Commensurability classes of $(-2,3,n)$ pretzel knot complements

Abstract

Let $K$ be a hyperbolic $\left(-2,3,n\right)$ pretzel knot and $M=S^3\setminus K$ its complement. For these knots, we verify a conjecture of Reid and Walsh: there are at most three knot complements in the commensurability class of $M$. Indeed, if $n\ne 7$, we show that $M$ is the unique knot complement in its class. We include examples to illustrate how our methods apply to a broad class of Montesinos knots.