Related papers: Khovanov homology and cobordisms between split lin…
Khovanov homology is a recently introduced invariant of oriented links in $\mathbb{R}^3$. It categorifies the Jones polynomial in the sense that the (graded) Euler characteristic of the Khovanov homology is a version of the Jones polynomial…
A handlebody-link is a disjoint union of embeddings of handlebodies in $S^3$ and an HL-homotopy is an equivalence relation on handlebody-links generated by self-crossing changes. The second author and Ryo Nikkuni classified the set of…
In the cabling procedure for HOMFLY polynomials colored HOMFLY polynomials of a knot are obtained from ordinary HOMFLY of the cabled knot with extra twists added. Thus colored polynomials can be seen as relation between HOMFLYs of cabled…
This is the author's diploma thesis. We describe a simplification in the construction of Khovanov-Rozansky's categorification of quantum sl(n) link homology using the theory of maximal Cohen-Macaulay modules over hypersurface singularities…
A revised proof of the author's earlier result is given. It is shown that a boundary surface-link in the 4-sphere is a ribbon surface-link if the surface-link obtained from it by surgery along a pairwise nontrivial fusion 1-handle system is…
Let L be a link in an thickened annulus. We specify the embedding of this annulus in the three sphere, and consider its complement thought of as the axis to L. In the right circumstances this axis lifts to a null-homologous knot in the…
In the present paper, we construct the Khovanov homology theory for virtual links. Besides the direct approach with Z_{2} coefficients we also describe the Khovanov homology for framed links and the Khovanov homology using ``double cover''.…
In the first part of this paper, we constructed a filtered U(r)-equivariant stable homotopy type called the spectrum of strict broken symmetries sB(L) of links L given by closing a braid with r strands. Evaluating this filtered spectrum on…
We prove a conjecture of Migdail and Wehrli regarding the odd Khovanov cobordism maps associated to knotted spheres. Our key tool is Daemi's plane Floer homology, which we use in place of a Lee deformation. Continuing the analogy with Lee…
We observe that Khovanov homology detects causality in $(2+1)$-dimensional globally hyperbolic spacetimes whose Cauchy surface is homeomorphic to $\mathbb R^2$
Given an $m$-periodic link $L\subset S^3$, we show that the Khovanov spectrum $\mathcal{X}_L$ constructed by Lipshitz and Sarkar admits a homology group action. We relate the Borel cohomology of $\mathcal{X}_L$ to the equivariant Khovanov…
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…
We give a simple, combinatorial construction of a unital, spherical, non-degenerate $\ast$-planar algebra over the ring $\mathbb{Z}[q^{1/2},q^{-1/2}]$. This planar algebra is similar in spirit to the Temperley-Lieb planar algebra, but…
We study the Khovanov complex of closed piecewise linear curves in the 3-space. A polygonal link representation endows the cube of resolutions with an additional combinatorial structure. The set of symmetries preserving this structure and…
In this paper, we discuss a proof of the isotopy invariance of a parametrized Khovanov link homology including categorifications of the Jones polynomial and the Kauffman bracket polynomial though it is a known fact. In order to present a…
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…
In this note, we prove the existence of a tri-graded Khovanov-type bicomplex (Theorem 1.2). The graded Euler characteristic of the total complex associated with this bicomplex is the colored Jones polynomial of a link. The first grading of…
Khovanov homology is a powerful invariant of oriented links that categorifies the Jones polynomial. Nevertheless, computing Khovanov homology of a given link remains challenging in general with current techniques. In this work we focus on…
We construct an algebra of non-trivial homological operations on Khovanov homology with coefficients in $\mathbb Z_2$ generated by two Bockstein operations. We use the unified Khovanov homology theory developed by the first author to lift…
We establish an isomorphism between the Khovanov-Rozansky triply graded link homology and the geometric triply graded homology due to the authors. Hence we provide an interpretation of the Khovanov-Rozansky homology of the closure of a…