The universal character ring of some families of one-relator 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 palindromic and, in particular, of the $\left(-2,2m+1,2n+1\right)$-pretzel knot for all integers $m$ and $n$. For the $\left(-2,3,2n+1\right)$-pretzel knot, we give a simple proof of a result in [Trans. AMS, to appear] on its universal character ring, and an elementary proof of a result in [J. Knot Theory Ramif. 11 (2002) 1251–1289] on the number of irreducible components of its character variety.