Distinguishing some genus one knots using finite quotients

TL;DR

This paper introduces a new criterion based on finite quotients of knot groups to distinguish certain genus one knots in the 3-sphere, applying recent advances to specific hyperbolic and pretzel knots.

Contribution

It provides a novel method for knot distinction using finite quotients, extending the toolkit for classifying genus one knots in 3-sphere.

Findings

Successfully distinguishes specific hyperbolic knots using the criterion.

Applies the criterion to an infinite family of pretzel knots.

Demonstrates the effectiveness of finite quotients in knot classification.

Abstract

We give a criterion for distinguishing a prime knot $K$ in $S^3$ from every other knot in $S^3$ using the finite quotients of $\pi_1(S^3\setminus K)$. Using recent work of Baldwin-Sivek, we apply this criterion to the hyperbolic knots $5_2$, $15n_{43522}$, and the three-strand pretzel knots $P(-3,3,2n+1)$ for every integer $n$.