Non-triviality of the Jones polynomial and the crossing numbers of amphicheiral knots

TL;DR

This paper investigates the Jones polynomial's role in determining crossing numbers of amphicheiral knots and explores its implications for various link classes, volume bounds, and knot properties.

Contribution

It establishes the crossing numbers of prime amphicheiral knots using the Jones polynomial and shows non-triviality of the polynomial for several link classes.

Findings

Crossing numbers of prime amphicheiral knots determined.

Semiadequate links and Whitehead doubles have non-trivial Jones polynomial.

Infinitely many positive knots lack positive minimal crossing diagrams.

Abstract

Using an involved study of the Jones polynomial, we determine, as our main result, the crossing numbers of (prime) amphicheiral knots. As further applications, we show that several classes of links, including semiadequate links and Whitehead doubles of semiadequate knots, have non-trivial Jones polynomial. We also prove that there are infinitely many positive knots with no positive minimal crossing diagrams. Some relations to the twist number of a link, Mahler measure and the hyperbolic volume are given, for example explicit upper bounds on the volume for Montesinos and 3-braid links in terms of their Jones polynomial.