Characterizing slopes for the $(-2,3,7)$-pretzel knot

TL;DR

This paper provides explicit examples of characterizing slopes for the hyperbolic $(-2,3,7)$-pretzel knot, addressing the scarcity of concrete examples despite known existence results.

Contribution

It offers the first explicit examples of characterizing slopes for the $(-2,3,7)$-pretzel knot, advancing understanding of knot characterization.

Findings

Explicit characterizing slopes for the $(-2,3,7)$-pretzel knot identified

Addresses the gap between existence proofs and explicit examples

Enhances knowledge of hyperbolic knot characterization

Abstract

In this note we exhibit concrete examples of characterizing slopes for the knot $12n242$, aka the $(-2,3,7)$-pretzel knot. Although it was shown by Lackenby that every knot admits infinitely many characterizing slopes, the non-constructive nature of the proof means that there are very few hyperbolic knots for which explicit examples of characterizing slopes are known.