Khovanov homology of torus links

In this paper we show that there is a cut-off in the Khovanov homology of (2k,2kn)-torus links, namely that the maximal homological degree of non-zero homology groups of (2k,2kn)-torus links is 2k2n. Furthermore, we calculate explicitly the homology group in homological degree 2k2n and prove that it coincides with the center of the ring Hk 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 S2×S1, in preparation]. Also we give an explicit formula for the ranks of the homology groups of (3,n)-torus knots for every n∈N.