A whittled complex for the Khovanov homology of torus links

TL;DR

This paper introduces an algorithm to simplify the Khovanov chain complex for torus links, significantly reducing generators and providing bounds on their number at fixed homological degrees.

Contribution

It presents a novel algorithm for reducing generators in the Khovanov complex of torus braids, creating the 'whittled complex' and establishing bounds on generator counts.

Findings

The algorithm effectively reduces the number of generators.

The 'whittled complex' simplifies Khovanov homology computations.

Bounds on generators are established at fixed homological degrees.

Abstract

We give an algorithm for reducing the number of generators of the Khovanov chain complex of the torus braid $ft^k_n = (\sigma_1\sigma_2\cdots \sigma_{n-1})^k$ on $n$ strands by applying Bar-Natan Gaussian elimination along a distinguished set of Gaussian elimination isomorphisms. We call the resulting complex $\mathcal{FT}^k_n$ 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.