Conjectures on the Khovanov Homology of Torus Knots, Twist Knots, and Legendrian Simple Knots

TL;DR

The paper explores the potential of Khovanov homology to uniquely identify certain classes of knots, providing numerical evidence and conjectures linking it to Legendrian simplicity.

Contribution

It conjectures that Khovanov homology can distinguish all torus and twist knots and may be connected to Legendrian simplicity.

Findings

All knots with the same Khovanov polynomial as a torus or twist knot are themselves such knots.

Numerical evidence supports the conjecture for knots with 20 or fewer crossings.

Khovanov homology may be able to distinguish Legendrian simple knots.

Abstract

A theorem of Kronheimer and Mrowka states that Khovanov homology is able to detect the unknot. That is, if a knot has the Khovanov homology of the unknot, then it is equivalent to it. Similar results hold for the trefoils and the figure-eight knot. We conjecture that Khovanov homology is able to distinguish all torus and twist knots. Numerical evidence has been gathered by examining all prime knots with 20 or fewer crossings, a total of 2,199,471,680 knots (not including mirrors). We found that all knots with the same Khovanov polynomial (the Poincar\'{e} polynomial of Khovanov homology) as a torus or twist knot are indeed torus or twist knots themselves. Since torus knots are known to be Legendrian simple, and since all twist knots $K_{m}$ with $m\geq{-3}$ are Legendrian simple, this provides evidence for the claim that Khovanov homology and Legendrian simplicity may be connected. We conjecture that indeed Khovanov homology is able to distinguish Legendrian simple knots and use the (conjectured) Legendrian simple knots from the Legendrian knot atlas to test this claim. A similar observation was made, and no knots with 20 or fewer crossing share their Khovanov polynomial with the knots in the Legendrian knot atlas (except for the knots that are a part of this atlas).