On the universal character ring and the character variety of the (-2,3,2n+1)-pretzel knot

TL;DR

This paper computes the universal character ring for certain two-generator groups and applies it to analyze the character variety of the (-2,3,2n+1)-pretzel knot, providing simpler proofs of existing results.

Contribution

It offers a less technical proof of the universal character ring for the pretzel knot and an elementary proof of its character variety, advancing understanding of knot invariants.

Findings

Universal character ring computed for the pretzel knot class

Simplified proofs of character ring and variety results

Enhanced accessibility of knot invariant analysis

Abstract

We calculate the universal character ring of a class of two-generator, one-relator groups. As an application we give a less technical proof of a result in [LT] on the universal character ring of the (-2,3,2n+1)-pretzel knot. We also give an elementary proof of a result in [Ma] on the character variety of the (-2,3,2n+1)-pretzel knot.