In 2000, Khovanov defined a link homology theory by categorifying the Jones polynomial of links. It is called Khovanov homology, now. In the paper under review, the author focuses on the torsion of Khovanov homology.
Based on extensive computations, the author gives some conjectures. In particular, the author conjectures that the Khovanov homology of every non-split link except for the trivial one, the Hopf link, and their connected sum has \(2\)-torsion (Conjecture 1).
A link is H-slim if its Khovanov homology groups with \(\mathbb{Z}\) coefficient are supported on two adjacent diagonals \(2i-j\)=const and there is no torsion on the upper diagonal. In the paper under review, the author proves that the Khovanov homology of every H-slim link has no torsion of order \(p^{k}\) for any odd prime \(p\) and \(k\geq \) (Theorem 1). As a corollary, we can conclude that the Khovanov homology of every non-split alternating link has no torsion of order \(p\) for any \(p\) other than a power of \(2\) (Corollary 2). Moreover, the author proves that every non-split alternating link except for the trivial one, the Hopf link, and their connected sum has \(2\)-torsion (Corollary 5).
In the original version of this paper, the author conjectured that no link has torsion of odd order in Khovanov homology. In the published version, this conjecture turned out to be false and the author presents a counterexample, the \((5,6)\)-torus knot whose Khovanov homology has \(3\)-torsion (p. 346, Remark).
There are some works related with this paper. For example, M. M. Asaeda and J. H. Przytycki also proved Corollary 5 in [J. M. Bryden, (ed.), Advances in topological quantum field theory. Proceedings of the NATO Advanced Research Workshop on new techniques in topological quantum field theory, Kananaskis Village, Canada, August 22–26, 2001. Dordrecht: Kluwer Academic Publishers. NATO Science Series II: Mathematics, Physics and Chemistry 179, 135–166 (2004; Zbl 1088.57012)]. M. D. Pabiniak et al. [Geom. Dedicata 140, 19–48 (2009; Zbl 1200.05088)] and [J. H. Przytycki and R. Sazdanović, Fundam. Math. 225, 277–303 (2014; Zbl 1295.57010)] discussed torsion of Khovanov homology groups for semi-adequate links.