Ozsvath, Rasmussen and Szabo constructed odd Khovanov homology. It is a link invariant which has the same reduction modulo 2 as (even) Khovanov homology. Szabo introduced a spectral sequence with mod 2 coefficients from mod 2 Khovanov homology to another link homology. He got his spectral sequence from a chain complex with a filtration. We give an integral lift of Szabo's complex, that provides a spectral sequence from odd Khovanov homology to a link homology, from which one can get Szabo's link homology with the Universal Coefficient Theorem. Szabo has constructed such a lift independently, but has not yet published it. This is my master thesis which I wrote under supervision of Thomas Schick at Georg August University Göttingen in summer 2011. It is in German. I will publish a reworked version in English later.
Let L be a null homologous link in $\mathbb{RP}^3$. We define Khovanov-type homologies of L which depend on an extra input $α= (V_0,V_1,f,g)$ consisting of two graded vectors spaces and two maps between them. With some specific choice of $α= α_{APS}$, we recover the categorification of the Kauffman bracket due to Asaeda-Przytycki-Sikora. With another choice of $α= α_{HF}$, we construct a spectral sequence from our theory converging to the Heegaard Floer homology of the even branched double cover of $\mathbb{RP}^3$.
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 $-k$.
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…
Given a knot, we ask how its Khovanov and Khovanov-Rozansky homologies change under the operation of introducing twists in a pair of strands. We obtain long exact sequences in homology and further algebraic structure which is then used to derive topological and computational results. Two of our applications include giving a new way to generate arbitrary numbers of knots with isomorphic homologies and finding an infinite number of mutant knot pairs with isomorphic reduced homologies.
Kanenobu has given infinite families of knots with the same HOMFLY polynomials. We show that these knots also have the same sl(n) and HOMFLY homologies, thus giving the first example of an infinite family of knots undistinguishable by these invariants. This is a consequence of a structure theorem about the homologies of knots obtained by twisting up the ribbon of a ribbon knot with one ribbon.
In this article we introduce a family of transverse invariants arising from the deformations of Khovanov homology. This family includes the invariants introduced by Plamenevskaya and by Lipshitz, Ng, and Sarkar. Then, we investigate the invariants arising from Bar-Natan's deformation. These invariants, called $β$-invariants, are essentially equivalent to Lipshitz, Ng, and Sarkar's invariants $ψ^\pm$. From the $β$-invariants we extract two non-negative integers which are transverse invariants (the $c$-invariants). Finally, we give several conditions which imply the non-effectiveness of the $c$-…
Quasi-alternating links are a natural generalization of alternating links. In this paper, we show that quasi-alternating links are "homologically thin" for both Khovanov homology and knot Floer homology. In particular, their bigraded homology groups are determined by the signature of the link, together with the Euler characteristic of the respective homology (i.e. the Jones or the Alexander polynomial). The proofs use the exact triangles relating the homology of a link with the homologies of its two resolutions at a crossing.
In arXiv:1308.3152, the author proved that the Khovanov-Rozansky homology $\mathcal{H}_N$ with potential $ax^{N+1}$ is an invariant for transverse links in the standard contact $3$-sphere. In the current paper, we study the $\mathbb{Z}_2 \oplus \mathbb{Z}^{\oplus 3}$-graded $\mathbb{Q}[a]$-module structure of $\mathcal{H}_N$, which leads to better understanding of the effect of stabilization on $\mathcal{H}_N$. As an application, we compute $\mathcal{H}_N$ for all transverse unknots.
We provide a finite dimensional categorification of the symmetric evaluation of $\mathfrak{sl}_N$-webs using foam technology. As an output we obtain a symmetric link homology theory categorifying the link invariant associated to symmetric powers of the standard representation of $\mathfrak{sl}_N$. In addition, the construction is actually made in an equivariant setting. We prove also that there is a spectral sequence from the Khovanov-Rozansky triply graded link homology to the symmetric one and provide along the way a foam interpretation of Soergel bimodules.
We show a spectral sequence for the rational Khovanov homology of an oriented link in terms of the rational Khovanov complexes and homologies of the link surgeries along an admissible cut. As a non trivial corollary, we give an explicit splitting formula for the Jones polynomial.
This is an expository paper discussing some parallels between the Khovanov and knot Floer homologies. We describe the formal similarities between the theories and give some examples which illustrate a somewhat mysterious correspondence between them.
An elementary introduction to Khovanov construction of superpolynomials. Despite its technical complexity, this method remains the only source of a definition of superpolynomials from the first principles and therefore is important for development and testing of alternative approaches. In this first part of the review series we concentrate on the most transparent and unambiguous part of the story: the unreduced Jones superpolynomials in the fundamental representation and consider the 2-strand braids as the main example. Already for the 5_1 knot the unreduced superpolynomial contains more items…
We continue to develop the tensor-algebra approach to knot polynomials with the goal to present the story in elementary and comprehensible form. The previously reviewed description of Khovanov cohomologies for the gauge group of rank N-1=1 was based on the cut-and-join calculus of the planar cycles, which are involved rather artificially. We substitute them by alternative and natural set of cycles, not obligatory planar. Then the whole construction is straightforwardly lifted from SL(2) to SL(N) and reproduces Khovanov-Rozansky (KR) polynomials, simultaneously for all values of N. No matrix fa…
We explain how Queffelec-Sartori's construction of the HOMFLY-PT link polynomial can be interpreted in terms of parabolic Verma modules for $\mathfrak{gl}_{2n}$. Lifting the construction to the world of categorification, we use parabolic 2-Verma modules to give a higher representation theory construction of Khovanov-Rozansky's triply graded link homology.
We consider colored operads and their actions on categories. As a special example we construct a cobordism category with a colored operad action arising from oriented planar arc diagrams. This is used to construct an invariant of oriented tangle diagrams with values in the homotopy category attached to the cobordism category. Motivated by Bar-Natan's categorification of the Jones polynomial, it categorifies the quantum $sl_n$ quantum invariants and is adapted to the categorification of the $sl_n$ quantum invariants by Khovanov and Rozansky using matrix factorizations. We conjecture to exist th…
A second part of detailed elementary introduction into Khovanov homologies. This part is devoted to reduced Jones superpolynomials. The story is still about a hypercube of resolutions of a link diagram. Each resolution is a collection of non-intersecting cycles, and one associates a 2-dimensional vector space with each cycle. Reduced superpolynomial arises when for all cycles, containing a "marked" edge of the link diagram, the vector space is reduced to 1-dimensional. The rest remains the same. Edges of the hypercube are associated with cut-and-join operators, acting on the cycles. Superpartn…
We extend Bar-Natan's cobordism based categorification of the Jones polynomial to virtual links. Our topological complex allows a direct extension of the classical Khovanov complex ($h=t=0$), the variant of Lee ($h=0,t=1$) and other classical link homologies. We show that our construction allows, over rings of characteristic two, extensions with no classical analogon, e.g. Bar-Natan's $\mathbb{Z}/2$-link homology can be extended in two non-equivalent ways. Our construction is computable in the sense that one can write a computer program to perform calculations, e.g. we have written a Mathemati…
This paper begins with a survey of some applications of Khovanov homology to low-dimensional topology, with an eye toward extending these results to $\mathfrak{sl}(n)$ homologies. We extend Levine and Zemke's ribbon concordance obstruction from Khovanov homology to $\mathfrak{sl}(n)$ homology for $n \geq 2$, including the universal $\mathfrak{sl}(2)$ and $\mathfrak{sl}(3)$ homology theories. Inspired by Alishahi and Dowlin's bounds for the unknotting number coming from Khovanov homology and relying on spectral sequence arguments, we produce bounds on the alternation number of a knot. Lee and B…
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 study the relationship between the HOMFLY and sl(N) knot homologies introduced by Khovanov and Rozansky. For each N>0, we show there is a spectral sequence which starts at the HOMFLY homology and converges to the sl(N) homology. As an application, we determine the KR-homology of knots with 9 crossings or fewer.
We define a homology $\mathcal{H}_N$ for closed braids by applying Khovanov and Rozansky's matrix factorization construction with potential $ax^{N+1}$. Up to a grading shift, $\mathcal{H}_0$ is the HOMFLYPT homology defined in arXiv:math/0505056. We demonstrate that, for $N \geq 1$, $\mathcal{H}_N$ is a $\mathbb{Z}_2\oplus\mathbb{Z}^{\oplus 3}$-graded $\mathbb{Q}[a]$-module that is invariant under transverse Markov moves, but not under negative stabilization/de-stabilization. Thus, for $N\geq 1$, this homology is an invariant for transverse links in the standard contact $S^3$, but not for smoo…
We apply the techniques of totally twisted Khovanov homology to the constructions by M. Asaeda, J. Przytycki, and A. Sikora of Khovanov type homologies for links and tangles in I-bundles over (orientable) surfaces. As a result we describe an invariant chain complex built out of resolutions with only non-contractible circles. We use these to understand the δ-graded homology for links with alternating projection to the surface.
We establish a direct map between refined topological vertex and sl(N) homological invariants of the of Hopf link, which include Khovanov-Rozansky homology as a special case. This relation provides an exact answer for homological invariants of the of Hopf link, whose components are colored by arbitrary representations of sl(N). At present, the mathematical formulation of such homological invariants is available only for the fundamental representation (the Khovanov-Rozansky theory) and the relation with the refined topological vertex should be useful for categorifying quantum group invariants a…
We show that the triply graded Khovanov-Rozansky homology of knots and links over a field of positive odd characteristic $p$ descends to an invariant in the homotopy category finite-dimensional $p$-complexes. A $p$-extended differential on the triply graded homology discovered by Cautis is compatible with the $p$-DG structure. As a consequence we get a categorification of the Jones polynomial evaluated at an odd prime root of unity
We use Khovanov-Rozansky gl(N) link homology to define invariants of oriented smooth 4-manifolds, as skein modules constructed from certain 4-categories with well-behaved duals. The technical heart of this construction is a proof of the sweep-around property, which makes these link homologies well defined in the 3-sphere.
Khovanov homology extends to singular links via a categorified analogue of Vassiliev skein relation. In view of Vassiliev theory, the extended Khovanov homology can be seen as Vassiliev derivatives of Khovanov homology. In this paper, we develop a new method to compute the first derivative. Namely, we introduce a complex, called a crux complex, and prove that the Khovanov homologies of singular links with unique double points are homotopic to cofibers of endomorphisms on crux complexes. Since crux complexes are actually small for some links, the result enables a direct computation of the first…
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…
Inspired by bordered Floer homology, we describe a type A structure on a Khovanov homology for a tangle, which complements the type D structure in a previous paper. The type A structure is a differential module over a certain algebra. This can be paired with the type D structure to recover the Khovanov chain complex. The homotopy type of the type A structure is a tangle invariant, and homotopy equivalences of the type A structure result in chain homotopy equivalences on the Khovanov chain complex. We can use this to simplify computations and introduce a modular approach to the computation of K…
We propose a framework for unifying the sl(N) Khovanov-Rozansky homology (for all N) with the knot Floer homology. We argue that this unification should be accomplished by a triply graded homology theory which categorifies the HOMFLY polynomial. Moreover, this theory should have an additional formal structure of a family of differentials. Roughly speaking, the triply graded theory by itself captures the large N behavior of the sl(N) homology, and differentials capture non-stable behavior for small N, including knot Floer homology. The differentials themselves should come from another variant o…
In this note we make an attempt to compare a cohomological theory of Hilbert spaces of ground states in the ${\cal N}=(2,2)$ 2d Landau-Ginzburg theory in models describing link embeddings in ${\mathbb{R}}^3$ to Khovanov and Khovanov-Rozansky homologies. To confirm the equivalence we exploit the invariance of Hilbert spaces of ground states for interfaces with respect to homotopy. In this attempt to study solitons and instantons in the Landau-Giznburg theory we apply asymptotic analysis also known in the literature as exact WKB method, spectral networks method, or resurgence. In particular, we …
We generalize our previous work on categorification of Kauffman bracket skein module of surfaces, by extending our homology to tangles in cylinders over surfaces, F x [0,1]. Our homology of 0-tangles and 1-tangles in D^3 coincides (up to normalization) with Khovanov link homology and the reduced Khovanov link homology. We prove the basic properties of our homology. In particular, the short exact sequence of homologies of skein related tangles and the Kunneth formula for the tensor product of tangles.
The SO(3) Kauffman polynomial and the chromatic polynomial of planar graphs are categorified by a unique extension of the Khovanov homology framework. Many structural observations and computations of homologies of knots and spin networks are included.
For every oriented surface of finite type, we construct a functorial Khovanov homology for links in a thickening of the surface, which takes values in a categorification of the corresponding gl(2) skein module. The latter is a mild refinement of the Kauffman bracket skein algebra, and its categorification is constructed using a category of gl(2) foams that admits an interesting non-negative grading. We expect that the natural algebra structure on the gl(2) skein module can be categorified by a tensor product that makes the surface link homology functor monoidal. We construct a candidate bifunc…
Given a band sum of a split two-component link along a nontrivial band, we obtain a family of knots indexed by the integers by adding any number of full twists to the band. We show that the knots in this family have the same Heegaard knot Floer homology and the same instanton knot Floer homology. In contrast, a generalization of the cosmetic crossing conjecture predicts that the knots in this family are all distinct. We verify this prediction by showing that any two knots in this family have distinct Khovanov homology. Along the way, we prove that each of the three knot homologies detects the …
We give a geometric interpretation of Bar-Natan's universal invariant for the class of tangles in the 3-ball with four ends: we associate with such 4-ended tangles $T$ multicurves $\widetilde{\operatorname{BN}}(T)$, that is, collections of immersed curves with local systems in the 4-punctured sphere. These multicurves are tangle invariants up to homotopy of the underlying curves and equivalence of the local systems. They satisfy a gluing theorem which recovers the reduced Bar-Natan homology of links in terms of wrapped Lagrangian Floer theory. Furthermore, we use $\widetilde{\operatorname{BN}}…
We conjecturally extract the triply graded Khovanov-Rozansky homology of the (m, n) torus knot from the unique finite dimensional simple representation of the rational DAHA of type A, rank n - 1, and central character m/n. The conjectural differentials of Gukov, Dunfield and the third author receive an explicit algebraic expression in this picture, yielding a prescription for the doubly graded Khovanov-Rozansky homologies. We match our conjecture to previous conjectures of the first author relating knot homology to q, t-Catalan numbers, and of the last three authors relating knot homology to H…
This thesis splits into two major parts. The connection between the two parts is the notion of "categorification" which we shortly explain/recall in the introduction. In the first part of this thesis we extend Bar-Natan's cobordism based categorification of the Jones polynomial to virtual links. Our topological complex allows a direct extension of the classical Khovanov complex (h=t=0), the variant of Lee (h=0,t=1) and other classical link homologies. We show that our construction allows, over rings of characteristic 2, extensions with no classical analogon, e.g. Bar-Natan's Z/2Z-link homology…
We construct complexes $P_{1^n}$ of Soergel bimodules which categorify the Young idempotents corresponding to one-column partitions. A beautiful recent conjecture of Gorsky-Rasmussen relates the Hochschild homology of categorified Young idempotents with the flag Hilbert scheme. We prove this conjecture for $P_{1^n}$ and its twisted variants. We also show that this homology is also a certain limit of Khovanov-Rozansky homologies of torus links. Along the way we obtain several combinatorial results which could be of independent interest.
We show that Rasmussen's invariant of knots, which is derived from Lee's variant of Khovanov homology, is equal to an analogous invariant derived from certain other filtered link homologies.
Besides offering a friendly introduction to knot homologies and quantum curves, the goal of these lectures is to review some of the concrete predictions that follow from the physical interpretation of knot homologies. In particular, this interpretation allows one to pose questions that would not have been asked otherwise, such as, "Is there a direct relation between Khovanov homology and the A-polynomial of a knot?" We will explain that the answer to this question is "yes," and introduce a certain deformation of the planar algebraic curve defined by the zero locus of the A-polynomial. This nov…
