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

TL;DR

This paper proves that hyperbolic (-2,3,n) pretzel knot complements are mostly unique within their commensurability classes, confirming a conjecture and illustrating the methods with broader Montesinos knots.

Contribution

It verifies a conjecture that most such knot complements are unique in their class, extending the result to a broad class of Montesinos knots.

Findings

Most (-2,3,n) pretzel knot complements are unique in their commensurability class.

The paper confirms the conjecture for all n except 7.

Methods are applicable to a broad class of Montesinos knots.

Abstract

Let K be a hyperbolic (-2,3,n) pretzel knot and M = S^3 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 \neq 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.