Ozsvath and Szabo have established an algebraic relationship, in the form of a spectral sequence, between the reduced Khovanov homology of (the mirror of) a link L in S^3 and the Heegaard Floer homology of its double-branched cover. This relationship has since been recast by the authors as a specific instance of a broader connection between Khovanov- and Heegaard Floer-type homology theories, using a version of Heegaard Floer homology for sutured manifolds developed by Juhasz. In the present work we prove the naturality of the spectral sequence under certain elementary TQFT operations, using a generalization of Juhasz's surface decomposition theorem valid for decomposing surfaces geometrically disjoint from an imbedded framed link.
We show that all versions of Heegaard Floer homology, link Floer homology, and sutured Floer homology are natural. That is, they assign concrete groups to each based 3-manifold, based link, and balanced sutured manifold, respectively. Furthermore, we functorially assign isomorphisms to (based) diffeomorphisms, and show that this assignment is isotopy invariant. The proof relies on finding a simple generating set for the fundamental group of the "space of Heegaard diagrams," and then showing that Heegaard Floer homology has no monodromy around these generators. In fact, this allows us to give s…
For a nullhomologous Legendrian knot in a closed contact 3-manifold Y we consider a contact structure obtained by positive rational contact surgery. We prove that in this situation the Heegaard Floer contact invariant of Y is mapped by a surgery cobordism to the contact invariant of the result of contact surgery. In addition we characterize the spin-c structure on the cobordism that induces the relevant map. As a consequence we determine necessary and sufficient conditions for the nonvanishing of the contact invariant after rational surgery when Y is the standard 3-sphere, generalizing previou…
We discuss a relationship between Khovanov- and Heegaard Floer-type homology theories for braids. Explicitly, we define a filtration on the bordered Heegaard-Floer homology bimodule associated to the double-branched cover of a braid and show that its associated graded bimodule is equivalent to a similar bimodule defined by Khovanov and Seidel.
The aim of this paper is to introduce and study a geometric spectral sequence in Khovanov homology. The construction was motivated by a similar spectral sequence from Khovanov homology to Heegaard Floer homology.
Ozsvath and Szabo show that there is a spectral sequence whose E^2 term is the reduced Khovanov homology of L, and which converges to the Heegaard Floer homology of the (orientation reversed) branched double cover of S^3 along L. We prove that the E^k term of this spectral sequence is an invariant of the link L for all k >= 2. If L is a transverse link in the standard tight contact structure on S^3, then we show that Plamenevskaya's transverse invariant psi(L) gives rise to a transverse invariant, psi^k(L), in the E^k term for each k >= 2. We use this fact to compute each term in the spectral …
We show that the only irreducible three-manifold with positive first Betti number and Heegaard Floer homology of rank two is homeomorphic to zero-framed surgery on the trefoil. We classify links whose branched double cover gives rise to this manifold. Together with a spectral sequence from Khovanov homology to the Floer homology of the branched double cover, our results show that Khovanov homology detects the unknot if and only if it detects the two component unlink.
We discuss generalizations of Ozsvath-Szabo's spectral sequence relating Khovanov homology and Heegaard Floer homology, focusing attention on an explicit relationship between natural Z (resp., 1/2 Z) gradings appearing in the two theories. These two gradings have simple representation-theoretic (resp., geometric) interpretations, which we also review.
We define the reduced Khovanov homology of an open book (S,h), and we identify a distinguished "contact element" in this group which may be used to establish the tightness or non-fillability of contact structures compatible with (S,h). Our construction generalizes the relationship between the reduced Khovanov homology of a link and the Heegaard Floer homology of its branched double cover. As an application, we give combinatorial proofs of tightness for several contact structures which are not Stein-fillable. Lastly, we investigate a comultiplication structure on the reduced Khovanov homology o…
We study a module structure on Khovanov homology, which we show is natural under the Ozsvath-Szabo spectral sequence to the Floer homology of the branched double cover. As an application, we show that this module structure detects trivial links. A key ingredient of our proof is that the H_1/Torsion module structure on Heegaard Floer homology detects S^1xS^2 connected summands.
Using the relation between Khovanov homology and the Heegaard Floer homology of branched double covers, we show how Khovanov homology can be used to establish tightness of branched double covers of certain transverse knots. We give examples of several infinite families of knots whose branched covers are tight for Khovanov-homological reasons, and show that some of these branched covers are not Stein fillable.
We exhibit an infinite family of knots with isomorphic knot Heegaard Floer homology. Each knot in this infinite family admits a nontrivial genus two mutant which shares the same total dimension in both knot Floer homology and Khovanov homology. Each knot is distinguished from its genus two mutant by both knot Floer homology and Khovanov homology as bigraded groups. Additionally, for both knot Heegaard Floer homology and Khovanov homology, the genus two mutation interchanges the groups in $δ$-gradings $k$ and $-k$.
These lecture notes, which were designed for the Summer School "Heegaard-Floer Homology and Khovanov Homology" in Marseilles, 29th May - 2nd June, 2006, provide an elementary introduction to Khovanov homology. The intended audience is graduate students with some minimal background in low-dimensional and algebraic topology. The lectures cover the basic definitions, important properties, a number of variants and some applications. At the end of each lecture the reader is referred to the relevant literature for further reading.
We define an invariant of transverse links in the standard contact 3-sphere as a distinguished element of the Khovanov homology of the link. The quantum grading of this invariant is the self-linking number of the link. For knots, this gives a bound on the self-linking number in terms of Rasmussen's invariant s(K). We prove that our invariant vanishes for transverse knot stabilizations, and that it is non-zero for quasipositive braids. We also discuss a connection to Heegaard Floer invariants.
Let $L\subset S^3$ be a link. We study the Heegaard Floer homology of the branched double-cover $Σ(L)$ of $S^3$, branched along $L$. When $L$ is an alternating link, $\HFa$ of its branched double-cover has a particularly simple form, determined entirely by the determinant of the link. For the general case, we derive a spectral sequence whose $E^2$ term is a suitable variant of Khovanov's homology for the link $L$, converging to the Heegaard Floer homology of $Σ(L)$.
A categorification of a polynomial link invariant is an homological invariant which contains the polynomial one as its graded Euler characteristic. This field has been initiated by Khovanov categorification of the Jones polynomial. Later, P. Ozsvath and Z. Szabo gave a categorification of Alexander polynomial. Besides their increased abilities for distinguishing knots, this new invariants seem to carry many geometrical informations. On the other hand, Vassiliev works gives another way to study link invariant, by generalizing them to singular links i.e. links with a finite number of rigid trans…
We define a link homology theory that is readily seen to be both isomorphic to reduced odd Khovanov homology and fully determined by data impervious to Conway mutation. This gives an elementary proof that odd Khovanov homology is mutation invariant, and therefore that mod 2 Khovanov homology is mutation invariant. We also establish mutation invariance for the entire Ozsvath-Szabo spectral sequence from reduced Khovanov homology to the Heegaard Floer homology of the branched double cover.
Bordered Heegaard Floer homology is a three-manifold invariant which associates to a surface F an algebra A(F) and to a three-manifold Y with boundary identified with F a module over A(F). In this paper, we establish naturality properties of this invariant. Changing the diffeomorphism between F and the boundary of Y tensors the bordered invariant with a suitable bimodule over A(F). These bimodules give an action of a suitably based mapping class group on the category of modules over A(F). The Hochschild homology of such a bimodule is identified with the knot Floer homology of the associated op…
Given a transverse link in the standard contact 3-sphere, we study the contact manifold that arises as a branched double cover of the sphere. We give a contact surgery description of such manifolds, which allows to determine the Heegaard Floer contact invariants for some of them. By example of the knots of Birman--Menasco, we show that these contact manifolds may fail to distinguish between non-isotopic transverse knots. We also investigate the relation between the Heegaard Floer contact invariants of the branched double covers and the Khovanov homology, in particular, the transverse link inva…
We define an invariant of based transverse links, as a well-defined element inside the equivariant Heegaard Floer cohomology of its branched double cover, defined by Lipschitz, Hendricks, and Sarkar. We prove the naturality and functoriality of equivariant Heegaard Floer cohomology for branched double covers of $S^3$ along based knots, and then prove that our transverse link invariant $c_{\mathbb{Z}_{2}}(ξ_{K})$ is an well-defined element which is always nonvanishing and functorial under certain classes of symplectic cobordisms, and describe its behavior under negative stabilization. It follow…
We describe how to formulate Khovanov's functor-valued invariant of tangles in the language of bordered Heegaard Floer homology. We then give an alternate construction of Lawrence Roberts' Type D and Type A structures in Khovanov homology, and his algebra $\mathcal{B}Γ_n$, in terms of Khovanov's theory of modules over the ring $H^n$. We reprove invariance and pairing properties of Roberts' bordered modules in this language. Along the way, we obtain an explicit generators-and-relations description of $H^n$ which may be of independent interest.
The (untwisted) oriented cube of resolutions for knot Floer homology assigns a complex $C_{F}(S)$ to a singular resolution $S$ of a knot $K$. Manolescu conjectured that when $S$ is in braid position, the homology $H_{*}(C_{F}(S))$ is isomorphic to the HOMFLY-PT homology of $S$. Together with a naturality condition on the induced edge maps, this conjecture would prove the spectral sequence from HOMFLY-PT homology to knot Floer homology. Using a basepoint filtration on $C_{F}(S)$, a recursion formula for HOMFLY-PT homology, and additional $sl_{n}$-like differentials on $C_{F}(S)$, we prove this …
We provide a unified framework for proving Reidemeister-invariance and functoriality for a wide range of link homology theories. These include Lee homology, Heegaard Floer homology of branched double covers, singular instanton homology, and \Szabo's geometric link homology theory. We follow Baldwin, Hedden, and Lobb (arXiv:1509.04691) in leveraging the relationships between these theories and Khovanov homology. We obtain stronger functoriality results by avoiding spectral sequences and instead showing that each theory factors through Bar-Natan's cobordism-theoretic link homology theory.
Given a diagram of a link K in S^3, we write down a Heegaard diagram for the branched-double cover Sigma(K). The generators of the associated Heegaard Floer chain complex correspond to Kauffman states of the link diagram. Using this model we make some computations of the homology \hat{HF}(Sigma(K)) as a graded group. We also conjecture the existence of a delta-grading on \hat{HF}(Sigma(K)) analogous to the delta-grading on knot Floer and Khovanov homology.
Given a link in the three-sphere, Z. Szabó and the second author constructed a spectral sequence starting at the Khovanov homology of the link and converging to the Heegaard Floer homology of its branched double-cover. The aim of this paper and its sequel is to explicitly calculate this spectral sequence, using bordered Floer homology. There are two primary ingredients in this computation: an explicit calculation of filtered bimodules associated to Dehn twists and a pairing theorem for polygons. In this paper we give the first ingredient, and so obtain a combinatorial spectral sequence from Kh…
This is a companion paper to earlier work of the authors, which interprets the Heegaard Floer homology for a manifold with torus boundary in terms of immersed curves in a punctured torus. We prove a variety of properties of this invariant, paying particular attention to its relation to knot Floer homology, the Thurston norm, and the Turaev torsion. We also give a geometric description of the gradings package from bordered Heegaard Floer homology and establish a symmetry under spin$^c$ conjugation; this symmetry gives rise to genus one mutation invariance in Heegaard Floer homology for closed t…
There are a number of homological knot invariants, each satisfying an unoriented skein exact sequence, which can be realized as the limit page of a spectral sequence starting at a version of the Khovanov chain complex. Compositions of elementary 1-handle movie moves induce a morphism of spectral sequences. These morphisms remain unexploited in the literature, perhaps because there is still an open question concerning the naturality of maps induced by general movies. In this paper we focus on the spectral sequences due to Kronheimer-Mrowka from Khovanov homology to instanton knot Floer homology…
This work has two goals. The first is to provide a conceptual introduction to Heegaard Floer homology, the second is to survey the current state of the field, without aiming for completeness. After reviewing the structure of Heegaard Floer homology, we list some of its most important applications. Many of these are purely topological results, not referring to Heegaard Floer homology itself. Then, we briefly outline the construction of Lagrangian intersection Floer homology. We construct the Heegaard Floer chain complex as a special case of the above, and try to motivate the role of the various…
It was recently proved by several authors that ribbon concordances induce injective maps in knot Floer homology, Khovanov homology, and the Heegaard Floer homology of the branched double cover. We give a simple proof of a similar statement in a more general setting, which includes knot Floer homology, Khovanov-Rozansky homologies, and all conic strong Khovanov-Floer theories. This gives a philosophical answer to the question of which aspects of a link TQFT make it injective under ribbon concordances.
We describe the first part of a gluing theory for the bigraded Khovanov homology with integer coefficients. This part associates a type D structure to a tangle properly embedded in a half-space and proves that the homotopy class of the type D structure is an invariant of the isotopy class of the tangle. The construction is modeled off bordered Heegaard-Floer homology, but uses only combinatorial/diagrammatic methods
We introduce an invariant of tangles in Khovanov homology by considering a natural inverse system of Khovanov homology groups. As application, we derive an invariant of strongly invertible knots; this invariant takes the form of a graded vector space that vanishes if and only if the strongly invertible knot is trivial. While closely tied to Khovanov homology -- and hence the Jones polynomial -- we observe that this new invariant detects non-amphicheirality in subtle cases where Khovanov homology fails to do so. In fact, we exhibit examples of knots that are not distinguished by Khovanov homolo…
Seidel and Smith introduced the graded fixed-point symplectic Khovanov cohomology group Kh_{symp,inv}(K) for a knot K inside S^{3}, as well as a spectral sequence converging to the Heegaard Floer homology-hat group for the connected sum of the double branched cover with a copy of S^{2}xS^{1}. The E^{1}-page of this spectral sequence is isomorphic to a factor of Kh_{symp,inv}(K). Seidel and Smith proved that Kh_{symp,inv} is a knot invariant. We show here that the higher pages of their spectral sequence are knot invariants also.
We introduce the notion of a Khovanov-Floer theory. Roughly, such a theory assigns a filtered chain complex over Z/2 to a link diagram such that (1) the E_2 page of the resulting spectral sequence is naturally isomorphic to the Khovanov homology of the link; (2) this filtered complex behaves nicely under planar isotopy, disjoint union, and 1-handle addition; and (3) the spectral sequence collapses at the E_2 page for any diagram of the unlink. We prove that a Khovanov-Floer theory naturally yields a functor from the link cobordism category to the category of spectral sequences. In particular, …
Given a Heegaard splitting of a three-manifold Y, we consider the SL(2,C) character variety of the Heegaard surface, and two complex Lagrangians associated to the handlebodies. We focus on the smooth open subset corresponding to irreducible representations. On that subset, the intersection of the Lagrangians is an oriented d-critical locus in the sense of Joyce. Bussi associates to such an intersection a perverse sheaf of vanishing cycles. We prove that in our setting, the perverse sheaf is an invariant of Y, i.e., it is independent of the Heegaard splitting. The hypercohomology of this sheaf …
In arXiv:1611.09927, we constructed a well-defined Lagrangian Floer invariant for any closed, oriented $3$-manifold $Y$ via the symplectic geometry of so-called traceless $\mathrm{SU}(2)$-character varieties. This invariant, $\mathrm{SI}(Y)$, which we refer to as the symplectic instanton homology of $Y$, was also shown to satisfy an exact triangle for Dehn surgeries on knots which is typical of Floer-theoretic invariants of $3$-manifolds. In this article, we demonstrate further structural properties of this symplectic instanton homology. For example, Floer theories are expected to roughly sati…
Seidel-Smith and Hendricks used equivariant Floer cohomology to define some spectral sequences from symplectic Khovanov homology and Heegaard Floer homology. These spectral sequences give rise to Smith-type inequalities. Similar-looking spectral sequences have been defined by Lee, Bar-Natan, Ozsváth-Szabó, Lipshitz-Treumann, Szabó, Sarkar-Seed-Szabó, and others. In this paper we give another construction of equivariant Floer cohomology with respect to a finite group action and use it to prove some invariance properties of these spectral sequences; prove that some of these spectral sequences ag…
In their recent preprint, Baldwin, Ozsváth and Szabó defined a twisted version (with coefficients in a Novikov ring) of a spectral sequence, previously defined by Ozsváth and Szabó, from Khovanov homology to Heegaard-Floer homology of the branched double cover along a link. In their preprint, they give a combinatorial interpretation of the $E_3$-term of their spectral sequence. The main purpose of the present paper is to prove directly that this $E_3$-term is a link invariant. We also give some concrete examples of computation of this invariant.
The Khovanov homology of a link in $S^3$ and the Heegaard Floer homology of its branched double cover are related through a spectral sequence constructed by Ozsváth and Szabó. This spectral sequence has topological applications but is difficult to compute. We build an isomorphic spectral sequence whose underlying filtered complex is as simple as possible: it has the same rank as the Khovanov chain group. We show that this spectral sequence is not isomorphic to Szabó's combinatorial spectral sequence, which Seed and Szabó conjectured to be equivalent to Ozsváth-\Szabo's. The discrepancy leads u…
We establish a relationship between Heegaard Floer homology and the fractional Dehn twist coefficient of surface automorphisms. Specifically, we show that the rank of the Heegaard Floer homology of a 3-manifold bounds the absolute value of the fractional Dehn twist coefficient of the monodromy of any of its open book decompositions with connected binding. We prove this by showing that the rank of Floer homology gives bounds for the number of boundary parallel right or left Dehn twists necessary to add to a surface automorphism to guarantee that the associated contact manifold is tight or overt…
A companion paper to "On knot Floer homology in branched double covers" applied to braided branched loci. We reprove the main result of that paper concerning alternating branched loci when projected to an annulus, without using Khovanov homology. This provides two advantages: 1) the results hold for integer coefficients and 2) the spin^c structures are more readily discernable. We apply this result to a branch locus which is a braid, and use the braid structure to find information about a fibered knot in the branched double cover. In some cases this provides all the information about the knot …
We prove the existence of a spectral sequence for Lagrangian Floer homology which converges to the Floer homology of the image of a Lagrangian submanifold under multiple fibred Dehn twists. The $E_1$ term of the sequence is given by the hypercube of "resolutions" of the Dehn twists involved. The proof relies on the exact triangle for fibered Dehn twists due to Wehrheim and Woodward. As applications we obtain a spectral sequences from Khovanov homology to symplectic Khovanov homology. Also when a 3-manifold $M$ is given by gluing two handlebodies by a surface diffeomorphism $ϕ$, we obtain a spe…
Given a link in the three-sphere, Ozsváth and Szabó showed that there is a spectral sequence starting at the Khovanov homology of the link and converging to the Heegaard Floer homology of its branched double cover. The aim of this paper is to explicitly calculate this spectral sequence in terms of bordered Floer homology. There are two primary ingredients in this computation: an explicit calculation of bimodules associated to Dehn twists, and a general pairing theorem for polygons. The previous part (arXiv:1011.0499) focuses on computing the bimodules; this part focuses on the pairing theorem …
Quasi-alternating links are a generalization of alternating links. They are homologically thin for both Khovanov homology and knot Floer homology. Recent work of Greene and joint work of the first author with Kofman resulted in the classification of quasi-alternating pretzel links in terms of their integer tassel parameters. Replacing tassels by rational tangles generalizes pretzel links to Montesinos links. In this paper we establish conditions on the rational parameters of a Montesinos link to be quasi-alternating. Using recent results on left-orderable groups and Heegaard Floer L-spaces, we…
This paper considers the invariance of knot Floer homology in a purely algebraic setting, without reference to Heegaard diagrams, holomorphic disks, or grid diagrams. We show that (a small modification of) Ozsváth and Szabó's cube of resolutions for knot Floer homology, which is assigned to a braid presentation with a basepoint, is invariant under braid-like Reidemeister moves II and III and under conjugation. All moves are assumed to happen away from the basepoint. We also describe the behavior of the cube of resolutions under stabilization. The techniques echo those employed to prove the inv…
Given a knot K in S^3, Seidel and Smith described in arXiv:1002.2648v3 a graded cohomology group Kh_{symp,inv}(K), a variant of their symplectic Khovanov cohomology group. They also constructed a spectral sequence converging to the Heegaard Floer-hat homology group for the connected sum of the double branched cover and a copy of S^{2}xS^{1} (with E^1-page isomorphic to a direct summand of Kh_{symp,inv}(K)). In a previous paper (arXiv:1004.2476v5), we showed that the higher pages of this spectral sequence are knot invariants. Here we discuss a reduced version of the spectral sequence which dire…
