Stephan M. Wehrli

Categorification of the Colored Jones Polynomial and Rasmussen Invariant of Links

We define a family of formal Khovanov bracketsof a colored link depending on two param- eters. The isomorphismclassesof these bracketsare invariants of framed colored links. The Bar-Natan functors applied to these brackets produce Khovanov and Lee homology theories categorifying the colored Jones polynomial. Further, we study conditions under which framed colored link cobordisms induce chain transformations between our formal brackets. We conjecture that for special choice of parameters, Khovanov and Lee homology theories of colored links are functorial (up to sign). Fi- nally, we extend the Rasmussen invariant to links and give examples where this invariant is a stronger obstruction to sliceness than the multivariable Levine-Tristram signature.

Sutured Khovanov homology, Hochschild homology, and the Ozsváth-Szabó spectral sequence

In 2001, Khovanov and Seidel constructed a faithful action of the (m+1)-strand braid group on the derived category of left modules over a quiver algebra, A_m. We interpret the Hochschild homology of the Khovanov-Seidel braid invariant as a direct summand of the sutured Khovanov homology of the annular braid closure.

An Elementary Fact About Unlinked Braid Closures

Let \(n \in \mathbb {Z}^+\). We provide two short proofs of the following classical fact, one using Khovanov homology and one using Heegaard–Floer homology: if the closure of an n-strand braid \(\sigma \) is the n-component unlink, then \(\sigma \) is the trivial braid.