The lexicographic degree of the first two-bridge knots

TL;DR

This paper determines the minimal polynomial degree representations of all two-bridge knots with up to 11 crossings, linking knot theory with real algebraic geometry and braid theory methods.

Contribution

It provides the lexicographic degree for these knots and introduces a novel approach connecting polynomial knot representations with algebraic trigonal curves and braid theory.

Findings

Lexicographic degrees for all two-bridge knots with ≤11 crossings determined.

Establishment of a method transforming polynomial knot problems into algebraic curve studies.

Application of Orevkov's braid theoretical method to knot polynomial representations.

Abstract

We study the degree of polynomial representations of knots. We give the lexicographic degree of all two-bridge knots with 11 or fewer crossings. First, we estimate the total degree of a lexicographic parametrisation of such a knot. This allows us to transform this problem into a study of real algebraic trigonal plane curves, and in particular to use the braid theoretical method developed by Orevkov.