Non-meridional epimorphisms of knot groups

TL;DR

This paper demonstrates that there are infinitely many pairs of prime knot groups where one is a homomorphic image of the other, but no epimorphism preserves meridians, challenging previous assumptions in knot theory.

Contribution

It provides counterexamples showing that not all homomorphic images of knot groups admit meridian-preserving epimorphisms, revealing new complexity in knot group mappings.

Findings

Existence of infinitely many prime knot group pairs without meridian-preserving epimorphisms.

Counterexamples to the assumption that homomorphic images always admit such epimorphisms.

Clarification of the relationship between homomorphisms and meridian-preserving epimorphisms in knot groups.

Abstract

In the literature of the study of knot group epimorphisms, the existence of an epimorphism between two given knot groups is mostly (if not always) shown by giving an epimorphism which preserves meridians. A natural question arises: is there an epimorphism preserving meridians whenever a knot group is a homomorphic image of another? We answer in the negative by presenting infinitely many pairs of prime knot groups (G,G') such that G' is a homomorphic image of G but no epimorphism of G onto G' preserves meridians.