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Excluding cosmetic surgeries on hyperbolic 3-manifolds

David Futer, Jessica S. Purcell, Saul Schleimer

TL;DR

The paper develops a practical, hyperbolic-geometry–based framework to test the cosmetic surgery conjecture for one-cusp 3-manifolds, combining rigid hyperbolic invariants with knot-invariant obstructions. It provides an explicit finite-slope procedure, implemented in open-source software, that successfully verifies the conjectures for large data sets: all nontrivial knots up to $19$ crossings and all one-cusped SnapPy census manifolds, with detailed results on purely and chirally cosmetic cases and on common fillings. The work also extends to arithmetic hyperbolic manifolds, proving exact volume equalities in specific cases and illustrating deep connections between hyperbolic geometry, topology, and number-theoretic invariants. Collectively, these results yield broad computational evidence supporting the cosmetic surgery conjecture in both knot and hyperbolic-manifold contexts and provide a robust, reproducible toolkit for future explorations. The methods balance rigorous hyperbolic calculations (volumes, length spectra, and covers) with algorithmic obstruction tests, enabling large-scale verification and offering practical pathways for extending the approach to more general settings.

Abstract

This paper employs knot invariants and results from hyperbolic geometry to develop a practical procedure for checking the cosmetic surgery conjecture on any given one-cusped manifold. This procedure has been used to establish the following computational results. First, we verify that all knots up to 19 crossings, and all one-cusped 3-manifolds in the SnapPy census, do not admit any purely cosmetic surgeries. Second, we check that a hyperbolic knot with at most 15 crossings only admits chirally cosmetic surgeries when the knot itself is amphicheiral. Third, we enumerate all knots up to 13 crossings that share a common Dehn fillings with the figure-8 knot. The code that verifies these results is publicly available on GitHub.

Excluding cosmetic surgeries on hyperbolic 3-manifolds

TL;DR

The paper develops a practical, hyperbolic-geometry–based framework to test the cosmetic surgery conjecture for one-cusp 3-manifolds, combining rigid hyperbolic invariants with knot-invariant obstructions. It provides an explicit finite-slope procedure, implemented in open-source software, that successfully verifies the conjectures for large data sets: all nontrivial knots up to crossings and all one-cusped SnapPy census manifolds, with detailed results on purely and chirally cosmetic cases and on common fillings. The work also extends to arithmetic hyperbolic manifolds, proving exact volume equalities in specific cases and illustrating deep connections between hyperbolic geometry, topology, and number-theoretic invariants. Collectively, these results yield broad computational evidence supporting the cosmetic surgery conjecture in both knot and hyperbolic-manifold contexts and provide a robust, reproducible toolkit for future explorations. The methods balance rigorous hyperbolic calculations (volumes, length spectra, and covers) with algorithmic obstruction tests, enabling large-scale verification and offering practical pathways for extending the approach to more general settings.

Abstract

This paper employs knot invariants and results from hyperbolic geometry to develop a practical procedure for checking the cosmetic surgery conjecture on any given one-cusped manifold. This procedure has been used to establish the following computational results. First, we verify that all knots up to 19 crossings, and all one-cusped 3-manifolds in the SnapPy census, do not admit any purely cosmetic surgeries. Second, we check that a hyperbolic knot with at most 15 crossings only admits chirally cosmetic surgeries when the knot itself is amphicheiral. Third, we enumerate all knots up to 13 crossings that share a common Dehn fillings with the figure-8 knot. The code that verifies these results is publicly available on GitHub.
Paper Structure (40 sections, 20 theorems, 57 equations, 2 figures)

This paper contains 40 sections, 20 theorems, 57 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Purely cosmetic surgeries on knots in $S^3$
  3. Cosmetic surgery conjecture for general 3-manifolds
  4. Chirally cosmetic surgeries
  5. Common fillings of distinct cusped manifolds
  6. Code on GitHub
  7. Acknowledgements
  8. Obstructions and tests from knot invariants
  9. Theoretical tools
  10. Practical tests
  11. Effectiveness of the tests
  12. Results
  13. Beyond the 3-sphere using hyperbolic tools
  14. Invariants based on homology
  15. Linearized Hodgson--Kerckhoff bounds
  16. ...and 25 more sections

Key Result

Theorem 2.1

Let $K \subset S^3$ be a nontrivial knot, with Seifert genus $g(K)$. Suppose that $s_1$ and $s_2$ are a purely cosmetic pair of slopes for $K$. Then one of the following holds: In particular, if $th(K) < 2g(K)(g(K)-2)$, then $K$ cannot have purely cosmetic surgeries.

Figures (2)

  • Figure 1: The two prime knots up to 19 crossings that were not ruled out by Tests \ref{['Test:Casson']}--\ref{['Test:HanselmanFull']}. Both exceptional knots have 18 crossings.
  • Figure 2: The three functions appearing in \ref{['Eqn:f_inverse_linear']}.

Theorems & Definitions (58)

  • Definition 1.1
  • Conjecture 1.2: Cosmetic surgery conjecture
  • Conjecture 1.3: Hyperbolic cosmetic surgery conjecture
  • Definition 1.4
  • Conjecture 1.6: Chirally cosmetic surgery conjecture for knots in $S^3$
  • Theorem 2.1: Hanselman
  • Theorem 2.2: Daemi, Lidman, Miller Eismeier
  • Definition 2.3
  • Theorem 2.11
  • proof
  • ...and 48 more