Search for torsion in Khovanov homology

Abstract

In the Khovanov homology of links, presence of ${\mathbb{Z}}_2$-torsion is a very common phenomenon. Finite number of examples of knots with ${\mathbb{Z}}_n$-torsion for $n>2$ were also known, none for $n>8$. In this paper, we prove that there are infinite families of links whose Khovanov homology contains ${\mathbb{Z}}_n$-torsion for $2<n<9$ and ${\mathbb{Z}}_{2^s}$-torsion for $s<24$. We also introduce $4$-braid links with ${\mathbb{Z}}_3$-torsion which are counterexamples to the PS braid conjecture. We also provide an infinite family of knots with ${\mathbb{Z}}_5$-torsion in reduced Khovanov homology and ${\mathbb{Z}}_3$-torsion in odd Khovanov homology.

33 pages, 8 figures, and 33 tables

References in corpus (3)

• Homology of torus links
• Homotopy type of circle graphs complexes motivated by extreme Khovanov homology
• On the first group of the chromatic cohomology of graphs