A whittled complex for the Khovanov homology of torus links

Abstract

We give an algorithm for reducing the number of generators of the Khovanov chain complex of the torus braid $f{t}_n^k=({\sigma }_1{\sigma }_2\cdots {\sigma }_{n-1}{)}^k$ on $n$ strands by applying Bar-Natan Gaussian elimination along a distinguished set of Gaussian elimination isomorphisms. We call the resulting complex ${\mathcal{F}\mathcal{T}}_n^k$ a \emph{whittled complex} for the Khovanov homology of torus braids. Using this algorithm, we provide a bound for the number of generators at a fixed homological degree in our whittled complex.

32 pages, 30 figures. Comments welcome