The shrinkage type of knots

TL;DR

This paper investigates the spectral gaps of cellular differentials in cyclic coverings of knot complements, classifying knots into three types based on irrationality exponents linked to Alexander polynomials, and explores their properties across various knot families.

Contribution

It introduces a new classification of knots into three types based on spectral gap asymptotics related to Alexander polynomials and demonstrates the prevalence of all types among different knot families.

Findings

All three knot types are abundant in examples.

Spectral gap asymptotics relate to irrationality exponents of algebraic number ratios.

Different behaviors observed for twist, torus, and low-crossing knots.

Abstract

We study spectral gaps of cellular differentials for finite cyclic coverings of knot complements. Their asymptotics can be expressed in terms of irrationality exponents associated with ratios of logarithms of algebraic numbers determined by the first two Alexander polynomials. From this point of view it is natural to subdivide all knots into three different types. We show that examples of all types abound and discuss what happens for twist and torus knots as well as knots with few crossings.