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Abstract
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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
homology, and thus obtained analogous stability properties of
homology of torus knots, also conjectured by Dunfield, Gukov and Rasmussen.
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Keywords
Khovanov homology, torus knots, thickness, stability
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Mathematical Subject Classification 2000
Primary: 57M25
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Publication
Received: 24 September 2006
Accepted: 22 November 2006
Published: 29 March 2007
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