On standard forms of 1--dominations between knots with same Gromov volumes

Abstract

Let $k$ and $k^{\prime }$ be two knots in 3-sphere. Say $k$ 1--dominates $k^{\prime }$, if there is a proper degree 1 map $f\co E(k)\to E(k')$, between knot exterior of $k_i$. Theorem: Suppose that any companion of $k$ is prime. If $k$ 1--dominates $k^{\prime }$ with the same Gromov volume, then $k^{\prime }$ can be obtained from $k$ by finitely many de-satellizations. The condition of "same Gromov volume" clearly can not be removed. We also give a new construction of 1-domination between knots with same Gromov volume to show that the condition "any companion of $k$ is prime" can not be removed.

15 pages

References in corpus (2)

• Epimorphisms between 2-bridge link groups
• Non-zero degree maps between $2n$-manifolds