Related papers: Notes on Khovanov homology
We modify the definition of the Khovanov complex for oriented links in a thickening of an oriented surface to obtain a triply graded homological link invariant with a new homotopical grading.
We study the relationship between Bar-Natan's perturbation in Khovanov homology and Szabo's geometric spectral sequence, and construct a link invariant that generalizes both into a common theory. We study a few properties of the new…
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…
This paper is a brief overview of recent results by the authors relating colored Jones polynomials to geometric topology. The proofs of these results appear in the papers [arXiv:1002.0256] and [arXiv:1108.3370], while this survey focuses on…
A discussion given to the question of extending Khovanov homology from links to embedded graphs, by using the Kauffman topological invariant of embedded graphs by associating family of links and knots to a such graph by using some local…
In two previous papers, the author showed how to decompose the Khovanov homology of a link $\mathcal{L}$ into the algebraic pairing of a type D structure and a type A structure (as defined in bordered Floer homology), whenever a diagram for…
Khovanov homology of a link and chromatic graph homology are known to be isomorphic in a range of homological gradings that depend on the girth of a graph. We discuss patterns shared by these two homology theories. In particular, we improve…
Quantum invariants like the colored Jones polynomial are algebraic in nature but are conjectured to detect important information about the geometry of links. In this thesis we explore these connections using an enhanced version of the RT…
We conjecture that the stable Khovanov homology of torus knots can be described as the Koszul homology of an explicit non-regular sequence of quadratic polynomials. The corresponding Poincare series turns out to be related to the…
We study homology groups of posets with functor coefficients and apply our results to give a novel approach to study Khovanov homology of knots and related homology theories.
This expository paper is based on the lectures given at the program `Modular Representation Theory of Finite and $p$-adic Groups' at the National University of Singapore. We are concerned with recent results on representation theory and…
Khovanov defined graded homology groups for links L in R^3 and showed that their polynomial Euler characteristic is the Jones polynomial of L. Khovanov's construction does not extend in a straightforward way to links in I-bundles M over…
A spatial refinement of Bar-Natan homology is given, that is, for any link diagram $D$ we construct a CW-spectrum $\mathcal{X}_{\mathit{BN}}(D)$ whose reduced cellular cochain complex gives the Bar-Natan complex of $D$. The stable homotopy…
A spectral sequence is established, from Bar-Natan's variant of Khovanov homology to a deformation of instanton homology for knots and links. This spectral sequence arises as a specialization of a spectral sequence from a characteristic-2…
This paper formulates a generalization of our work on quantum knots to explain how to make quantum versions of algebraic, combinatorial and topological structures. We include a description of previous work on the construction of Hilbert…
This is the second of a series of two technical papers devoted to the analysis of holonomy invariants in strict higher gauge theory with end applications in higher Chern--Simons theory. We provide a definition of trace over a crossed module…
Let $A$ be a noetherian connected graded algebra. We introduce and study homological invariants that are weighted sums of the homological and internal degrees of cochain complexes of graded $A$-modules, providing weighted versions of…
We review the construction and context of a stable homotopy refinement of Khovanov homology.
Into this note we collect topics related to homogeneous vector bundles, elliptic adjoint orbits and so forth.
We construct a bicomplex for the categorification of the colored Jones polynomial. This work is motivated by the problem suggested by Anna Beliakova and Stephan Wehrli who discussed the categorification of the colored Jones polynomial in…