#### Volume 13, issue 6 (2013)

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

### Lukas Lewark

Algebraic & Geometric Topology 13 (2013) 3661–3686
##### 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, $|{s}_{3}|<|{s}_{2}|$. 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 ${\mathfrak{s}\mathfrak{l}}_{3}$–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.

##### Keywords
webs, foams, pretzel knots, four-ball genus, Khovanov–Rozansky homologies, Rasmussen invariant, $\mathfrak{sl}_N$ concordance invariants
Primary: 57M25
Secondary: 81R50
##### Publication
Revised: 27 May 2013
Accepted: 18 June 2013
Published: 16 October 2013
##### Authors
 Lukas Lewark Institut de Mathématiques de Jussieu (IMJ) – Paris Rive Gauche Bâtiment Sophie Germain Case 7012 75205 Paris Cedex 13 France http://www.math.jussieu.fr/~lewark/