On the lexicographic degree of two-bridge knots

TL;DR

This paper investigates the polynomial representation degrees of two-bridge knots, providing explicit lexicographic degrees for certain classes and establishing a lower bound for all knots using real polynomial curve properties.

Contribution

It introduces the first explicit lexicographic degree calculations for two-bridge torus and twist knots and derives a universal lower bound for the degree of polynomial knot representations.

Findings

Lexicographic degree for two-bridge torus knots determined

Lexicographic degree for generalized twist knots determined

A sharp lower bound for the lexicographic degree of any knot established

Abstract

We study the degree of polynomial representations of knots. We obtain the lexicographic degree for two-bridge torus knots and generalized twist knots. The proof uses the braid theoretical method developed by Orevkov to study real plane curves, combined with previous results from [KP10] and [BKP14]. We also give a sharp lower bound for the lexicographic degree of any knot, using real polynomial curves properties.