Verification Of The Jones Unknot Conjecture Up To 22 Crossings

TL;DR

This paper confirms that the Jones polynomial distinguishes the unknot for all knots with up to 22 crossings through extensive computer enumeration and novel computational strategies.

Contribution

It introduces new methods to efficiently compute Jones polynomials and verify unknottedness for knots up to 22 crossings, extending previous results.

Findings

Jones polynomial distinguishes the unknot up to 22 crossings

Developed novel computational strategies for knot analysis

Discovered new unknot diagrams without crossing-reducing moves

Abstract

We proved by computer enumeration that the Jones polynomial distinguishes the unknot for knots up to 22 crossings. Following an approach of Yamada, we generated knot diagrams by inserting algebraic tangles into Conway polyhedra, computed their Jones polynomials by a divide-and-conquer method, and tested those with trivial Jones polynomials for unknottedness with the computer program SnapPy. We employed numerous novel strategies for reducing the computation time per knot diagram and the number of knot diagrams to be considered. That made computations up to 21 crossings possible on a single processor desktop computer. We explain these strategies in this paper. We also provide total numbers of algebraic tangles up to 18 crossings and of Conway polyhedra up to 22 vertices. We encountered new unknot diagrams with no crossing-reducing pass moves in our search. We report one such diagram in this paper.