Genera of two-bridge knots and epimorphisms of their knot groups

TL;DR

This paper establishes a precise condition relating the genera of two-bridge knots for the existence of an epimorphism between their knot groups, advancing understanding of their algebraic relationships.

Contribution

It provides a necessary and sufficient condition based on genera for when an epimorphism exists between the knot groups of two-bridge knots.

Findings

Derived a condition linking genera of two-bridge knots for epimorphisms

Characterized when one knot group's algebraic structure maps onto another

Enhanced understanding of algebraic relationships among two-bridge knots

Abstract

Let $K,K'$ be two-bridge knots of genus $n,k$ respectively. We show the necessary and sufficient condition of $n$ in terms of $k$ that there exists an epimorphism from the knot group of $K$ onto that of $K'$.