$\mathfrak{sl}_3$–foam homology calculations

Abstract

We exhibit a certain infinite family of three-stranded quasi-alternating pretzel knots, which are counterexamples to Lobb’s conjecture that the ${\mathfrak{s}\mathfrak{l}}_3$–knot concordance invariant $s_3$ (suitably normalised) should be equal to the Rasmussen invariant $s_2$. For this family, $\left|s_3\right|<\left|s_2\right|$. However, we also find other knots for which $\left|s_3\right|>\left|s_2\right|$. The main tool is an implementation of Morrison and Nieh’s algorithm to calculate Khovanov’s ${\mathfrak{s}\mathfrak{l}}_3$–foam link homology. Our C++ program is fast enough to calculate the integral homology of, eg, the $\left(6,5\right)$–torus knot in six minutes. Furthermore, we propose a potential improvement of the algorithm by gluing sub-tangles in a more flexible way.