1-domination of knots

TL;DR

This paper investigates the concept of 1-domination among knots in the 3-sphere, proposing that a knot that 1-dominates another is generally more complex, supported by various evidence.

Contribution

It introduces the notion of 1-domination between knots via degree 1 maps of exteriors and provides evidence that such domination correlates with knot complexity.

Findings

Evidence supporting the idea that 1-dominating knots are more complex.

Development of a framework to compare knots based on 1-domination.

Insights into the hierarchy of knots through degree 1 maps.

Abstract

We say that a knot $k_1$ in the $3$-sphere {\it $1$-dominates} another $k_2$ if there is a proper degree 1 map $E(k_1) \to E(k_2)$ between their exteriors, and write $k_1 \ge k_2$. When $k_1 \ge k_2$ but $k_1 \ne k_2$ we write $k_1 > k_2$. One expects in the latter eventuality that $k_1$ is more {\it complicated}. In this paper we produce various sorts of evidence to support this philosophy.