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

TL;DR

This paper investigates the conditions under which one knot 1--dominates another with the same Gromov volume, establishing that such domination implies a finite sequence of de-satellizations under certain conditions.

Contribution

It proves that if a knot 1--dominates another with equal Gromov volume and all companions are prime, then the latter can be obtained via finitely many de-satellizations, and provides a new construction illustrating the necessity of conditions.

Findings

1--domination implies de-satellization sequence under prime companion condition

Same Gromov volume is essential for the main theorem

New construction shows the prime companion condition cannot be removed

Abstract

Let $k$ and $k'$ be two knots in 3-sphere. Say $k$ 1--dominates $k'$, 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 $k'$ with the same Gromov volume, then $k'$ 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.