Related papers: Structures in HOMFLY-PT homology
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…
Khovanov--Rozansky's HOMFLY homology is determined for all prime knots with up to 11 crossings by direct computations.
We compute the reduced version of Khovanov and Rozansky's sl(N) homology for two-bridge knots and links. The answer is expressed in terms of the HOMFLY polynomial and signature.
Motivated by the $y$-ification of HOMFLY--PT homology by Gorsky and Hogancamp, and the $\mathfrak{sl}_2$-action of Gorsky, Hogancamp, and Mellit, we construct $y$-ifications of Khovanov homology and its equivariant versions within…
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…
The aim of this paper is two-fold. First, we give a fully geometric description of the HOMFLYPT homology of Khovanov-Rozansky. Our method is to construct this invariant in terms of the cohomology of various sheaves on certain algebraic…
We define a wall-crossing morphism for Khovanov-Rozansky homology; that is, a map between the KR homology of knots related by a crossing change. Using this map, we extend KR homology to an invariant of singular knots categorifying the…
We construct an $\mathfrak{sl}_2$-action on equivariant $\mathfrak{gl}_N$-link homologies. As a consequence we obtain an action of $\mathfrak{sl}_2$ on these homologies as well as a $p$-DG structures for $p$ a prime number. We explore…
We define a deformation of the triply graded Khovanov-Rozansky homology of a link $L$ depending on a choice of parameters $y_c$ for each component of $L$, which satisfies link-splitting properties similar to the Batson-Seed invariant.…
Khovanov has given a construction of the Khovanov-Rozansky link invariants (categorifying the HOMFLYPT invariant) using Hochschild cohomology of 2-braid groups. We give a direct proof that his construction does give link invariants. We show…
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…
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…
We conjecture a relation between the sl(N) knot homology, recently introduced by Khovanov and Rozansky, and the spectrum of BPS states captured by open topological strings. This conjecture leads to new regularities among the sl(N) knot…
M. Khovanov and L. Rozansky gave a categorification of the HOMFLY-PT polynomial. This study is a generalization of the Khovanov-Rozansky homology. We define a homology associated to the quantum $(sl_n,\land V_n)$ link invariant, where…
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…
In "Homfly polynomial via an invariant of colored plane graphs", Murakami, Ohtsuki, and Yamada provide a state-sum description of the level $n$ Jones polynomial of an oriented link in terms of a suitable braided monoidal category whose…
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…
The HOMFLY-PT polynomial is a link invariant which is effective in determining chiral knot and link types with small crossing numbers. In this chapter, we concentrate on knots. We provide a guide for computing the knot types of…
We give a simple recursion which computes the triply graded Khovanov-Rozansky homology of several infinite families of knots and links, including the $(n,nm\pm 1)$ and $(n,nm)$ torus links for $n,m\geq 1$. We interpret our results in terms…
By means of Rasmussen's formulation of Khovanov-Rozansky homology originally given over $\mathbb{Q}$ in arXiv:math/0607544, we compare different flavors of $\mathfrak{sl}(n)$ link homology with the link invariants obtained by Kitchloo in…