A note on alternating knots in handlebodies

TL;DR

This paper extends classical knot theory results to knots in handlebodies, proving minimal crossing number and writhe invariance for alternating diagrams using a generalized Jones polynomial.

Contribution

It introduces a new theorem for alternating knots in handlebodies and generalizes the Jones polynomial to this setting, expanding previous work.

Findings

Dotted-reduced alternating diagrams realize minimal crossing number

Diagrams of the same knot have identical writhe

Generalized Jones polynomial is a key proof tool

Abstract

We establish a Kauffman-Murasugi-Thistlethwaite-type theorem for alternating knots in a solid torus. Specifically, we show that any dotted-reduced alternating diagram of a knot in a handlebody realizes the minimal crossing number, and that any two such diagrams of the same knot have identical writhe. The proof relies on a generalization of the Jones polynomial to the setting of handlebodies. A stronger version of this result was already proved by Boden, Karimi, and Sikora using a different generalized Jones polynomial; therefore, this text largely expands on one of the main proof tools.