Knot Group Epimorphisms, II

TL;DR

This paper investigates the relationships between knots based on epimorphisms of their groups, identifying specific conditions under which certain classes of knots can map onto others, especially focusing on torus and 2-bridge knots.

Contribution

It characterizes the possible epimorphic images of torus and 2-bridge knots, revealing constraints on the types and determinants of knots that can occur as images.

Findings

For torus knots, the epimorphism relation preserves the class, and the target must also be a torus knot.

In 2-bridge knots, epimorphisms with peripheral system preservation imply the target knot has a determinant dividing the original.

Only finitely many 2-bridge knots can be epimorphic images of a given 2-bridge knot under the peripheral-preserving relation.

Abstract

We consider the relations $\ge$ and $\ge_p$ on the collection of all knots, where $k \ge k'$ (respectively, $k \ge_p k'$) if there exists an epimorphism $\pi k \to \pi k'$ of knot groups (respectively, preserving peripheral systems). When $k$ is a torus knot, the relations coincide and $k'$ must also be a torus knot; we determine the knots $k'$ that can occur. If $k$ is a 2-bridge knot and $k \ge_p k'$, then $k'$ is a 2-bridge knot with determinant a proper divisor of the determinant of $k$; only finitely many knots $k'$ are possible.