Some families of minimal elements for the partial ordering on prime knots

TL;DR

This paper identifies specific families of prime knots, including twist and double twist knots, as minimal elements in the partial ordering of prime knots using character variety presentations and polynomial irreducibility criteria.

Contribution

It introduces an elementary method based on Chebyshev polynomials and polynomial irreducibility to determine minimal elements in the prime knot partial order.

Findings

All twist knots are minimal elements.

Certain double twist and 2-bridge knots are minimal.

Method uses character varieties and polynomial irreducibility.

Abstract

We show that all twist knots, certain double twist knots and some other 2-bridge knots are minimal elements for the partial ordering on the set of prime knots. The key to these results are presentations of their character varieties using Chebyshev polynomials and a criterion for irreducibility of a polynomial of two variables. These give us an elementary method to discuss the number of irreducible components of the character varieties, which concludes the result essentially.