On minimality of two-bridge knots

TL;DR

This paper investigates the minimality of two-bridge knots by analyzing their knot groups and determining which knots are minimal within specific parameter ranges.

Contribution

It provides a classification of minimal two-bridge knots for certain parameters, expanding understanding of their algebraic properties.

Findings

Identifies minimal two-bridge knots with q ≤ 6 or p ≤ 100.

Establishes criteria for minimality based on knot group epimorphisms.

Contributes to the algebraic classification of two-bridge knots.

Abstract

A knot is called minimal if its knot group admits epimorphisms onto the knot groups of only the trivial knot and itself. In this paper, we determine which two-bridge knot $\mathfrak{b}(p,q)$ is minimal where $q \leq 6$ or $p \leq 100$.