Excluding cosmetic surgeries on hyperbolic 3-manifolds

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.

31 pages, 2 figures. v3 incorporates the use of verified length spectra in SnapPy, making the main results unconditional. It also incorporates the work of Daemi-Lidman-Miller Eismeier into an Alexander polynomial test for cosmetic surgeries. To appear in the Journal of Computational Geometry