Classical invariants of spiral knots

TL;DR

This paper introduces recursive formulas for Alexander polynomials of spiral knots, a generalization of torus knots, and explores their properties and implications for classification in knot theory.

Contribution

It provides the first general recursive formula for the Alexander polynomials of spiral knots and derives new properties and a genus formula for this family.

Findings

Recursive Alexander polynomial formula for spiral knots

Derived genus formula for spiral knots

Insights into classification of spiral knots

Abstract

Torus knots are an important family of knots about which much is understood; invariants of torus knots often exhibit nice formulas, making them convenient and fundamental building blocks for examples in knot theory. Spiral knots, defined and first studied by Brothers et al., are a braid-theoretic generalization of torus knots, but comparatively not much is known about this broader family of knots. We give a general recursive formula for the Alexander polynomials of spiral knots, and from this we derive several properties of spiral knots, including a simple genus formula. Additionally, we investigate the consequences these results have on classification questions.