Homology of torus links

Abstract

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 group of $(2k,2kn)$-torus link is $2k^{2}n$. Furthermore, we calculate explicitely the homology groups in homological degree $2k^{2}n$ and prove that it coincides with the centre of the ring $H^k$ of crossingless matchings, introduced by M. Khovanov in \cite{tan}. Also we give an explicit formula for the ranks of the homology groups of $(3,n)$-torus knots for every $n\in \mathbb{N}$.

18 pages; expanded version

References in corpus (1)

• Khovanov-Rozansky homology of two-bridge knots and links

Cited by in corpus (5)

• State Cycles, Quasipositive Modification, and Constructing H-thick Knots in Khovanov Homology
• The dealternating number and the alternation number of a closed 3-braid
• The Universal sl_2 Link Homology Theory
• Graphical methods establishing nontriviality of state cycle Khovanov homology classes
• A categorification of the quantum sl(N)-link polynomials using foams