Khovanov homology of torus links

Abstract

In this paper we show that there is a cut-off in the Khovanov homology of ( 2 k , 2 k n ) -torus links, namely that the maximal homological degree of non-zero homology groups of ( 2 k , 2 k n ) -torus links is 2 k 2 n . Furthermore, we calculate explicitly the homology group in homological degree 2 k 2 n and prove that it coincides with the center of the ring H k of crossingless matchings, introduced by M. Khovanov in [M. Khovanov, A functor-valued invariant for tangles, Algebr. Geom. Topol. 2 (2002) 665–741, arXiv:math.QA/0103190]. This gives the proof of part of a conjecture by M. Khovanov and L. Rozansky in [M. Khovanov, L. Rozansky, A homology theory for links in S 2 × S 1 , in preparation]. Also we give an explicit formula for the ranks of the homology groups of ( 3 , n ) -torus knots for every n ∈ N .