Torsion of the Khovanov homology

Details Torsion of the Khovanov homology Torsion of the Khovanov homology

2004-05-25

arxiv.org/abs/math/0405474v1

Mathematics

Geometric Topology

18 pages, 5 figures, and 4 tables

Scientific paper

Khovanov homology is a recently introduced invariant of oriented links in S^3. Its (graded) Euler characteristic is a version of the Jones polynomial of the link. In this paper we study torsion of the Khovanov homology. Based on our calculations, we conjecture that * every link except the trivial knot, the Hopf link, their connected sums and disjoint unions has torsion of order 2; * no link has torsion of odd order; * all homologically thin links are torsion thin, in particular, their torsion is completely determined by the free part of the Khovanov homology; * a knot is torsion rich if and only if its reduced Khovanov homology has torsion; * two knots have the same ranks of the Khovanov homology if and only if they have the same torsion. We prove the first two conjectures for all non-split alternating links. We also prove a weakened version of the third conjecture, namely, that all homology slim links are weakly torsion thin.

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