The universal character ring of some families of one-relator groups

TL;DR

This paper investigates the universal character rings of specific one-relator groups, including certain pretzel knots, providing explicit calculations and simpler proofs for known results.

Contribution

It computes the universal character ring for families of one-relator groups with palindromic relators and offers elementary proofs for properties of the (-2,3,2n+1)-pretzel knot.

Findings

Calculated universal character rings for specific one-relator groups.

Provided elementary proofs for the number of irreducible components of certain character varieties.

Extended understanding of the algebraic structure of pretzel knot groups.

Abstract

We study the universal character ring of some families of one-relator groups. As an application, we calculate the universal character ring of two-generator one-relator groups whose relators are palindrome, and, in particular, of the (-2,2m+1,2n+1)-pretzel knot for all integers m and n. For the (-2,3,2n+1)-pretzel knot, we give a less technical proof of a result in [LT1] on its universal character ring, and an elementary proof of a result in [Ma] on the number of irreducible components of its character variety.