Diagram genus, generators and applications

TL;DR

This paper advances the understanding of knot diagram genus by classifying generators for genus 4, applying these results to polynomial invariants, and confirming several conjectures for knots up to this genus.

Contribution

It provides a new description of generators using Hirasawa's algorithm, complete classification for genus 4, and applications to polynomial invariants and conjectures.

Findings

Classified all genus 4 knot generators.

Established non-triviality of skein and Jones polynomials for certain knots.

Proved conjectures of Hoste and Fox for genus up to 4.

Abstract

We continue the study of the genus of knot diagrams, deriving a new description of generators using Hirasawa's algorithm. This description leads to good estimates on the maximal number of crossings of generators and allows us to complete their classification for knots of genus 4. As applications of the genus 4 classification, we establish non-triviality of the skein polynomial on $k$-almost positive knots for $k\le 4$, and of the Jones polynomial for $k\le 3$. For $k\le 4$, we classify the occurring achiral knots, and prove a trivializability result for $k$-almost positive unknot diagrams. This yields also estimates on the number of unknotting Reidemeister moves. We describe the positive knots of signature (up to) 4. Using a study of the skein polynomial, we prove the exactness of the Morton-Williams-Franks braid index inequality and the existence of a minimal string Bennequin surface for alternating knots up to genus 4. We also prove for such knots conjectures of Hoste and Fox about the roots and coefficients of the Alexander polynomial.