Alexander-Beck modules detect the unknot

TL;DR

This paper introduces the Alexander-Beck module, a new knot invariant that refines the classical Alexander module and uniquely detects the unknot, enhancing knot classification methods.

Contribution

The authors define the Alexander-Beck module and prove it is an effective unknot detector, providing a novel tool for knot theory.

Findings

Alexander-Beck module refines the classical Alexander module.

It uniquely detects the unknot among all knots.

The invariant enhances knot classification techniques.

Abstract

We introduce the Alexander-Beck module of a knot as a canonical refinement of the classical Alexander module, and we prove that this new invariant is an unknot-detector.