Volume 7, issue 1 (2007)

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Homological thickness and stability of torus knots

Marko Stošić

Algebraic & Geometric Topology 7 (2007) 261–284

arXiv: math.GT/0511532


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 sl(n) homology, and thus obtained analogous stability properties of sl(n) homology of torus knots, also conjectured by Dunfield, Gukov and Rasmussen.

Khovanov homology, torus knots, thickness, stability
Mathematical Subject Classification 2000
Primary: 57M25
Received: 24 September 2006
Accepted: 22 November 2006
Published: 29 March 2007
Marko Stošić
Instituto de Sistemas e Robótica and CAMGSD
Instituto Superior Técnico, TU Lisbon
Av Rovisco Pais 1
1049-001 Lisbon