Nonfibered knots and representation shifts

TL;DR

This paper proves a conjecture linking nonfibered knots to the uncountability of finite covers of their infinite cyclic cover for a class including all genus 1 knots, using symbolic dynamics techniques.

Contribution

It establishes the conjecture for a broad class of knots, including all genus 1 knots, advancing understanding of knot fiberedness and covering space properties.

Findings

Confirmed the conjecture for genus 1 knots

Connected knot fiberedness to properties of infinite cyclic covers

Applied symbolic dynamics to knot theory

Abstract

The authors conjectured previously that a knot is nonfibered if and only if its infinite cyclic cover has uncountably many finite covers. We prove the conjecture for a class of knots that includes all knots of genus 1, using techniques from symbolic dynamics.