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Profinite rigidity for two-bridge links and 3-tangle Montesinos links

Tamunonye Cheetham-West, Xiaoyu Xu

Abstract

For any two-bridge link or 3-tangle Montesinos link $L\subset S^3$ (including knot), this paper proves that $π_1(S^3-L)$ is profinitely rigid among the fundamental groups of compact orientable 3-manifolds.

Profinite rigidity for two-bridge links and 3-tangle Montesinos links

Abstract

For any two-bridge link or 3-tangle Montesinos link (including knot), this paper proves that is profinitely rigid among the fundamental groups of compact orientable 3-manifolds.
Paper Structure (25 sections, 36 theorems, 7 equations, 3 figures)

This paper contains 25 sections, 36 theorems, 7 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries.
  3. Profinite completion
  4. Profinite completions of 3-manifold groups
  5. Cohomology of profinite groups
  6. Dehn filling
  7. Peripheral regularity
  8. Aligning Dehn fillings
  9. Link complements
  10. Perfect isomorphism
  11. Branched coverings
  12. Two-bridge links
  13. Schubert normal form
  14. Double branched cover
  15. Burde's classification theorem
  16. ...and 10 more sections

Key Result

Theorem 1.1

For any two-bridge link (including knot) $L\subseteq S^3$, $\pi_1(S^3-L)$ is profinitely rigid in $\mathscr{M}$.

Figures (3)

  • Figure 1: The oriented Schubert diagram for $\mathbf{b}(8,3)$ (the Whitehead link)
  • Figure 2: Montesinos links
  • Figure 3: Oer84

Theorems & Definitions (67)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.1
  • proof
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1: DFPR
  • Proposition 2.1: RZ10
  • Definition 2.1
  • Theorem 2.2: WZ17WZ17bWZ2
  • ...and 57 more