Related papers: Annular Evaluation and Link Homology
We introduce extensions of Khovanov homology and the Lee and Bar-Natan spectral sequences for links in $ \mathbb{RP}^3 $. These extensions are distinct to those previously defined by Asaeda-Przytycki-Sikora (and Gabrov\v{s}ek's…
We compare eight versions of finite-dimensional categorifications of the colored Jones polynomial and show that they yield isomorphic results over a field of characteristic zero. As an application, we verify a physics-motivated conjectural…
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.…
We introduce a class of links strictly containing quasi-alternating links for which mod 2 reduced Khovanov homology is always thin. We compute the framed instanton homology for double branched covers of such links. Aligning certain dotted…
We show that there is a spectral sequence with $E^2$-page given by the Khovanov homology of a link in $S^1\times S^2$, as defined by Rozansky in arXiv:1011.1958, which converges to the Hochschild homology of an $A_\infty$-bimodule defined…
In the first of these two lectures, I use a comparison to symplectic Khovanov homology to motivate the idea that the Jones polynomial and Khovanov homology of knots can be defined by counting the solutions of certain elliptic partial…
In this article we study the Heegaard Floer link homology of $(n, n)$-torus links. The Alexander multigradings which support non-trivial homology form a string of $n-1$ unit hypercubes in $\mathbb{R}^{n}$, and we compute the ranks and…
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 refine Khovanov homology in the presence of an involution on the link. This refinement takes the form of a triply-graded theory, arising from a pair of filtrations. We focus primarily on strongly invertible knots and show, for instance,…
Let K in S^3 be a knot, and let \widetilde{K} denote the preimage of K inside its double branched cover, \Sigma(K). We prove, for each integer n > 1, the existence of a spectral sequence from Khovanov's categorification of the reduced…
In arXiv:math/0508510, Rasmussen observed that the Khovanov-Rozansky homology of a link is a finitely generated module over the polynomial ring generated by the components of this link. In the current paper, we study the module structure of…
In this paper we analyze the quantum homological invariants (the Poincar\'e polynomials of the $\mathfrak{sl}_N$ link homology). In the case when the dimensions of homologies of appropriate topological spaces are precisely known, the…
The Jones polynomial can be expressed in terms of spanning trees of the graph obtained by checkerboard coloring a knot diagram. We show there exists a complex generated by these spanning trees whose homology is the reduced Khovanov…
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…
Lobb observed in [arXiv:1103.1412] that each equivariant sl(N) Khovanov-Rozansky homology over C[a] admits a standard decomposition of a simple form. In the present paper, we derive a formula for the corresponding Lee-Gornik spectral…
We define an annular version of odd Khovanov homology and prove that it carries an action of the Lie superalgebra $\mathfrak{gl}(1|1)$ which is preserved under annular Reidemeister moves.
This expository note outlines why it is sometimes useful to consider the bigraded type A link homology theories as associated with the Lie algebras gl(N) instead of sl(N).
We prove that E_n-homology of non-unital commutative algebras can be described as functor homology when one considers functors from a certain category of planar trees with n levels. For different n these homology theories are connected by…
We prove the Categorified Wrapping Number Conjecture for large classes of annular links, including alternating annular links and tangle closures exhibiting plumbed link phenomena. We do so by characterizing when a resolution is sufficient…
We prove that the detection rate of n-crossing alternating links by many standard link invariants decays exponentially in n, implying that they detect alternating links with probability zero. This phenomenon applies broadly, in particular…