Volume 13, issue 6 (2013)

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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
$\mathfrak{sl}_3$–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 sl3–knot concordance invariant s3 (suitably normalised) should be equal to the Rasmussen invariant s2. For this family, |s3| < |s2|. However, we also find other knots for which |s3| > |s2|. The main tool is an implementation of Morrison and Nieh’s algorithm to calculate Khovanov’s sl3–foam link homology. Our C++ program is fast enough to calculate the integral homology of, eg, 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
webs, foams, pretzel knots, four-ball genus, Khovanov–Rozansky homologies, Rasmussen invariant, $\mathfrak{sl}_N$ concordance invariants
Mathematical Subject Classification 2010
Primary: 57M25
Secondary: 81R50
References
Publication
Received: 16 February 2013
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/