The Cosmetic Surgery Conjecture
Let be a closed, oriented three-manifold and a knot whose exterior is boundary-irreducible. A purely cosmetic surgery is a pair of distinct surgery slopes on producing orientation-preserving homeomorphic manifolds. The Cosmetic Surgery Conjecture. If there exist two surgery slopes for which produce orientation-preserving homeomorphic manifolds, then there is a homeomorphism of the exterior sending one slope to the other.
This conjecture concerns when distinct Dehn fillings on a knot exterior can yield the same oriented three-manifold. The source notes that it subsumes the knot complement problem and the nugatory crossing conjecture; in the stated paper, the main result proves the absence of purely cosmetic surgeries for homotopically essential knots in the Poincaré homology sphere, but the general conjecture remains open.
Equivalent formulations 1Other wordings
Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.
The cosmetic surgery conjecture
Let be a knot in a closed oriented three-manifold . Assume that the knot exterior is irreducible and is not homeomorphic to the solid torus. Two Dehn surgeries on are purely cosmetic if they have homeomorphic results via an orientation-preserving homeomorphism. Cosmetic surgery conjecture. If two different Dehn surgeries on are purely cosmetic, then there is a homeomorphism of that takes one slope to the other. This conjecture asserts that purely cosmetic surgeries arise only from a symmetry of the knot exterior; it is a central question about the rarity of cosmetic Dehn surgeries in three-manifold topology.
source: Alan Du, “Heegaard Floer Surgery Formula and Cosmetic Surgeries”, arXiv:2401.10395 (2026).
References
Primary source
Tye Lidman, “Cosmetic surgeries and the Poincare homology sphere”, arXiv:1902.06801 (2019).
Additional references
3 papers in this index state this conjecture (2010–2019). The statement above is taken from the most recent of them; the others are arXiv:1504.06180, arXiv:1001.3926.
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