sl3-foam homology calculations

TL;DR

This paper presents new counterexamples to a conjecture relating two knot invariants, using an efficient computational tool for calculating sl_3-foam homology, and discusses potential improvements to the algorithm.

Contribution

It provides counterexamples to Lobb's conjecture and introduces a fast computational implementation for sl_3-foam homology calculations.

Findings

Counterexamples to Lobb's conjecture where |s_3| < |s_2|

Discovery of knots with |s_3| > |s_2|

Efficient C++ implementation for calculating Khovanov's sl_3-foam homology

Abstract

We exhibit a certain infinite family of three-stranded quasi-alternating pretzel knots which are counterexamples to Lobb's conjecture that the sl_3-knot concordance invariant s_3 (suitably normalised) should be equal to the Rasmussen invariant s_2. For this family, |s_3| < |s_2|. However, we also find other knots for which |s_3| > |s_2|. The main tool is an implementation of Morrison and Nieh's algorithm to calculate Khovanov's sl_3-foam link homology. Our C++-program is fast enough to calculate the integral homology of e.g. the (6,5)-torus knot in six minutes. Furthermore, we propose a potential improvement of the algorithm by gluing sub-tangles in a more flexible way.