Related papers: A type A structure in Khovanov Homology
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…
In this note, we construct a closed model structure on the category of $\mathbb{Z}/2\mathbb{Z}$-graded complexes of projective systems of ind-Banach spaces. When the base field is the fraction field $F$ of a complete discrete valuation ring…
We show that basic homotopical notions such as homotopy sets and groups, connected and truncated maps, cellular constructions and skeleta, etc., extend to the setting of $(\infty,\infty)$-categories, as well as to presentable categories…
We introduce a Poisson version of the graded twist of a graded associative algebra and prove that every graded Poisson structure on a connected graded polynomial ring $A:=\Bbbk[x_1,\ldots,x_n]$ is a graded twist of a unimodular Poisson…
In this paper, we discuss degree 0 crossing change on Khovanov homology in terms of cobordisms. Namely, using Bar-Natan's formalism of Khovanov homology, we introduce a sum of cobordisms that yields a morphism on complexes of two diagrams…
The alpha complex is a subset of the Delaunay triangulation and is often used in computational geometry and topology. One of the main drawbacks of using the alpha complex is that it is non-monotone, in the sense that if ${\cal…
We introduce the notion of a template for discrete Morse theory. Templates provide a memory efficient approach to the computation of homological invariants (e.g., homology, persistent homology, Conley complexes) of cell complexes. We…
This chapter lays out a framework for discussing (\ast)-structures on module-algebras over a Hopf (\ast)-algebra (H). We define a complex conjugation functor (V \mapsto \bar{V}), which is an involution on the module category (\hmod), and…
We explain how to calculate link homology for a Lie algebra $\mathfrak{g}$ using the Fukaya category associated to a 2d A-model. Links are represented as configurations of particular A-branes and link homology is given by Homs between these…
The aim of this paper is to explain how, through the work of a number of people, some algebraic structures related to groupoids have yielded algebraic descriptions of homotopy n-types. Further, these descriptions are explicit, and in some…
The standard methods for calculating Khovanov homology rely either on long exact/spectral sequences or on the algorithmic "divide and conquer" approach developed by Bar-Natan. In this paper, we employ an alternative and arguably simpler…
Using Bar-Natan's Khovanov homology we define a homology theory for coloured, oriented, framed links. We then compute this explicitly.
The goal of this paper is to address A. Shumakovitch's conjecture about the existence of $\Z_2$-torsion in Khovanov link homology. We analyze torsion in Khovanov homology of semi-adequate links via chromatic cohomology for graphs which…
We prove that the Khovanov spectra associated to links and tangles are functorial up to homotopy and sign.
We survey several mathematical developments in the holonomy approach to gauge theory. A cornerstone of this approach is the introduction of group structures on spaces of based loops on a smooth manifold, relying on certain homotopy…
Inspired by the $S^n$ colored version of Khovanov and Khovanov-Rozansky homology, we define a colored version of knot Floer homology by studying the colimit of a directed system of link Floer homology with infinite full twists.…
We consider families of reductive complexes related by level-raising operators and originating from an associative algebra. In the main theorem it is shown that the multiple cohomology of that complexes is given by the factor space of…
We show that the rational homotopy type of the complement of a toric arrangement is completely determined by two sets of combinatorial data. This is obtained by introducing a differential graded algebra over Q whose minimal model is…
Let F denote the homotopy fiber of a map f:K-->L of 2-reduced simplicial sets. Using as input data the strongly homotopy coalgebra structure of the chain complexes of K and L, we construct a small, explicit chain algebra, the homology of…
We construct a spectral sequence relating the Khovanov homology of a strongly invertible knot to the annular Khovanov homologies of the two natural quotient knots. Using this spectral sequence, we re-prove that Khovanov homology…