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On the Alexander polynomials of modular knots

Soon-Yi Kang, Toshiki Matsusaka, Kyungbae Park

TL;DR

The paper investigates Alexander polynomials of modular knots, a class obtained by lifting SL_2(Z) geodesics to knot complements via Ghys’ modular framework. It presents two complementary descriptions of modular knots—via Bernoulli-shift braids and via words from continued fraction data—grounded in the Burau representation to derive explicit Alexander polynomials. Key findings include a finite number of degree-n Alexander polynomials for modular knots, along with the striking result that every integer occurs as a coefficient, and negative-coefficient runs of arbitrary length, highlighting the richness of modular knots beyond torus or L-space knots. The work connects modular knots to Lorenz and torus knots through Christoffel words, braid-index computations, and arithmetic aspects of quadratic fields, demonstrating both deep structure and wide diversity within this knot family.

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.

On the Alexander polynomials of modular knots

TL;DR

The paper investigates Alexander polynomials of modular knots, a class obtained by lifting SL_2(Z) geodesics to knot complements via Ghys’ modular framework. It presents two complementary descriptions of modular knots—via Bernoulli-shift braids and via words from continued fraction data—grounded in the Burau representation to derive explicit Alexander polynomials. Key findings include a finite number of degree-n Alexander polynomials for modular knots, along with the striking result that every integer occurs as a coefficient, and negative-coefficient runs of arbitrary length, highlighting the richness of modular knots beyond torus or L-space knots. The work connects modular knots to Lorenz and torus knots through Christoffel words, braid-index computations, and arithmetic aspects of quadratic fields, demonstrating both deep structure and wide diversity within this knot family.

Abstract

Closed geodesics associated with indefinite binary quadratic forms, or equivalently with real quadratic irrationals, have long been studied as geometric -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 -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.

Paper Structure

This paper contains 20 sections, 18 theorems, 96 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Modular braids and modular knots
  3. Braid groups
  4. Continued fraction expansions
  5. Modular knots using the Bernoulli shift
  6. Modular knots using words
  7. Lorenz braids and Lorenz knots
  8. Lorenz braids
  9. Some remarks on Lorenz knots
  10. Torus knots as modular knots
  11. Christoffel words
  12. Torus knots
  13. Some invariants of modular knots
  14. Braid index
  15. Class numbers of imaginary quadratic fields
  16. ...and 5 more sections

Key Result

Theorem 1.1

For a positive integer $n$, we define Then we have $A_n = \emptyset$ for every odd $n$, and $\# A_n \le p(n)$ for every even $n$, where $p(n)$ denotes the number of partitions of $n$.

Figures (18)

  • Figure 1: Braid diagrams for generators $\sigma_i, \sigma_i^{-1}$.
  • Figure 2: Braid diagrams for the braid relations.
  • Figure 3: The closure of a braid.
  • Figure 4: A braid diagram defined from the periodic orbit of the Bernoulli shift for $11/31$.
  • Figure 5: A braid diagram for the Lyndon word $LRLRR$.
  • ...and 13 more figures

Theorems & Definitions (65)

  • Theorem 1.1: \ref{['thm:fin-Alexander']}
  • Theorem 1.2: \ref{['thm:coeff-Alexander']}
  • Theorem 1.3: \ref{['thm:neg-long']}
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Example 2.6
  • ...and 55 more