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A new twist on modular links from an old perspective

Khanh Le

TL;DR

The paper identifies the complements of arithmetic modular links, as constructed by Migueles–Pinsky–Purcell, with the complements of augmented chainlinks in S^3, showing each M_Γ is homeomorphic to S^3 \setminus C_n for n = |Γ|. This is achieved by realizing UT(Σ_{Mod}) as the trefoil complement and lifting modular geodesics through a 6-fold cover from UT(Σ_{1,1}), then demonstrating that the n-component complements arise as n-fold cyclic covers of the Whitehead link complement. Consequently, every M_Γ fibers and, except for a small set of component counts, contains a closed embedded essential surface via Cooper–Long criteria. The results connect the dynamics of modular geodesics with explicit 3-manifold topology, embedding arithmetic modular links into the well-studied topology of augmented chainlinks and their cyclic covers.

Abstract

We show that the complement of arithmetic modular links found in arXiv:2307.09409 is homeomorphic to the complement of augmented chainlinks. In particular, these link complements arise as n-fold cyclic covers of the Whitehead link complement.

A new twist on modular links from an old perspective

TL;DR

The paper identifies the complements of arithmetic modular links, as constructed by Migueles–Pinsky–Purcell, with the complements of augmented chainlinks in S^3, showing each M_Γ is homeomorphic to S^3 \setminus C_n for n = |Γ|. This is achieved by realizing UT(Σ_{Mod}) as the trefoil complement and lifting modular geodesics through a 6-fold cover from UT(Σ_{1,1}), then demonstrating that the n-component complements arise as n-fold cyclic covers of the Whitehead link complement. Consequently, every M_Γ fibers and, except for a small set of component counts, contains a closed embedded essential surface via Cooper–Long criteria. The results connect the dynamics of modular geodesics with explicit 3-manifold topology, embedding arithmetic modular links into the well-studied topology of augmented chainlinks and their cyclic covers.

Abstract

We show that the complement of arithmetic modular links found in arXiv:2307.09409 is homeomorphic to the complement of augmented chainlinks. In particular, these link complements arise as n-fold cyclic covers of the Whitehead link complement.
Paper Structure (8 sections, 7 theorems, 13 equations, 7 figures)

This paper contains 8 sections, 7 theorems, 13 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Acknowledgement
  3. Preliminaries
  4. Definitions and background
  5. A construction of arithmetic modular links
  6. Proof of the main results
  7. Proof of \ref{['thm:ComplementArithmeticModularLinks']}
  8. Proof of \ref{['cor:MostArithmeticModularLinksAreLarge']}

Key Result

Theorem 1.1

Let $\Gamma \in \mathcal{F}\Sigma_{\mathop{\mathrm{Mod}}\nolimits}$ be an $n$-component modular link. The complement $M_\Gamma$ is homeomorphic to the complement $S^3 \setminus C_{n}$.

Figures (7)

  • Figure 1: The Whitehead link $S^3\setminus C_1$, a 3-fold cyclic cover $S^3 \setminus C_3$ (top left) and a 4-fold cyclic cover $S^3\setminus C_4$ (top right) both branched over the blue component. Forgetting the dotted components in $S^3\setminus C_3$ and $S^3 \setminus C_4$, we obtain two split links whose complement are handlebodies of genus 2 and 3 respectively. Consequently, we obtain surjective homomorphisms $\pi_1(S^3\setminus C_3) \rightarrow F_2$ and $\pi_1(S^3\setminus C_4) \to F_3$ in each case.
  • Figure 2: A fundamental domain of $\Sigma_{\mathop{\mathrm{Mod}}\nolimits}$ in $\mathbb{H}^2$.
  • Figure 3: The modular template $\mathcal{T}$ together with a flow
  • Figure 4: The 3-component modular link $\{\textcolor{linkcolor1}{LR},\textcolor{linkcolor2}{L^2R^2},\textcolor{linkcolor3}{L^2RLR^2}\}$ and the trefoil knot $\textcolor{linkcolor0}{T_{2,3}}$. The $\textcolor{linkcolor1}{LR}$-component corresponds to the sequence of periodic orbit $\textcolor{linkcolor1}{\{1/3,2/3\}}$ on $(0,1)$. The $\textcolor{linkcolor2}{L^2R^2}$-component corresponds to the sequence of periodic orbit $\textcolor{linkcolor2}{\{1/5,2/5,4/5,3/5\}}$ on $(0,1)$. Finally, the $\textcolor{linkcolor3}{L^2RLR^2}$-component corresponds to the sequence of periodic orbit $\textcolor{linkcolor3}{\{11/63,22/63,44/63,25/63,50/63,37/63\}}$ on $(0,1)$.
  • Figure 5: The once-punctured torus $\Sigma_{1,1}$ with a punctured removed (blue). The $0/1$ curve is shown in the middle. The $1/0$ curve is shown on the right.
  • ...and 2 more figures

Theorems & Definitions (10)

  • Theorem 1.1
  • Corollary 1.2
  • Lemma 2.1: MiguelesPinskyPurcell
  • Theorem 2.2: MiguelesPinskyPurcell
  • Lemma 2.3: MiguelesPinskyPurcell
  • Remark 2.4
  • Lemma 3.1
  • Lemma 3.2
  • Remark 3.3
  • Remark 3.4