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.

17 pages, 5 figures. For associated C++-program, see http://www.math.jussieu.fr/~lewark/foamho.html

References in corpus (4)

• Some differentials on Khovanov-Rozansky homology
• Rasmussen's spectral sequences and the sl(N)-concordance invariants
• The Khovanov-Rozansky Cohomology and Bennequin Inequalities
• Khovanov-Rozansky Homology and Conway Mutation

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• Computations of HOMFLY homology