Some minimal elements for a partial order of prime knots

TL;DR

This paper investigates a partial order on prime knots based on epimorphisms between their groups, establishing minimality for all prime knots with up to six crossings and for fibered knots with irreducible Alexander polynomials.

Contribution

It proves minimality of prime knots with up to six crossings and fibered knots with irreducible Alexander polynomials within this partial order.

Findings

All prime knots with up to 6 crossings are minimal.

Fibered knots with irreducible Alexander polynomial are minimal.

The partial order is defined by epimorphisms between knot groups.

Abstract

A partial order on the set of prime knots can be defined by the existence of an epimorphism between knot groups. We prove that all the prime knots with up to $6$ crossings are minimal. We also show that each fibered knot with the irreducible Alexander polynomial is minimal.