Excluding cosmetic surgeries on hyperbolic 3-manifolds

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 filling with the figure-8 knot. The code that verifies these results is publicly available on GitHub.

Publication Details

Journal
Journal of Computational Geometry (Carleton University)
Published
2025-11-25
DOI
https://doi.org/10.20382/jocg.v16i1a19
Primary Topic
Geometric and Algebraic Topology
Type
article
Field-Weighted Citation Impact
0.00
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article

Excluding cosmetic surgeries on hyperbolic 3-manifolds

Journal of Computational Geometry (Carleton University)
Geometric and Algebraic Topology
article

Excluding cosmetic surgeries on hyperbolic 3-manifolds

article en

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 filling with the figure-8 knot. The code that verifies these results is publicly available on GitHub.

Journal of Computational Geometry (Carleton University)
Openalex Percentile: Top 25%
Geometric and Algebraic Topology
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Excluding cosmetic surgeries on hyperbolic 3-manifolds · Journal of Computational Geometry (Carleton University) (2025) | TGRS Research Map | TGRS