Further Study of Kanenobu Knots

TL;DR

This paper computes the rational Khovanov homology groups for all Kanenobu knots and determines their crossing numbers in many cases, providing new insights into their topological properties.

Contribution

It fully determines the Khovanov homology for all Kanenobu knots and establishes crossing number results, including a conjecture for a specific class.

Findings

Khovanov homology groups for all Kanenobu knots are determined.

Crossing numbers are established for many Kanenobu knots.

A conjecture is proposed for the crossing number when pq < 0 and |pq| > max{|p|, |q|}.

Abstract

We determine the rational Khovanov bigraded homology groups of all Kanenobu knots. Also, we determine the crossing number for all Kanenobu knots $K(p,q)$ with $pq > 0$ or $|pq|\leq \max \{|p|, |q|\}$. In the case where $pq < 0$ and $|pq| > \max \{|p|, |q|\}$, we conjecture that the crossing number is $|p| + |q| + 8$.