sl3-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 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.
Keywords
Cite
@article{arxiv.1212.2553,
title = {sl3-foam homology calculations},
author = {Lukas Lewark},
journal= {arXiv preprint arXiv:1212.2553},
year = {2014}
}
Comments
17 pages, 5 figures. For associated C++-program, see http://www.math.jussieu.fr/~lewark/foamho.html