arXiv research

We introduce some chain maps between Khovanov complexes. Each of the chain maps commutes with a chain homotopy map and a retraction maps which obtain a Reidemeister invariance of Khovanov homology.

Khovanov homology for links in S^3 via 1-tangle diagrams in annulus.

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. T…

This paper explores Khovanov adequacy in knot theory.

New homomorphism from Khovanov homology for knot concordance.

We simplify Khovanov homology for torus braids using Gaussian elimination.

Khovanov homology detects essential surfaces in knot complements.

Extends Jones' construction to Thompson's group F and link homology.

We show that the limiting Khovanov chain complex of any infinite positive braid categorifies the Jones-Wenzl projector. This result extends Lev Rozansky's categorification of the Jones-Wenzl projectors using the limiting complex of infinite torus braids. We also show a similar result for the limiting Lipshitz-Sarkar-Kh…

Proof of isotopy invariance in Khovanov link homology.

We give chain homotopy maps of Khovanov-type link homology of a universal differential. The universal differential, discussed by Mikhail Khovanov, Marco Mackaay, Paul Turner and Pedro Vaz, contains the original Khovanov's differential and Lee's differential. We also consider the conditions of any differential ensuring …

New proof found for Khovanov's assertion about Frobenius algebra twists.

Refines Khovanov homology using signed Burnside categories.

The Temperley-Lieb algebra is a fundamental component of SU(2) topological quantum field theories. We construct chain complexes corresponding to minimal idempotents in the Temperley-Lieb algebra. Our results apply to the framework which determines Khovanov homology. Consequences of our work include semi-orthogonal deco…

Motivated by the works of Krasner [arXiv:0801.4018] and Lobb [arXiv:1103.1412], we simplify the Khovanov-Rozansky chain complexes of open 2-braids. As an application, we show that, for a knot containing a "long" 2-braid, the sl(N) Rasmussen invariant of this knot depends linearly on the length of this 2-braid. We refin…

Ozsváth, Rasmussen and Szabó constructed odd Khovanov homology. It is a link invariant which has the same reduction modulo 2 as (even) Khovanov homology. Szabó 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 …

Study of Khovanov homology invariants from $U(1) imes U(1)$-equivariant algebra.

Fix an integer N>1. To each diagram of a link colored by 1,...,N, we associate a chain complex of graded matrix factorizations. We prove that the homotopy type of this chain complex is invariant under Reidemeister moves. When every component of the link is colored by 1, this chain complex is isomorphic to the chain com…

Defines odd Khovanov homology via categorification of q-Schur algebra.

Defines cobordism maps connecting Khovanov and instanton homologies.

Khovanov homology ist a new link invariant, discovered by M. Khovanov, and used by J. Rasmussen to give a combinatorial proof of the Milnor conjecture. In this thesis, we give examples of mutant links with different Khovanov homology. We prove that Khovanov's chain complex retracts to a subcomplex, whose generators are…

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 …

Mikhail Khovanov in math.QA/9908171 defined, for a diagram of an oriented classical link, a collection of groups numerated by pairs of integers. These groups were constructed as homology groups of certain chain complexes. The Euler characteristics of these complexes are coefficients of the Jones polynomial of the link.…

We show that the limiting unicolored $\mathfrak{sl}(N)$ Khovanov-Rozansky chain complex of any infinite positive braid categorifies a highest-weight projector. This result extends an earlier result of Cautis categorifying highest-weight projectors using the limiting complex of infinite torus braids. Additionally, we sh…

Using the diagrammatic calculus for Soergel bimodules developed by B. Elias and M. Khovanov, we show that Rouquier complexes are functorial over braid cobordisms. We explicitly describe the chain maps which correspond to movie move generators.

Khovanov homology for pro-tangles and spectral sequences

In this thesis we work with Khovanov homology of links and its generalizations, as well as with the homology of graphs. Khovanov homology of links consists of graded chain complexes which are link invariants, up to chain homotopy, with graded Euler characteristic equal to the Jones polynomial of the link. Hence, it can…

We define several homology theories for central hyperplane arrangements, categorifying well-known polynomial invariants including the characteristic polynomial, Poincare polynomial, and Tutte polynomial. We consider basic algebraic properties of such chain complexes, including long-exact sequences associated to deletio…

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 ci…

For a diagram of a 2-stranded tangle in the 3-ball we define a twisted complex of compact Lagrangians in the triangulated envelope of the Fukaya category of the smooth locus of the pillowcase. We show that this twisted complex is a functorial invariant of the isotopy class of the tangle, and that it provides a factoriz…

We use a special kind of 2-dimensional extended Topological Quantum Field Theories (TQFTs), so-called open-closed TQFTs, in order to extend Khovanov homology from links to arbitrary tangles, not necessarily even. For every plane diagram of an oriented tangle, we construct a chain complex whose homology is invariant und…

This paper reinterprets Khovanov-Sano symmetries using BV formalism.

For each graph, we construct a bigraded chain complex whose graded Euler characteristic is a version of the Tutte polynomial. This work is motivated by earlier work of Khovanov, Helme-Guizon and Rong, and others.

A homological invariant of 3-manifolds is defined, using abelian Yang-Mills gauge theory. It is shown that the construction, in an appropriate sense, is functorial with respect to the families of 4-dimensional cobordisms. This construction and its functoriality are used to define several link invariants. The strongest …

New spanning tree model connects knot homology, s-invariant, and exotic discs.

New torsion patterns found in Khovanov homology of link diagrams.

Given a connect sum of link diagrams, there is an isomorphism which decomposes unnormalized Khovanov chain groups for the product in terms of normalized chain groups for the factors; this isomorphism is straightforward to see on the level of chains. Similarly, any plumbing $x*y$ of Kauffman states carries an isomorphis…

Two categorifications are given for the arrow polynomial, an extension of the Kauffman bracket polynomial for virtual knots. The arrow polynomial extends the bracket polynomial to infinitely many variables, each variable corresponding to an integer {\it arrow number} calculated from each loop in an oriented state summa…

Quantum algorithm approximates Khovanov homology ranks.

We prove that the Khovanov-Lee complex of an oriented link, L, in a thickened annulus, A x I, has the structure of a bifiltered complex whose filtered chain homotopy type is an invariant of the isotopy class of L in A x I. Using ideas of Ozsvath-Stipsicz-Szabo as reinterpreted by Livingston, we use this structure to de…

We create a framework for odd Khovanov homology in the spirit of Bar-Natan's construction for the ordinary Khovanov homology. Namely, we express the cube of resolutions of a link diagram as a diagram in a certain 2-category of chronological cobordisms and show that it is 2-commutative: the composition of 2-morphisms al…

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…

The paper classifies links with low rank knot Floer and Khovanov homologies.

In this paper we give a quantum statistical interpretation for the bracket polynomial state sum <K> and for the Jones polynomial. We use this quantum mechanical interpretation to give a new quantum algorithm for computing the Jones polynomial. This algorithm is useful for its conceptual simplicity, and it applies to al…

In this paper we introduce a chain complex $C_{1 \pm 1}(D)$ where D is a plat braid diagram for a knot K. This complex is inspired by knot Floer homology, but it the construction is purely algebraic. It is constructed as an oriented cube of resolutions with differential d=d_0+d_1. We show that the E_2 page of the assoc…

We describe a modification of Khovanov homology (math.QA/9908171), in the spirit of Bar-Natan (math.GT/0410495), which makes the theory properly functorial with respect to link cobordisms. This requires introducing `disorientations' in the category of smoothings and abstract cobordisms between them used in Bar-Natan's …

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 filte…

New method computes first Vassiliev derivative of Khovanov homology.