[Paper Review] All Known Principal Congruence Links

Mark D. Baker, Matthias Goerner+1 more|arXiv (Cornell University)|Feb 11, 2019

Geometric and Algebraic Topology9 references4 citations

TL;DR

This paper compiles and catalogs all known link diagrams in $S^3$ for principal congruence link complements arising from Bianchi groups $\mathrm{PSL}(2,\mathcal{O}_d)$, with $d$ square-free and $I$ an ideal in the ring of integers $\mathcal{O}_d$. It provides explicit diagrams for 39 cases across $d = 1, 2, 3, 5, 7, 11, 15, 19, 23, 31, 47, 71$, and includes one non-principal congruence regular tessellation link complement, all accessible via SnapPy-compatible files.

ABSTRACT

This report lists the link diagrams in S^3 for all principal congruence link complements for which such a link diagram is known. Several unpublished link diagrams are included. Related to this, we also include one link diagram for an arithmetic regular tessellation link complement.

Motivation & Objective

• To compile and preserve all known link diagrams for principal congruence link complements in $S^3$ arising from Bianchi groups $\mathrm{PSL}(2,\mathcal{O}_d)$.
• To serve as a living repository for these diagrams, updating as new examples are discovered.
• To include diagrams for cases where such representations were previously unpublished or unavailable.
• To provide a reference for citing specific link complements using the pair $(d, I)$ rather than figure numbers, ensuring stability in future citations.
• To include one non-principal congruence link that is a regular tessellation link complement, for thematic and arithmetic completeness.

Proposed method

• The authors collect and present link diagrams for all known principal congruence link complements in $S^3$, derived from quotients $\mathbb{H}^3 / \Gamma(I)$, where $\Gamma(I)$ is the kernel of the reduction map $\mathrm{PSL}(2,\mathcal{O}_d) \to \mathrm{PSL}(2,\mathcal{O}_d/I)$.
• Diagrams are generated using known constructions from the literature, including those by Baker (1981), Thurston (1979), Goerner (2011), and Dörn, Ocel, and Reid (2022), as well as original diagrams by the second author.
• For cases with multiple symmetric representations, only one diagram is shown, typically the most symmetric or canonical one.
• All diagrams are made available in a machine-readable format compatible with SnapPy, enabling computational exploration of the link complements.
• The paper includes a non-principal congruence example: a regular tessellation link complement arising from a Coxeter group action on $\mathbb{H}^3$, tessellating the complement into 6 regular ideal cubes.
• The authors emphasize citing via the pair $(d, I)$ rather than figure numbers to ensure long-term stability of references.

Experimental results

Research questions

• RQ1 Which principal congruence link complements in $S^3$ have known diagrammatic representations?
• RQ2 How can a comprehensive, stable, and computationally accessible repository of such diagrams be maintained?
• RQ3 What is the role of symmetry and canonical representation in selecting a single diagram per $(d, I)$ pair?
• RQ4 Are there non-principal congruence links that are arithmetically significant and worth including for completeness?
• RQ5 What is the geometric and arithmetic significance of the regular tessellation link complement in Figure 39?

Key findings

• The paper provides 39 known link diagrams for principal congruence link complements across $d = 1, 2, 3, 5, 7, 11, 15, 19, 23, 31, 47, 71$, with $d=1$ having 6 diagrams and $d=23$ having 3.
• For $d=1$, the pair $(1, \langle 4 + \sqrt{-1} \rangle)$ is missing from the known diagrams, indicating an open case.
• For $d=7$, the ideal $(7, \langle 2 + \sqrt{-7} \rangle)$ is missing, and similarly for $d=11$, $(11, \langle (5 + \sqrt{-11})/2 \rangle)$ is not yet diagrammed.
• The paper includes a non-principal congruence link complement (Figure 39) that is a regular tessellation of $\mathbb{H}^3$ into 6 regular ideal cubes, with flag-transitive symmetry and arithmetic structure.
• All diagrams are available in a SnapPy-compatible format at [Goe18, prinCong/Links/], enabling computational verification and exploration.
• The authors emphasize that future citations should reference the pair $(d, I)$ rather than figure numbers to ensure persistence as the list evolves.