sl3–foam homology calculations

Lewark, Lukas

doi:10.2140/agt.2013.13.3661

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the Journal

Editorial Board

Subscriptions

Submission Guidelines

Submission Page

Policies for Authors

Ethics Statement

ISSN 1472-2739 (online)

ISSN 1472-2747 (print)

Author Index

To Appear

Other MSP Journals

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

Lukas Lewark

Algebraic & Geometric Topology 13 (2013) 3661–3686

DOI: 10.2140/agt.2013.13.3661

Abstract

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.

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)