Non-meridional epimorphisms of knot groups

Abstract

In 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 $\left(G,G^{\prime }\right)$ such that $G^{\prime }$ is a homomorphic image of $G$ but no epimorphism of $G$ onto $G^{\prime }$ preserves meridians.