Knot Group Epimorphisms, II

Abstract

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

14 pages, 1 figure