − ( x − x ) x ◦ , m F ( x • , x • ) = ( x − x ) x • ,(2) m F ( x ◦ , x • ) = ǫ F ( x ◦ ) = ǫ F ( x • ) = R F .Furthermore, by setting e x ∗ = ( ∗ = ◦− ∗ = • ,and x ◦ = x • , x • = x ◦ , Given a Frobenius algebra F we can twist its co-multiplication and its co-unit by pre-composingthem with the multiplication by an invertible element in A F . Two Frobenius algebras are twist equi-valent if the can be obtained one from the other via twist. It has been proven by Khovanov [13,Proposition 3 and Corollary 1] that two twist equivalent Frobenius algebras of rank 2 produce iso-morphic Khovanov-type homologies. N TRANSVERSE INVARIANTS FROM KHOVANOV-TYPE HOMOLOGIES 9 we obtain(4) x = x − e x ( x − x ) A F , x ∈ { x ◦ , x • } .The involution x in the set { x ◦ , x • } will be called conjugation .Given a Frobenius algebra ( A , m , ι , ∆ , ǫ ) with x ∈ A , write(5) ∆ ( x ) = ∑ i x ′ i ⊗ x ′′ i where the elementary tensors x ′ i ⊗ x ′′ i are totally determined by the equations:(6) m ( x , y ) = ∑ i ( x ′′ i , y ) x ′ i , ∀ y ∈ A ,and ( · , · ) indicates the (non-degenerate) bi-linear pairing ǫ ( m ( · , · )) ([11, Proposi-tions 4.3 and 4.8], see also [26, Chapter 2]). It follows immediately from Equations(1), (2), (5) and (6) that(7) ∆ F ( x • ) = x • ⊗ x • , ∆ F ( x ◦ ) = x ◦ ⊗ x ◦ .Finally, we need to check the de-cupped torus map , that is the R F -linear map t F : R F −→ A F : 1 R F m F ( ∆ F ( A F )) Simple computations show that ∆ F ( A F ) = x ◦ ⊗ A F + A F ⊗ x • = x • ⊗ A F + A F ⊗ x ◦ ,from which it follows(8) t F ( R F ) = x • + x ◦ .To conclude this section, we shall list some special elements of the family F . Let R be a ring. Define A BIG to be the (graded) R [ U , V ] -algebra A BIG = R [ U , V ][ X ]( X − UX + V ) ,where x − : = X , and x + : =
1, have degrees, respectively, − +
1. In order todefine the structure of Frobenius algebra, define a co-multiplication ∆ = ∆ BIG , asfollows ∆ ( x + ) = x + ⊗ R x − + x − ⊗ R x + − Ux + ⊗ x + , ∆ ( x − ) = x − ⊗ R x − − Vx + ⊗ x + .Finally, the co-unit map is defined by ǫ : A BIG → R [ U , V ] : a ( U , V ) x + + b ( U , V ) x − b ( U , V ) .Many familiar examples are obtained by specifying U , V or both, in elements u or v of R (that is, applying the functor · ⊗ R [ U , V ] R [ U , V ] / ( U − u , V − v ) ). Inparticular(1) Khovanov theory Kh , is obtained by setting U = V = OLee , is obtained by setting U = V = TLee , is obtained by setting U = V = BN , is obtained by setting V = V -theory, denoted by VT , is obtained by setting U = deg ( U ) = −
2, and deg ( V ) = − BIG becomes a graded Frobenius algebra and hence BN , VT , and Kh , inherit thisstructure, while TLee and
OLee become filtered Frobenius algebras.
3. K hovanov - type homologies The aim of this section is to recall the definition of Khovanov-type homology.We shall review first how to associate to each Frobenius algebra F (over thering R ) and to each link diagram D , a chain complex C •F ( D , R ) . The homotopytype of this chain complex depends only on the link and not on the diagram D representing it. The second part of this section shall be concerned on reviewinghow to endow the complex C •F ( D , R ) with either a second grading or a filtration,depending on whether F is a graded or filtered Frobenius algebra. Finally, theend of the section shall be dedicated to fix some notation used in the rest of thepaper.3.1. Definition of Khovanov-type homologies.
Let D be an oriented link dia-gram. A local resolution of a crossing is its replacement with either a 0-resolutionor with a 1-resolution . Definition 3.1. A resolution of D is the set of circles, embedded in R , obtainedfrom D by performing a local resolution at each crossing. The total number of1-resolutions performed in order to obtain a resolution s is denoted by | s | .Let R D be the set of all the possible resolutions of D . Define an elementaryrelation on R D as follows r ≺ s ⇐⇒ | r | < | s | and r , s differ by a single local resolution.A square [ s , s , s , s ] is a collection of four (distinct) resolutions such that: s ≺ s , s ≺ s , s ≺ s , and s ≺ s . Definition 3.2. A sign function is a map sgn : R D × R D → {
0, 1, − } ,satisfying the following two properties:(1) sgn ( r , s ) = r ⊀ s ;(2) for each square [ s , s , s , s ] , we have sgn ( s , s ) sgn ( s , s ) = − sgn ( s , s ) sgn ( s , s ) .Given a Frobenius algebra F = ( R , A , m , ι , ∆ , ǫ ) define C i F ( D , R ) = M | r |− n − ( D )= i A r , A r = O γ ∈ r A γ ,where A γ is just a copy of A indexed by a circle γ ∈ r , n − ( D ) (resp. n + ( D ) )denotes the number of negative (resp. positive) crossings in D , and r runs through R D . These are the R -modules which play the role of (co)chain groups. In orderto define a (co)chain complex, all that is left to do is to define a differential. Thisis done in two steps. Start by defining d sr : A r → A s , r ≺ s .Given s such that r ≺ s , then r and s differ by a single local resolution. Hencethere is an identification of all the circles in the two resolutions, except the onesinvolved in the change of local resolution. There are only two cases to consider:(a) two circles of r , say γ , γ are merged in a single circle γ ′ in s , or (b) a circle γ belonging to r is split in into two circles, say γ ′ and γ ′ , in s . Consider theelementary tensor x = N γ ∈ r α γ ∈ A r , then d sr is defined as follows d sr ( x ) = ( N γ ∈ r ∩ s α γ ⊗ m ( α γ , α γ ) in case (a) N γ ∈ r ∩ s α γ ⊗ ∆ ( α γ ) in case (b) N TRANSVERSE INVARIANTS FROM KHOVANOV-TYPE HOMOLOGIES 11 and extended by R -linearity. Finally, fix a sign function sgn and define d i F : C i F ( D , R ) → C i + F ( D , R ) : x ∈ A r ∑ r ≺ s sgn ( r , s ) d sr ( x ) . Remark . Notice that d sr is well-defined because of the commutativity of m ,and of the co-commutativity of ∆ . On the other hand, d F depends on the choicesign function sgn . In particular, the existence of d F depends on the existence ofsuch a function. Proposition 3.1 (Khovanov, [12]) . There exists a sign function sgn such that the com-plex ( C •F ( D , R ) , d •F ) is a (co)chain complex. Furthermore, ( C •F ( D , R ) , d •F ) does notdepend, up to isomophism, on the choice of the sign function sgn , or on the order of thecircles in each resolution. Finally, given a sequence of Reidemeister moves between twodiagrams D and D ′ , there exists a chain homotopy equivalence Φ : C •F ( D , R ) −→ C •F ( D ′ , R ) induced by this sequence. Remark . The map associated to the sequence of Reidemeister moves is notuniquely defined (see [10]). In Section 4 we specify the maps associated to eachReidemeister move used in this paper. These maps are the maps commonly usedthrough the literature (for example, see [1]).
Remark . It is immediate from the definition of Khovanov-type homology that C •F ( D ⊔ D ′ , R ) ≃ C •F ( D , R ) ⊗ R C •F ( D ′ , R ) ,as complexes of R -modules. Moreover, if F is a graded (resp. filtered) Frobeniusalgebra the above isomorphism respects the quantum grading (resp. the filtra-tion) defined in the next subsection.3.2. Gradings and some notation.
There is a unique condition on F for the ho-mology of the complex ( C •F ( D , R ) , d •F ) to yeld a link invariant: the rank of A F being 2 (cf. [13, Proposition 3, Theorems 5 and 6]). Moreover, this condition isalso sufficient to get functoriality (up to sign and up to boundary fixing isotopy[1, 10]).As we are concerned only with link invariant theories, from now on all Frobeniusalgebras are supposed to be free of rank 2. Once a basis of A is fixed, say x + , x − ,the elements of C i F ( D , R ) of the form N γ ∈ r α γ , with α γ ∈ { x + , x − } and r ∈ R D ,will be called states ; while those of the form N γ ∈ r α γ , with α γ ∈ A , will be called enhanced states . Notice that the states are an R -basis of C •F ( D , R ) , while the en-hanced states are a system of generators. Remark . If F is a graded (resp. filtered) Frobenius algebra, then the basis { x + , x − } will be taken to be composed of homogeneous elements (resp. to be afiltered basis). Under these conditions, it is possible to define another grading(resp. filtration) over the complex ( C •F ( D , R ) , d •F ) , as follows qdeg ( O γ ∈ r α γ ) = ∑ γ ∈ r deg A ( α γ ) − n − ( D ) + n + ( D ) + | r | ,for each state N γ ∈ r α γ . (Then the filtration is given by considering all the ele-ments which can be written as combination of states of degree greater or lowerthan a fixed qdeg , depending on whether the multiplication is non-decreasing ornon-increasing with respect to the qdeg .) Moreover, by definition of graded (resp.filtered) Frobenius algebra, the differential d •F is homogeneous (resp. filtered)with respect to the qdeg degree (resp. induced filtration), and the resultinghomology theory is hence doubly-graded (resp. filtered). Let F be a filtered Frobenius algebra, the filtration induced on the complex C •F ( D , R ) is denoted by F ◦ C •F ( D , R ) . Theorem 3.2.
Let D be an oriented link diagram. If F and G are isomorphic (graded,resp. filtered) Frobenius algebras, then ( C •F ( D , R F ) , d •F ) and ( C •G ( D , R G ) , d •G ) are iso-morphic as (doubly-graded, resp. filtered) complexes of both R F and R G modules.Proof. Let F = ( R F , A , m , ι , ∆ , ǫ ) and G = ( R G , B , n , , γ , η ) be two isomorphic(graded, resp. filtered) Frobenius algebras, and let ( f , F ) the (graded, resp.filtered) isomorphism between them. Then, for each resolution r we have theisomorphism of (graded, resp. filtered) R F -modules O γ ∈ r F : A r → B r ,where B is seen as an R F -module with the induced structure. This naturally in-duces an isomorphism of (bi-graded, resp. filtered) chain modules that commutes(by definition of morphism between Frobenius algebras) with differentials. Thesame reasoning works if R F is replaced by R G . (cid:3) Remark . Until now we have required the diagrams to be oriented: this isessential for the invariance. As the reader may have noticed, the orientationcomes up in the degree shift. The homological degree has been shifted by thenumber of negative crossings. Without this shift the homology is not invariant asgraded module (much less as bi-graded or filtered module).Let D be an oriented link diagram, and let a ⊆ R be a segment joining twostrands of D (i.e. edges of the underlying graph), and meeting D only at theendpoints. Therefore a does not intersect a crossing of D . Let D ′ be the unori-ented link diagram formed by replacing a small neighbourhood of a as shown inFigure 2. Given a resolution r of D , denote by γ and γ the circles in r (possibly γ = γ ) containing the endpoints of a . γ γ a D D ′ F igure
2. The diagrams D and D ′ .Notice that there is not, in general, a canonical way to endow D ′ with anorientation compatible with the one of D . The existence of such an orientationdepends on the relative orientations of the arcs containing the endpoints of a in D . We shall assume D ′ to be given such an orientation if it exists, otherwise weshall randomly orient D ′ .The saddle move along a is the map S : C •F ( D , R ) −→ C •− ω ( D , D ′ ) F ( D ′ , R ) ,where ω ( D , D ′ ) = n − ( D ) − n − ( D ′ ) , and S is defined on enhance states as fol-lows: S ( O γ ∈ r α γ , a ) = O γ ∈ r \{ γ , γ } α γ ⊗ ς ( α γ , α γ ) , This is injective because A , B are both flat R F -modules, and is obviously surjective. N TRANSVERSE INVARIANTS FROM KHOVANOV-TYPE HOMOLOGIES 13 and ς ( α γ , α γ ) = ( m F ( α γ , α γ ) if γ = γ ∆ F ( α γ ) if γ = γ Remark . Notice that S is well defined because of the commutativity of m F and of the co-commutativity of ∆ F . Moreover, given an enhanced state x , thechain S ( x , a ) is a sum of enhanced states and not necessarily a single enhancedstate. Remark . If F is a graded (or a filtered) Frobenius algebra, then S is a graded(resp. filtered) map of (filtered) degree − ω ( D , D ′ ) − ransverse invariants in K hovanov - type homologies The aim of this section is dual: to define β F -invariant, and prove some of theirproperties. This section is divided into six subsections. In the first subsectionwe introduce the β F -cycles. The second, third and fourth subsections are ded-icated to the study of the behaviour of the β F -cycles with respect to the mapsinduced by the Reidemeister moves. In the fifth subsection the β F -cycles areused to define the β F -invariants. Moreover, we prove Theorem 1.3 and discuss indetail in which sense the β F -invariants are invariants for transverse links. Finally,the last subsection is dedicated to the proof of Theorem 1.4, which concerns theuniqueness of the β F -invariants.4.1. The definition of the β F -cycles. Let D be an oriented link diagram, anddenote by r the oriented resolution of D . Mark a point p γ on each circle γ in r , and let q γ be the point in S obtained by pushing p γ slightly to the left withrespect to the orientation on r induced by D . The nesting number N ( γ ) is thenumber, counted modulo 2, of intersection points between the circles in r and ageneric segment between q γ and the point at the infinity in S = R ∪ { ∞ } .Define β F -cycles as follows: β F ( D , R ) ∈ C • , •F ( D , R ) is the enhanced state withunderlying resolution the oriented resolution, where each circle γ has label (i.e.the factor corresponding to A γ in the tensor product) b γ = b γ ( F ) = ( x ◦ if N ( γ ) ≡ mod x • if N ( γ ) ≡ mod β F ( D , R ) is defined exactly as β F ( D , R ) but exchanging the roles of x ◦ and x • . Sometimes the reference to R will be omitted from both the notation forthe β F -cycles and notation for the Khovanov-type chain complexes. Remark . Notice that in general β F ( D , R ) and β F ( D , R ) are not distinct. Proposition 4.1.
Let D be an oriented link diagram. The enhanced states β F ( D , R ) , β F ( D , R ) ∈ C •F ( D , R ) are cycles.Proof. Since two circles in the oriented resolution share a crossing only if theyhave distinct nesting numbers (see [23, Corollary 2.5]), which also implies thateach change of a local resolution in the oriented resolution merges two circles,the proposition follows directly from Equation (2). (cid:3)
Henceforth all the results will be stated for β F ( D , R ) , with the understandingthat the same results hold by replacing β F ( D , R ) with β F ( D , R ) . The unique resolution which inherits an orientation from D . First Reidemeister move.
Let D be an oriented link diagram. Denote by D ′ + the oriented link diagram obtained from D via a positive first Reidemeister move(i.e. the addition of a positive curl, see Figure 3) on an arc a . Finally, denote by c + the crossing appearing only in D ′ + . D a D ′ + D ′− R − ⇌ R + ⇋ c + c − F igure
3. The first Reidemeister move.The complex C F ( D ′ + , R ) can be identified (as a graded R -module) with thecomplex(9) C •F ( D ∪ (cid:13) ) ⊕ C •F ( D )( − ) ≃ ( C •F ( D ) ⊗ R A ) ⊕ C •F ( D )( − ) ,where ( · ) denotes the (homological) degree shift; that is, given a Z n graded mod-ule M • , then ( M ( J )) I = M I + J for each I , J ∈ Z n .Each resolution of D ′ + obtained by performing a 0-resolution of c + can be iden-tified with a resolution of D ∪ (cid:13) , while each of the remaining resolutions can beidentified with a resolution of D . To turn this identification into an isomorphismof R -complexes, we identify the complex with the mapping cone of the map S F associated to a saddle move. Concretely, we endow the graded R -module on theleft-hand side of (9) with the differential d ′F = (cid:18) d F ⊗ R id A S F d F (cid:19) ,where S F is the map associated to a saddle move merging the unknotted com-ponent with the circle γ ′ containing a . More explicitly, S F : C •F ( D ) ⊗ R A → C •F ( D ) : O γ ∈ r α γ ! ⊗ α O γ ∈ r \{ γ ′ } α γ ⊗ m F ( α γ ′ , α ) .Now, we are ready to define the map associated to the addition of the (positive)curl. This map, denoted by Φ + ( F ) or just Φ + , is defined as follows Φ + : C •F ( D ) −→ ( C •F ( D ) ⊗ R A ) ⊕ C •F ( D )( − ) O γ ∈ r α γ O γ ∈ r \{ γ ′ } α γ ⊗ (cid:0) α γ ′ ⊗ t F ( R ) − ∆ F ( α γ ′ ) (cid:1) ⊕ t F is the de-cupped torus map. To conclude the positive version of thefirst Reidemeister move, we need to define the map associated to the removal ofa positive curl. This map, denoted by Ψ + ( F ) or simply Ψ + , is given by Ψ + : ( C •F ( D ) ⊗ R A ) ⊕ C •F ( D )( − ) −→ C •F ( D ) O γ ∈ r α γ ! ⊗ a ! ⊕ O γ ∈ s δ γ ǫ F ( a ) O γ ∈ r α γ .Now, let us turn to the negative version of the first Reidemeister move. Forour scope it is sufficient to define only the map associated to the creation of anegative curl. Let us denote by D ′− the diagram obtained from D by adding anegative curl on the arc a (see Figure 3). Denote by c − the crossing of D ′− createdby the addition of the curl. Similarly to the case of the positive Reidemeister N TRANSVERSE INVARIANTS FROM KHOVANOV-TYPE HOMOLOGIES 15 move, there is an identification of the resolutions of D ′− where c − is replacedwith is 0-resolution and the resolutions of D . All the remaining resolutions of D ′− can be identified with the resolutions of D ∪ (cid:13) . These identifications induce thefollowing isomorphisms of (graded) R -modules(10) C •F ( D ′− ) ≃ C •F ( D )( − ) ⊕ C •F ( D ∪ (cid:13) ) ≃ C •F ( D )( − ) ⊕ ( C •F ( D ) ⊗ R A ) . Remark . Suppose F is a graded Frobenius algebra. Then the complex C •F ( D , R ) can be endowed with a second grading (see Subsection 3.2). To turn the iso-morphisms in (10) into isomorphisms of bi- graded R -modules it is necessary tointroduce an appropriate quantum degree shift (cf. [1, Section 6]). This shift isnot necessary in the case of the positive version of the first Reidemeister move.As in the case of R + , we wish to turn the isomorphisms in (10) into isomorph-isms of chain complexes. In order to do so it is sufficient to endow the rightmost R -module in (10) with the differential d ′F = (cid:18) d F S ′F d F ⊗ R id A (cid:19) ;where S ′F is the map associated to a saddle move splitting the circle γ ′ containingthe arc a . More explicitly, S ′F : C •F ( D ) → C •F ( D ) ⊗ R A : O γ ∈ r α γ O γ ∈ r \{ γ ′ } α γ ⊗ ∆ ( α γ ′ ) . Remark . There is no ambiguity in the labels given to the circles by ∆ ( α γ ′ ) because of the co-commutativity of ∆ .Finally, we can define the map associated to the addition of a negative curl,denoted by Φ − ( F ) or just Φ − , as follows Φ − : C •F ( D ) −→ C •F ( D )( − ) ⊕ ( C •F ( D ) ⊗ R A ) O γ ∈ r α γ ⊕ O γ ∈ r α γ ! ⊗ ι F ( R ) Now we are finally ready to state (and prove) a result describing the behaviour of β F ( D , R ) with respect to the maps associated to the first Reidemeister move(s). Remark . The map induced by the negative first Reidemeister move has beenobtained by composition; the map Φ − (resp. its homotopy inverse Ψ − ) can be ob-tained by composing (resp. pre-composing) the map associated to a non-coherentsecond Reidemeister move (see [7, Chapter 3, Section 3] or [1, Section 4]) and Ψ + (resp. Φ + ). With these definitions we have that Ψ − ◦ Φ − = − Id . Proposition 4.2.
Let D be an oriented link diagram. If D ′ + (resp. D ′− ) is the diagramobtained from D via a positive (resp. negative) first Reidemeister move (with the inducedorientation), then (R1p) Ψ + ( F )( β F ( D )) = β F ( D ′ + ) , Φ + ( F )( β F ( D ′ + )) = β F ( D ) , and (R1n) ( x − x )( Φ − ) ∗ ( F )([ β F ( D )]) = − e x [ β F ( D ′− )] ; where x is the label in β F ( D , R ) of the circle containing the arc where the first move isperformed. Proof.
Let us start from the addition of a positive curl. Suppose α γ ′ = x ∈{ x ◦ , x • } . It follows from Equations (7) and (8) that α γ ′ ⊗ t F ( R F ) − ∆ F ( α γ ′ ) = x ⊗ ( x + x ) − x ⊗ x = x ⊗ x ,where ¯ x denotes the conjugation on the set { x ◦ , x • } . Identify the oriented resolu-tion of D ′ + with the oriented resolution of D ∪ (cid:13) as in the definition of Φ + . Fromthe previous considerations it follows that the label of the un-knotted componentwhich does not belong to D in Φ + ( β F ( D )) is x , the label of γ ′ is x , and all theother labels remain unchanged. Thus, it follows immediately that Φ + ( β F ( D )) = β F ( D ′ + ) .To conclude the case of the positive R move, we must verify that β F ( D , R ) ispreserved by Ψ + . The claim follows from the following facts: (a) if a = b γ ′ then ǫ F ( a ) =
1, (b) the direct summand in C •F ( D ′ + ) corresponding to the orientedresolution of D ′ is mapped onto the direct summand in C •F ( D ) corresponding tothe oriented resolution, and (c) the labels on the circles that are not involved inthe move and in the circle γ ′ are left invariant by Ψ + .Now, let us turn to the behaviour of β F ( D , R ) with respect to the map associ-ated to the addition of a negative curl. Immediately from the definition it followsthat(11) Φ − ( β ( D , R )) = O γ ∈ r b γ ! ⊗ ι ( R ) ,where r denotes the oriented resolution of D , and the oriented resolution of D ′− is identified with the oriented resolution of D ∪ (cid:13) . Consider the chain η = ⊕ O γ ∈ r b γ ! ⊗ x ! = ⊕ ( β F ( D ) ⊗ x ) in C •F ( D ′− ) . Directly from the definition of d ′F follows d ′F ( β F ( D ) ⊕ ) = η .By Equation (4) we have β F ( D ′− ) = ⊕ O γ ∈ r b γ ! ⊗ ¯ x ! = η − e x ( x − x ) Φ − ( β F ( D )) .and the claim follows. (cid:3) Second Reidemeister move.
Let D be an oriented link diagram. Let a and b be two (un-knotted) arcs of D lying in a small ball. Performing a second Re-idemeister move on these arcs inserts two adjacent crossings, say c and c , ofopposite type (see Figure 4). ab ⇋ R c c D ′′ D F igure
4. The (un-oriented) second Reidemeister move.Denote by D ′′ the oriented link diagram obtained from D by performing asecond Reidemeister move on the arcs a and b . There are four possible resolutionsof the pair of crossings c and c . Let D ′′ ij , with i , j ∈ {
0, 1 } , be the link obtained N TRANSVERSE INVARIANTS FROM KHOVANOV-TYPE HOMOLOGIES 17 from D ′′ by performing a i -resolution on c and a j -resolution on c (Figure 5).Notice that there is a natural identification of the link D ′′ with D . D ′′ D ′′ D ′′ D ′′ F igure
5. The possible resolutions of c and c . Remark . Only one among the links D ′′ , D ′′ , D ′′ and D ′′ inherits the orient-ation from D ′′ , and this is either D ′′ or D ′′ .Similarly to the case of the first Reidemeister move, there is an isomorphismof graded R -modules(12) C •F ( D ′′ ) ≃ C •F ( D ′′ ) ⊕ C •F ( D ′′ )( − ) ⊕ C •F ( D ′′ )( − ) ⊕ C •F ( D ′′ )( − ) .given by the identification of each resolution of D ′′ with a resolution of D ′′ ij (for asuitable choice of i and j ). Remark . Assume F to be a graded Frobenius algebra. To turn the isomorph-ism in (12) into an isomorphism of bi-graded R -modules a suitable shift of thequantum degree has to be taken into account (cf. [1, Section 4]).The isomorphism in (12) is not an isomorphism of R -complexes. To obtainsuch an isomorphism it is necessary to modify the differential of the complex onthe right-hand-side of (12). This modified differential can be (roughly) defined asfollows d ′′F = d F S ′′ d F S ′′ d F S ′′ S ′′ d F where d ij F is the differential of the complex C •F ( D ′′ ij ) , and S ′′ ij , hk : C •F ( D ′′ ij , R ) −→ C •F ( D ′′ hk , R ) is the map corresponding to a saddle move from D ′′ ij to D ′′ hk . This description ismore than sufficient for our scope. The interested reader may consult [12, Section5] or [1, Section 4] for a more detailed description of d ′′F .Now, consider the diagram D ′′ . Denote by c and d the two arcs appearing inthe local picture in Figure 5 (see also Figure 6). Fix an arc g , meeting D ′′ onlyat the endpoints, joining c and d . Finally, fix an arc e , meeting D only at theendpoints, joining the arcs a and b . D ′′ γ ′′ g dcabe D = D ′′ F igure With the notation defined above, and using the notation introduced in Subsec-tion 3.2, we can finally define the map Ψ : C •F ( D ) −→ C •F ( D ′′ ) ⊕ C •F ( D ′′ )( − ) ⊕ C •F ( D ′′ )( − ) ⊕ C •F ( D ′′ )( − ) as follows Ψ ( x ) = ⊕ x ⊕ ( S ( x , e ) ⊗ ι ( R )) ⊕ x is an enhanced state, D and D ′′ have been identified, and ι ( R ) is thelabel of γ ′′ (cf. Figure 6). Similarly, the up-to-chain-homotopy inverse of Ψ Φ : C •F ( D ′′ ) ⊕ C •F ( D ′′ )( − ) ⊕ C •F ( D ′′ )( − ) ⊕ C •F ( D ′′ )( − ) −→ C •F ( D ) is given by Φ ( x ⊕ x ⊕ x ⊕ x ) = x + ǫ ( x γ ′′ ) S ( x , g ) ,where x ij denotes a (possibly trivial) enhanced state in C •F ( D ′′ ij ) , and x γ ′′ denotesthe label of γ ′′ in x (cf. Figure 6).Before stating the results concerning β F ( D , R ) recall that a R move is coherentif the arcs a and b involved are oriented as in the right of Figure 7. ⇋⇋ F igure
7. Non-coherent (left) and coherent (right) versions of thesecond Reidemeister move.
Proposition 4.3.
Let D be an oriented link diagram. Let D ′′ be the oriented link diagramobtained from D via a coherent second Reidemeister move. Then (R2c) Ψ ( β F ( D , R )) = β F ( D ′′ , R ) and Φ ( β F ( D ′′ , R )) = β F ( D , R ) . Proof.
Throughout this proof we will keep the notation shown in Figure 6. Let r be the oriented resolution of D . First, let us investigate the behaviour of β F ( D , R ) with respect to the map Ψ . It is a simple consequence of the Jordan curve the-orem that if the move is coherent then a and b do not belong to the same circle in r . Let γ a and γ b be the circles to which a and b , respectively, belong to. It followsdirectly from the definitions that Ψ ( β F ( D )) = ⊕ β F ( D ) ⊕ O γ ∈ r \{ γ a , γ b } b γ ⊗ m ( b γ a , b γ b ) ⊗ ι ( R ) ⊕ β F ( D , R ) of γ a and of γ b are conjugate.Thus, by Equation (2) we have m ( b γ a , b γ b ) = m ( b γ a , b γ a ) = D ′′ is identified (via the isomorphism in (12)) with the oriented resolution of D ′′ . Thus, it follows that Ψ ( β F ( D , R )) = β F ( D ′′ , R ) .As we argued before, the isomorphism in (12) sends β F ( D ′′ , R ) to0 ⊕ β F ( D , R ) ⊕ ⊕ γ a = γ b and b γ b = b γ a .From Equation (1), and from the considerations just made, we obtain S ( β F ( D , R ) , g ) = N TRANSVERSE INVARIANTS FROM KHOVANOV-TYPE HOMOLOGIES 19
Since Φ ( ⊕ β F ( D , R ) ⊕ ⊕ ) = β ( D , R ) + S ( β F ( D , R ) , g ) the claim follows. (cid:3) Third Reidemeister move.
We have arrived at the case of the third Re-idemeister move. This move is the hardest to deal with because it comes inseveral versions. Moreover, the number of crossings is equal across both sides ofthe move, so there is no loss of complexity.We shall avoid giving an explicit description of these maps, and instead de-scribe the procedure used to associate a map to each version of the third Re-idemeister move.It is necessary to remark that the set of all versions of third Reidemeister movescan be seen as generated by a sequence of (coherent versions of the) second Re-idemeister moves and the moves in Figure 8 (see [22, Lemma 2.6]). ⇐⇒⇐⇒ R ◦ R • D D ′ D D ′ F igure
8. Two version of the third Reidemeister move.Recall that a third Reidemeister move is braid-like (or coherent) if it can berealized as a relation in the braid group. All braid-like third Reidemeister movescan be obtained from the R ◦ move via a sequence of coherent R moves ([22,Lemma 2.6]). So, it is sufficient to prove the invariance of the β F -cycles withrespect to R ◦ move.In order to define the map associated to R ◦ and R • we make use of the so-called categorified Kauffman trick ([1, Section 4]). All the maps associated to theother third Reidemeister moves will be defined as a composition. Since we areconcerned only with braid-like moves, we shall describe explicitly only the mapassociated of the R ◦ move. The map associated to the R • move can be describedsimilarly.First, write the complexes associated to both sides of the Reidemeister move ascones. More precisely, C F ( D ) = Cone ( S : C F ( D ) → C F ( D )) ,where D i is the diagram obtained by performing the i -resolution on the crossinghighlighted in Figure 8, and S is the map associated to the saddle connecting thediagrams D and D . An analogous reasoning works for D ′ . Remark . One can define the maps associated to the braid-like third Reidemeistermoves directly using the categorified Kauffman trick, instead of defining themby composition. These maps may differ from those obtained by composition.Nonetheless, Lemma 4.4 applies almost verbatim, and one obtains that the mapsdefined via the categorified Kauffman trick still preserve the β F -cycles.One notices that the links D and D ′ (with the obvious notation) are related tothe link D ′′ (Figure 9) by a coherent R . This implies that there are maps inducedby the two R moves, f : C F ( D ) → C F ( D ′′ ) f ′ : C F ( D ′ ) → C F ( D ′′ ) , D D ′′ D ′ F igure
9. The links D , D ′′ and D ′ .which are quasi-isomorphisms. Denote the respective up-to-homotopy inversesby g and g ′ .The main point is that these maps are respectively a strong deformation re-tract and an inclusion in a deformation retract ([1, Definition 4.3]). Hence, by [1,Lemma 4.5] we have the quasi-isomorphisms Φ : C F ( D ) = Cone ( S : C F ( D ) → C F ( D )) → Cone (cid:0) S ◦ g : C F ( D ′′ ) → C F ( D ) (cid:1) and Φ ′ : C F ( D ′ ) = Cone (cid:0) S : C F ( D ′ ) → C F ( D ′ ) (cid:1) → Cone (cid:0) S ◦ g ′ : C F ( D ′′ ) → C F ( D ) (cid:1) ,as well as their up to homotopy inverses Ψ and Ψ ′ . Moreover, these maps canbe explicitly computed in terms of f , f ′ and their up-to-homotopy inverses g and g ′ . Finally, one notices that the cones over S ◦ g and over S ◦ g ′ can be identified.Using this identification, the maps associated to the R ◦ can be defined as follows Ψ = Φ ◦ Ψ ′ , and Φ = Φ ′ ◦ Ψ .The key point of the invariance of the β F -cycles is the following (technical) lemmawhich is left as an exercise (or see [7, Proposition 3.15]). Lemma 4.4.
Given a cone over a chain map S Γ = Cone ( S : C −→ D ) and an inclusion in a deformation retract (resp. a strong deformation retract)f : C → C ′ ( resp. g : C ′ → C ),denote by F (resp. G) the quasi-isomorphismF : Γ −→ Cone ( S ◦ g : C ′ −→ D ) (resp. G : Cone ( S ◦ g : C ′ −→ D ) −→ Γ )induced by f (resp. g). If f ( x ) = x ′ and g ( x ′ ) = x, thenF ( x ⊕ ) = x ′ ⊕ and G ( x ′ ⊕ ) = x ⊕ Proposition 4.5.
Let D and D ′ be two oriented link diagrams related by a coherent thirdReidemeister move. Then (R3c) Ψ ( β F ( D , R )) = β F ( D ′ , R ) and Φ ( β F ( D ′ , R )) = β F ( D , R ) . Proof.
The claim is an immediate consequence of Lemma 4.4 and Proposition4.3. (cid:3)
The β F -invariants. Recall that, as said in the introduction, the map associ-ated to a Markov move is the map associated to the corresponding Reidemeistermove on the braid closure. Now we can prove Theorem 1.3. of Theorem 1.3.
Every sequence of transverse Markov moves translates into a se-quence of positive first Reidemeister moves, and braid-like second and third Re-idemeister moves. A negative stabilization translates into a negative first Re-idemeister move. The theorem follows immediately from Propositions 4.2, 4.3and 4.5. (cid:3)
N TRANSVERSE INVARIANTS FROM KHOVANOV-TYPE HOMOLOGIES 21
Definition 4.1.
Given a braid diagram B , the β F -invariants associated to B are thecycles β F ( B ) = β F ( b B , R ) and β F ( B ) = β F ( b B , R ) . Remark . Since the β F -cycles are defined for all oriented link diagrams, onemay wonder if the β F -cycles define invariants for other representations of trans-verse links. For example, one may ask if the β F -cycles define invariants for trans-verse front projections (see [8] for a definition). However, this is not true. Thereader may consult [7, Chapter 4, Subsection 2.3].Let B and B ′ be two braids representing the transverse links T and T ′ , respect-ively. Given a sequence Σ of Markov (resp. Reidemeister) moves from B to B ′ (resp. from b B to b B ′ ), denote by Φ Σ : C •F ( b B , R ) −→ C •F ( b B ′ , R ) the map associated to Σ . If B and B ′ represent the same transverse link, then Σ can be taken to be a sequence of transverse Markov moves. It follows that Φ Σ sends the pair ( β F ( B ) , β F ( B )) to the pair ( β F ( B ′ ) , β F ( B ′ )) . Remark . Not all the sequences of Reidemeister moves between two braidclosures can be obtained as composition of Markov moves. In particular, the setof maps associated to sequences of Markov moves is a priori different from the setof maps associated to sequences of Reidemeister moves.Suppose that T and T ′ are two distinct transverse knots. We will say that T and T ′ are weakly (resp. strongly ) distinguished by the β F -invariants if it does notexist a sequence Σ of Markov (resp. Reidemeister) moves such that Φ Σ ( β F ( B )) = β F ( B ′ ) , and Φ Σ ( β F ( B )) = β F ( B ′ ) .If the elements of a non-simple pair are weakly (resp. strongly) distinguished bythe β F -invariants we will say that the β F -invariants are weakly (resp. strongly ) effective . Remark . We do not know whether the strong effectiveness implies the weakeffectiveness or not. This is related to the study of the maps induced by theMarkov moves in a Khovanov-type theory. It goes beyond the scope of the presentpaper to explore this subject.
Remark . We expect the strong (resp. weak) effectiveness to depend on theFrobenius algebra and on the base ring R . Since we do not have the means tostudy the effectiveness of the β F -invariants in full generality, this subject will notbe explored further in this paper.In Theorem 1.3 it is stated that the β F -invariants are preserved by the maps in-duced by a sequence of transverse Markov moves. Comparing it with Proposition1.2 one may notice that the signs of the β F -invariants are also preserved. How-ever, for each oriented link diagram D , the map − Id C •F ( D , R ) can be always realisedas the map associated to a sequence of Reidemeister moves from D to itself (see[1, 10]). Moreover, with our choices for the maps associated to a generating set ofReidemeister moves, − Id can be realised with a sequence of Markov moves (seeRemark 4.4). So, if there is a sequence Σ of Markov (resp. Reidemeister) moves,from B to B ′ (resp. from b B to b B ′ ), such that Φ Σ ( β F ( B ′ , R )) = − β F ( B ′ , R ) , and Φ Σ ( β F ( B ′ , R )) = − β F ( B ′ , R ) ,then there is also a sequence of Markov (resp. Reidemeister) moves Σ ′ , from B to B ′ (resp. from b B to b B ′ ), such that Φ Σ ′ ( β F ( B ′ )) = β F ( B ′ ) , and Φ Σ ′ ( β F ( B ′ )) = β F ( B ′ ) . a b F igure
10. Two coherently oriented arcs.
Remark . More in general, there is an action of the group of the chain homo-topy equivalences induced by sequences of Reidemeister (resp. Markov) movesfrom b B to itself, on C •F ( b B , R ) . Following [10], we call this group the monodromygroup (resp. the braid monodromy group ) and denote it by M b B (resp. BM B ). Ithas been shown in [10] that the monodromy group contains more than just theidentity and its opposite. Let Σ be a sequence of Reidemeister (resp. Markov)moves from B to B ′ . If Φ Σ sends the β F -invariants of B to a pair of elementsin the M b B ′ -orbit (resp. BM B ′ -orbit) of the β F -invariants of B ′ , then the corres-ponding transverse links cannot be strongly (resp. weakly) distinguished by the β F -invariants.At this point the following question arises naturally; how to use the β F -invariants to distinguish two transverse links? In other words, how can we provethat two cycles, belonging to different chain complexes, are not mapped one ontothe other by a given set of chain homotopy equivalences? An approach is to lookat their homology classes. For example, if one of the cycles has trivial homo-logy class while the other does not, then the two cycles cannot be mapped oneonto the other. Another approach is to use some extra structure on the complex(e.g. a filtration) which is preserved by the given set of chain homotopies. Thefirst approach was originally used by Plamenevskaya, while the latter approachwas used by Lipshitz, Ng and Sarkar. In the next section, we shall make use ofthe F [ U ] -module structure of Bar-Natan’s homology to extract some informationfrom the β BN -invariants.To conclude this section we shall prove a “uniqueness” result for the β F -invariants (i.e. Theorem 1.4).4.6. A uniqueness property.
Assume R = R F to be a unique factorization do-main (UFD). Even though not every Frobenius algebra which belongs to the fam-ily described in Subsection 2.2 satisfies this hypothesis, a large class of them does.In particular, this hypothesis is satisfied by the algebras BIG , BN , VT , Kh , OLee and
TLee .Suppose that we have a way to assign to each oriented link diagram D anenhanced state x ( D ) ∈ C •F ( D , R ) , whose underlying resolution is the orientedone. Recall that given a circle γ in the oriented resolution of D , b γ (resp. ¯ b γ )denotes the label of γ in β F ( D , R ) (resp. β F ( D , R ) ). Lemma 4.6.
Let D be an oriented link diagram. Denote by a and b two unknotted arcs ofD as in Figure 10, and by D ′ the link diagram obtained by performing a coherent secondReidemeister move along a and b . Finally, denote by γ a and γ b are the circles int theoriented resolution of D containing a and b , respectively. Suppose that Φ ( x ( D )) = x ( D ′ ) and Ψ ( x ( D ′ )) = x ( D ) , where Ψ and Φ are the maps associated to the second Reidemeister move and its inverse,respectively. Then, the labels of the circles γ a and γ b in x ( D ) are, respectively, (R-)multiples of either b γ a and b γ b , or of ¯ b γ a and ¯ b γ b . N TRANSVERSE INVARIANTS FROM KHOVANOV-TYPE HOMOLOGIES 23
Proof.
Denote by r and r ′ the oriented resolutions of D and D ′ , respectively. Fi-nally, denote by s the resolution of D ′ where all crossings but the two added bythe second Reidemeister move are resolved as in the oriented resolution.Let a and b the labels of the circles γ a and γ b in x ( D ) . Since Ψ ( x ( D )) = x ( D ′ ) ∈ A r ′ , it is immediate that m ( a , b ) =
0. Thus, a and b must be zero divisorsin A . It follows (since R [ X ] is an UFD) that a and b belong to either the idealgenerated by x ◦ or to the ideal generated by x • in A BN . Moreover, the two labelsshould belong to different ideals. Since b γ a is either x ◦ or x • and b γ b = ¯ b γ a theclaim follows. (cid:3) Lemma 4.7.
Let D be a non-split oriented link diagram (i.e. D is connected as a planargraph). If x ( D ) is invariant under coherent Reidemeister moves of the second type, theneither x ( D ) = r β ( D , R ) or x ( D ) = r ¯ β ( D , R ) , for some r ∈ R.Proof.
If two circles in the oriented resolution share a crossing it is possible toperform a coherent R involving those circles. Thus, by Lemma 4.6 each pairof circles sharing a crossing should be labeled either as in β or ¯ β , up to themultiplication by an element of R . Since D has only one split component, theSeifert graph is connected. So, if the label of a single circle is chosen, all the otherlabels are determined up to multiplication by an element of R , and the claimfollows. (cid:3) Let D be an oriented link diagram, and let D , ..., D k be its split components(i.e. the connected components of D seen as a 4-valent graph). We will say that D i and D j have compatible orientations if there exists ball B intersecting D in twounknotted arcs a and b , with a belonging to D i and b belonging to D j , which isambient isotopic in R to the ball in Figure 10.The diagram D is said to be coherently oriented if for each pair of split com-ponents of D , say D and D , there exists a sequence D = D i , ..., D i k = D ofsplit components of D such that the components D i j and D i j + have compatibleorientations for each j ∈ {
1, ..., k − } . Proposition 4.8.
Let D be a coherently oriented link diagram. If x ( D ) is invariantunder coherent Reidemeister moves of the second type, then x ( D ) is a (R-)multiple ofeither β F ( D , R ) or ¯ β F ( D , R ) .Proof. Let D be a coherently oriented diagram and D ,..., D k be its split com-ponents. By Lemma 4.6 the labels of x ( D ) on the components of the orientedresolution of a split component are exactly as in β F ( D , R ) or as in ¯ β F ( D , R ) , upto multiplication by an element of R . By the definition of coherently orientedlink diagram, given two split component, say D i and D j , there exists a sequence D = D i , ..., D i k = D of split components of D such that the components D i j and D i j + have compatible orientations for each j ∈ {
1, ..., k − } . By definitionof compatible orientation it is possible to perform a second type coherent Re-idemeister move using an arc of D i j and and arc of D i j + . Thus, again by Lemma4.6, if the labels of the circles corresponding to D i j in x ( D ) are as in β F ( D , R ) (up to multiplication by an element of R ), then also the labels of the circles cor-responding to D i j + in x ( D ) are as in β F ( D , R ) . Similarly, if the the labels of thecircles corresponding to D i j in x ( D ) are as in ¯ β F ( D , R ) , then also the labels ofthe circles corresponding to D i j + in x ( D ) are as in ¯ β F ( D , R ) . So, if the label ofa circle γ in x ( D ) is a R -multiple of b γ (resp. ¯ b γ ), then x ( D ) is an R -multiple of β F ( D , R ) (resp. ¯ β F ( D , R ) ). (cid:3) of Theorem 1.4. All the braid closures are coherently oriented link diagrams. The-orem 1.4 now follows straightforwardly from Proposition 4.8 . (cid:3)
5. S pecialising to B ar -N atan theory In this section we specialise our construction to the case β = β BN . First weexplore the relationship between the β -invariants, the Plamenevskaya invariantand the LNS-invariants. In the second part of this section we extract new trans-verse invariants from the homology classes of the β -invariants. Furthermore, weprovide some sufficient condition for these new invariants to be non-effective,and prove that also the vanishing of the Plamenvskaya invariant is non-effectivefor small knots.5.1. Relationship with other invariants.
Recall that the base ring for the Frobeniusalgebra BN is F [ U ] . Moreover, the resulting Khovanov-type homology is bi-graded, and the multiplication by U lowers the quantum degree by 2. It followsimmediately from the definitions that the β -invariants are bi-homogeneous cyclesof bi-degree ( sl ( B )) .Let us start by pointing out the relationship between Bar-Natan, Khovanov andLee theories. This relationship follows directly from the definitions and can becondensed into the following exact sequences(13) 0 → C • , • + BN ( D ) U · −→ C • , • BN ( D ) π Kh −→ C • , • Kh ( D ) → → C • BN ( D ) ( U − ) · −→ C • BN ( D ) π TLee −→ C • TLee ( D ) → π TLee ( β ( B )) = ψ − ( B ) , π TLee ( β ( B )) = ψ + ( B ) , π Kh ( β ( B )) = π Kh ( β ( B )) = ψ ( B ) .Since the LNS invariants are representatives of the canonical generators of twistedLee theory (cf. [23, 17]), their homology classes are linearly independent over F .The following proposition follows immediately Proposition 5.1.
Given an oriented link diagram D, then [ β ( D )] and [ β ( D )] generatea rank F [ U ] -submodule of H • , • BN ( D ) . In particular, [ β ( D )] and [ β ( D )] are alwaysnon-trivial and non-torsion. This is in stark contrast to the behaviour of the homology class of ψ . In fact, [ ψ ( B )] tends to vanish quite easily (cf. [20, Proposition 3]). However, the vanish-ing of [ ψ ] can be detected directly from [ β ] . More precisely, we have the followingresult. Proposition 5.2.
Given a braid B, the following conditions are equivalent(1) [ ψ ( B )] = ;(2) exists x ∈ H sl ( B )+ BN ( b B ) such that Ux = [ β ( B )] ;(3) exists x ∈ H sl ( B )+ BN ( b B ) such that Ux = [ β ( B )] ;Proof. Is immediate from the exact sequence in Equation (13) and from the factthat π Kh ( β ) = π Kh ( β ) = ψ . (cid:3) In order to prove the equivalence between the β - and the LNS-invariants weneed a technical lemma. The proof of this lemma is quite easy and is left as anexercise. N TRANSVERSE INVARIANTS FROM KHOVANOV-TYPE HOMOLOGIES 25
Lemma 5.3 (Unique homogeneous lift) . Let R be a PID, and let M be a graded R [ U ] -module, where deg ( U ) = − and the graded structure on R [ U ] is the natural one. Definea filtration on M / ( U − ) M as follows F i = (cid:10) [ x ] ∈ M / ( U − ) M | x ∈ M j , j ≤ i (cid:11) R . If M is non-trivial only in either even or odd degree and if [ x ] ∈ F i , then there exists aunique ˜ x ∈ M i such that [ ˜ x ] = [ x ] .of Proposition 1.5. Denote by ℓ the number of components of the Alexander clos-ure of B . It is well known that Bar-Natan homology of B and B ′ is supported inquantum degrees which are congruent to ℓ modulo 2 (cf. [12, Proposition 24], orsee [7, Corollary 2.24]). Notice that β (resp. β ) is the unique homogeneous liftof ψ + (resp. ψ − ) of quantum degree sl . Thus we can apply Lemma 5.3, and thestatement follows. (cid:3) Proposition 5.4.
The homology classes of the β -invariants are flype invariant in thefollowing sense; if B and B ′ are related by a negative flype, then there exists a (bi-graded)isomorphism of F [ U ] -modules F ∗ : H • , • BN ( b B ) → H • , • BN ( b B ′ ) sending [ β ( B )] and [ β ( B )] to [ β ( B ′ )] and [ β ( B ′ )] , respectively.Proof. In [16, Theorem 4.15] it has been shown that there is a filtered chain homo-topy equivalence f : C • TLee ( b B ) → C • TLee ( b B ′ ) , such that f ( ψ ⋆ ( B )) = ± ψ ⋆ ( B ′ ) + d TLee x ⋆ for some x ⋆ ∈ F sl C − TLee ( b B ′ ) ,where ⋆ ∈ { + , −} and sl = sl ( B ) = sl ( B ′ ) . Let D be an oriented link diagram.Thanks to Lemma 5.3 one can define, for each j , an isomorphism ρ j of F -chaincomplexes, whose inverse is π TLee , which fits into the following commutativesquare C • , jBN ( D ) L C • , j + BN ( D ) (cid:18) U (cid:19) (cid:15) (cid:15) π TLee - - F j C • TLee ( D ) ρ j n n (cid:127) _ (cid:15) (cid:15) C • , j − BN ( D ) L C • , jBN ( D ) π TLee . . F j − C • TLee ( D ) ρ j n n Recall that all the enhanced states in the Bar-Natan complex have quantum degreecongruent modulo 2 to the number of components of the link represented by D .It follows that one of the two summands on the right hand side of the square isalways trivial. Now, we can use the ρ j ’s to define a lift F of f to Bar-Natan theory.Since f is a filtered chain homotopy equivalence, the map F will be graded, F [ U ] -linear and a chain homotopy equivalence. Furthermore, Lemma 5.3 ensures usthat F ( β ( B )) = ± β ( B ′ ) + d BN f x + , and F ( β ( B )) = ± β ( B ′ ) + d BN f x − , where f x ± isthe unique lift of x ± of quantum degree sl . Since the sign changes are coherent([16, Proof of Theorem 4.5]), the claim follows. (cid:3) Remark . If the filtered degrees of x ± were strictly lower than sl , we couldhave only proved that U k times the homology classes of the β -invariants wereflype invariant, for some k > Remark . We have proved a slightly stronger statement; every time that theLNS-invariant are preserved as in Proposition 1.2, then the homology classes ofthe β -invariants are invariant as in Proposition 5.4. Proposition 5.4 implies that it is really difficult to use the homology classes ofthe β -invariants to distinguish flypes; even if one manages prove that the chainhomotopy F is not the chain homotopy equivalence associated to a sequence ofMarkov (or Reidemeister) moves, there will be the problem of how to extract in-formation from the homology classes of the β -invariants. Moreover, the argumentof Lipshitz, Ng and Sarkar can be modified to prove that the homology classes ofthe β F -invariants, for a large choice of F , cannot be used to distinguish flypes.5.2. The c -invariants. Let B be a braid, and F be a field. Define the c-invariants of B (over F ) as follows c F ( B ) = max n k | [ β ( B )] = U k x , for some x ∈ H • BN ( B , F [ U ]) o and c F ( B ) = max n k | [ β ( B )] = U k x , for some x ∈ H • BN ( B , F [ U ]) o The c -invariants are, of course, transverse braid invariants. Moreover, they providethe same or less amount of transverse information as the homology classes of the β -invariants. In particular, we have the flype invariance. Proposition 1.6.
The proposition follows directly from Proposition 5.4. (cid:3)
Notice that [ ψ ( B )] = c F ( B ) > c -invariants determine the vanishing of the homology class of ψ . Definition 5.1.
An oriented link λ is called c-simple if each non-simple pair oftransverse representatives of λ have the same c -invariants.The non-effectiveness of the c -invariants is equivalent to all links being c -simple. We wish to address the following question: let λ be an oriented link.What are the homological conditions which λ should satisfy to be c -simple?This question is intentionally vague. For example, we did not specify whichhomology one should consider, or which type of condition one should look for.However, we manage to give some sufficient conditions for a knot to be c -simple.First, we need to look more closely at the β -invariants. Let B be a braid repres-enting the knot κ . Denote by s ( κ ) = s ( κ , F ) the Rasmussen invariant of κ ([23]).Fix an isomorphism of bi-graded F [ U ] -modules(15) φ : H • , • BN ( b B , F [ U ]) → m M i = F [ U ]( U t i ) ( h i , q i ) ⊕ F [ U ]( s ( κ ) + ) ⊕ F [ U ]( s ( κ ) − ) ,which exists, for some choices of h i , q i ∈ Z and t i ∈ N , by the structure theoremfor graded modules over a PID (see, for example, [28, Theorem 3.19]) and by [13].Consider the natural generators of the module on the right hand side of (15), thatis e i = ( i − th place ↓
0, ..., 0, [ ] , 0, ..., 0 ) f = (
0, ..., 1, 0 ) and f = (
0, ..., 0, 1 ) ,where i ∈ {
1, ..., m } , and set˜ e i = φ − ( e i ) and ˜ f j = φ − ( f j ) .Notice that for each i we have ( hdeg ( ˜ e i ) , qdeg ( ˜ e i )) = ( h i , q i ) .Denote by I the set of all i ∈ {
1, ..., m } such that h i =
0. From the definitions ofthe ˜ e i ’s, ˜ f , ˜ f and c F ( B ) it follows immediately that(16) [ β ( B , F )] = U c F ( B ) α U r ˜ f + α U r ˜ f + ∑ i ∈ I γ i U k i ˜ e i ! , N TRANSVERSE INVARIANTS FROM KHOVANOV-TYPE HOMOLOGIES 27 where at least one among r , r , and the k i ’s (such that γ i U k i ˜ e i =
0) is zero.Moreover, as the homology classes of the β -invariants generate a rank 2 F [ U ] -sub-module of H • , • BN ( κ , F [ U ]) , it follows that at least one among α and α isnon-trivial. Since the β -invariants are homogeneous, it follows that q i − k i = s ( κ ) − − r = s ( κ ) + − r = sl ( B ) + c F ( B ) .In particular, we get that r = r + r equals 0 we obtain that s ( κ ) − = sl ( B ) + c F ( B ) .Thus, under the above assumption c F would be (half of) the difference between aknot invariant and the self linking, and hence non-effective. A similar reasoningapplies to ¯ c F . Making use of these considerations we can prove the followingproposition. Proposition 5.5.
Let κ be an oriented knot. If q i is greater than or equal to s ( κ ) − foreach i ∈ I , then κ is c-simple, where s ( κ ) denotes the Rasmussen invariant. Moreover,in this case we have s ( κ ) − = sl ( B ) + c F ( B ) , for each braid representative B of κ .Proof. If q i ≥ s ( κ ) −
1, then k i ≥ r . Thus, if r >
0, then the k i ’s are also strictlygreater than 0. It follows that r must be equal to 0, and the claim follows. (cid:3) Remark . the proof of Proposition 5.5 works also if κ is a link such that H • BN ( κ , F [ U ]) / Tor ( H • BN ( κ , F [ U ])) is supported in two quantum degrees. Theselinks are called pseudo-thin in [5]. of Proposition 1.7. Directly from the definitions, it follows that C • , • Kh ( D , F ) = C • , • BN ( D , F [ U ]) ⊗ F [ U ] F [ U ] / ( U ) , for each oriented link diagram D . From the Künneth theorem followsimmediately that if (1), (2) or (3) are satisfied, then q i ≥ s ( κ ) − (cid:3) From the analysis of the Bar-Natan and Khovanov homologies of all primeknots with less than 12 crossings it follows that
Corollary 5.6.
Let F be a field such that char ( F ) = . All prime knots with less than 12crossings and their mirror images satisfy the conditions in Proposition 1.7. In particular,they are c-simple over F .Proof. For the computation of integral Khovanov homology the reader may referto the KnotAtlas ([2]). Since there is only 2-torsion in the integral Khovanov ho-mology of the prime knots with less than 12 crossings, their Khovanov homologyover F is concentrated in the same bi-degrees as their rational Khovanov homo-logy. A well-known theorem due to Lee ([15]) states that alternating knots are Kh -thin. As a consequence of Proposition 1.7, all alternating knots are c -simple.So we may restrict our attention to the non-alternating knots. According to Knot-Info ([6]), among the 249 prime knots with less than 11 crossings those which arenon-alternating are the following8 The ones marked in red are the Kh -thin knots, while those in blue are the Kh -pseudo-thin knots. If a knot is Kh -thin or Kh -pseudo-thin, then also its mirrorimage is Kh -thin or Kh -pseudo-thin. Thus, by Proposition 1.7, all prime knots inthe list above (and also their mirrors) are c -simple, except 9 and 10 . Theseknots satisfy condition (2) of Proposition 1.7 and hence they are c -simple.Finally, among the non-alternating prime knots with 11 crossings and theirmirrors the ones which are neither pseudo-thin nor satisfy the condition (2) ofProposition 1.7 are m ( n ) m ( n ) n m ( n ) n m ( n ) m ( n ) m ( n ) n m ( n ) It is know that if char ( F ) = H • , • BN ( κ , F [ U ]) is iso-morphic to the F [ U ] -module M = m M i = F [ U ]( U k i ) ,for some m , k , ..., k m ∈ N \ { } ([7, Corollary 2.33]). The links listed above satisfypoint (3) of Proposition 1.7. Hence they are c -simple and the claim follows. (cid:3) The reader should take into account that knots with less than 13 crossingsseem to have pretty “simple” Khovanov homology. For example, the first primeknot to have Khovanov homology supported in more than three diagonals, whichis also the first with thick torsion, is the knot 13 n c -invariants (and of ψ ).R eferences [1] D. Bar-Natan. Khovanov homology for tangles and cobordisms. Geometry & Topology , 9:1443–1499, 2005.[2] D. Bar-Natan and S. Morrison et al. The Knot Atlas. http://katlas.org/wiki/.[3] D. Bennequin. Entrelacements et équations de Pfaff.
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