A characterisation of alternating knot exteriors

TL;DR

This paper provides a topological characterization of alternating knot exteriors using special spanning surfaces, establishing that 'alternating' is a property of the exterior, and introduces an algorithm to determine if a knot is prime and alternating.

Contribution

It offers the first topological characterization of alternating knot exteriors and presents a normal surface algorithm to identify prime and alternating knots from triangulations.

Findings

Alternating property is topologically characterized by special spanning surfaces.

The paper introduces an algorithm to decide if a knot is prime and alternating.

It extends the characterization to alternating link exteriors with marked meridians.

Abstract

We give a topological characterisation of alternating knot exteriors based on the presence of special spanning surfaces. This shows that alternating is a topological property of the knot exterior and not just a property of diagrams, answering an old question of Fox. We also give a characterisation of alternating link exteriors which have marked meridians. We then describe a normal surface algorithm which can decide if a knot is prime and alternating given a triangulation of its exterior as input.