Homological thickness and stability of torus knots

Stošić, Marko

doi:10.2140/agt.2007.7.261

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Homological thickness and stability of torus knots

Marko Stošić

Algebraic & Geometric Topology 7 (2007) 261–284

DOI: 10.2140/agt.2007.7.261

arXiv: math.GT/0511532

Abstract

In this paper we show that the nonalternating torus knots are homologically thick, ie that their Khovanov homology occupies at least three diagonals. Furthermore, we show that we can reduce the number of full twists of the torus knot without changing certain part of its homology, and consequently, there exists stable homology of torus knots conjectured by Dunfield, Gukov and Rasmussen in [Experiment. Math. 15 (2006) 129–159]. Since our main tool is the long exact sequence in homology, we have applied our approach in the case of the Khovanov–Rozansky $s l (n)$ homology, and thus obtained analogous stability properties of $s l (n)$ homology of torus knots, also conjectured by Dunfield, Gukov and Rasmussen.

Keywords

Khovanov homology, torus knots, thickness, stability

Mathematical Subject Classification 2000

Primary: 57M25

References

Publication

Received: 24 September 2006

Accepted: 22 November 2006

Published: 29 March 2007

Authors

Marko Stošić
Instituto de Sistemas e Robótica and CAMGSD Instituto Superior Técnico, TU Lisbon Av Rovisco Pais 1 1049-001 Lisbon Portugal

Volume 7, issue 1 (2007)