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Symmetry groups of hyperbolic links and their complements

Christian Millichap, Rolland Trapp

TL;DR

The paper addresses how symmetry groups of hyperbolic links relate to isometries of their complements, and demonstrates that complement symmetries can grow linearly with a parameter while link-symmetries stay bounded. It introduces ml-swaps on minimally twisted chain links and uses them to construct a family of complement partners $L_{4n}$ to $C_{4n}$ with $[\mathrm{Sym}(\mathbb{S}^3 \setminus L_{4n}) : \mathrm{Sym}(\mathbb{S}^3, L_{4n})] = 8n$. A general framework is developed (via Theorem \textit{thm:symcontrol}) to control symmetry groups through the linking-graph $\Gamma(L)$ and annular Dehn twists, enabling explicit realizations of groups such as $(\mathbb{Z}_2 \times \mathbb{Z}_2) \rtimes D_k$ with $k|4n$ and, in particular, an explicit linear growth in the index. The results illuminate how to engineer large complement symmetries while keeping link symmetries small, and suggest broad applications to fully augmented links and related geometric structures, along with several open questions about possible symmetry-subgroup realizations for complement partners.

Abstract

We explicitly construct a sequence of hyperbolic links $\{ L_{4n} \}$ where the number of symmetries of each $\mathbb{S}^{3} \setminus L_{4n}$ that are not induced by symmetries of the pair $(\mathbb{S}^{3}, L_{4n})$ grows linearly with n. Specifically, $[Sym(\mathbb{S}^{3} \setminus L_{4n}) : Sym(\mathbb{S}^{3}, L_{4n})] =8n \rightarrow \infty$ as $n \rightarrow \infty$. For this construction, we start with a family of minimally twisted chain links, $\{ C_{4n} \}$, where $Sym(\mathbb{S}^{3}, C_{4n})$ and $Sym(\mathbb{S}^{3} \setminus C_{4n})$ coincide and grow linearly with $n$. We then perform a particular type of homeomorphism on $\mathbb{S}^{3} \setminus C_{4n}$ to produce another link complement $\mathbb{S}^{3} \setminus L_{4n}$ where we can uniformly bound $|Sym(\mathbb{S}^{3}, L_{4n})|$ using a combinatorial condition based on linking number. A more general result highlighting how to control symmetry groups of hyperbolic links is provided, which has potential for further application.

Symmetry groups of hyperbolic links and their complements

TL;DR

The paper addresses how symmetry groups of hyperbolic links relate to isometries of their complements, and demonstrates that complement symmetries can grow linearly with a parameter while link-symmetries stay bounded. It introduces ml-swaps on minimally twisted chain links and uses them to construct a family of complement partners to with . A general framework is developed (via Theorem \textit{thm:symcontrol}) to control symmetry groups through the linking-graph and annular Dehn twists, enabling explicit realizations of groups such as with and, in particular, an explicit linear growth in the index. The results illuminate how to engineer large complement symmetries while keeping link symmetries small, and suggest broad applications to fully augmented links and related geometric structures, along with several open questions about possible symmetry-subgroup realizations for complement partners.

Abstract

We explicitly construct a sequence of hyperbolic links where the number of symmetries of each that are not induced by symmetries of the pair grows linearly with n. Specifically, as . For this construction, we start with a family of minimally twisted chain links, , where and coincide and grow linearly with . We then perform a particular type of homeomorphism on to produce another link complement where we can uniformly bound using a combinatorial condition based on linking number. A more general result highlighting how to control symmetry groups of hyperbolic links is provided, which has potential for further application.
Paper Structure (5 sections, 13 theorems, 4 equations, 3 figures)

This paper contains 5 sections, 13 theorems, 4 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Symmetries of Minimally Twisted Chain Links
  3. Non-overlapping Nested Partners
  4. Symmetries of Non-overlapping Nested Partners of $C_{4n}$
  5. Concluding Remarks and Open Questions

Key Result

Theorem 1.1

For each $n \geq 6$, there exists a hyperbolic link $L_{4n}$ with $Sym(\mathbb{S}^3 \setminus L_{4n}) \cong (D_{4n} \times {\mathbb Z}_2) \rtimes {\mathbb Z}_2$ and $Sym(\mathbb{S}^3, L_{4n}) \cong {\mathbb Z}_2 \times {\mathbb Z}_2$. In particular, $[Sym(\mathbb{S}^{3} \setminus L_{4n}) : Sym(\math

Figures (3)

  • Figure 1: A symmetric diagram for $C_{8}$.
  • Figure 2: An $ml$-swap as a product of annular Dehn twists
  • Figure 3: A non-overlapping nested partner of $C_{24}$ and its corresponding linking graph.

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 2.1: Theorem 6.1 and Corollary 6.3, MeMiTr2020
  • Proposition 2.2
  • proof
  • Corollary 2.3
  • proof
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3
  • proof
  • ...and 16 more