sl3–foam homology calculations

We exhibit a certain infinite family of three-stranded quasi-alternating pretzel knots, which are counterexamples to Lobb’s conjecture that the ${s l}_{3}$ –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 ${s l}_{3}$ –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.

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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/

Volume 13, issue 6 (2013)

Lukas Lewark

Abstract