On the Alexander polynomials of modular knots

TL;DR

This paper explores the Alexander polynomials of modular knots, revealing their finite diversity at fixed degrees and the unbounded variability of their coefficients, thus deepening understanding of their topological invariants.

Contribution

It introduces a detailed analysis of Alexander polynomials of modular knots, showing finiteness at fixed degrees and unbounded coefficient variability, advancing the study of SL_2(Z)-invariants.

Findings

Finitely many Alexander polynomials per fixed degree.

Any integer can appear as a coefficient in some Alexander polynomial.

Coefficients of the same sign can form arbitrarily long runs.

Abstract

Closed geodesics associated with indefinite binary quadratic forms, or equivalently with real quadratic irrationals, have long been studied as geometric $\mathrm{SL}_2(\mathbb{Z})$-invariants. Building on the Birman-Williams approach to Lorenz knots and following the notion of modular knots introduced by Ghys, this article investigates the topological $\mathrm{SL}_2(\mathbb{Z})$-invariants arising from modular knots. Our main focus is the Alexander polynomial of modular knots. Using the Burau representation, we highlight two contrasting features of this family. On the one hand, for each fixed degree, only finitely many Alexander polynomials of modular knots occur. On the other hand, any integer appears as a coefficient of the Alexander polynomial of some modular knot, and coefficients of the same sign can occur in runs of arbitrarily long length.