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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 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 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.