On the braid index of a two-bridge knot

TL;DR

This paper investigates the braid index of two-bridge knots, establishing inequalities related to knot group epimorphisms and calculating average braid indices for knots with fixed crossing numbers.

Contribution

It introduces new inequalities linking braid indices and knot group epimorphisms and computes average braid indices for two-bridge knots with specific crossing numbers.

Findings

Proved an inequality on braid indices based on knot group epimorphisms.

Calculated average braid index for all two-bridge knots with a given crossing number.

Provided theoretical insights into the structure of two-bridge knots.

Abstract

In this paper, we consider two properties on the braid index of a two-bridge knot. We prove an inequality on the braid indices of two-bridge knots if there exists an epimorphism between their knot groups. Moreover, we provide the average braid index of all two-bridge knots with a given crossing number.