Math Conjectures

Purely cosmetic surgery conjecture

OPENMajorConjectureProposed 1990 · Canonical special case

Canonical statement

Let K⊂S3K\subset S^{3} be a nontrivial knot and, for each slope t∈Q∪{∞}t\in\mathbb Q\cup\{\infty\} in the meridian--longitude basis, let St3(K)S^{3}_{t}(K) denote the result of tt-Dehn filling its exterior. If r,s∈Q∪{∞}r,s\in\mathbb Q\cup\{\infty\} and Sr3(K)S^{3}_{r}(K) and Ss3(K)S^{3}_{s}(K) are orientation-preservingly homeomorphic, then r=sr=s.
View source LaTeX
Let \(K\subset S^{3}\) be a nontrivial knot and, for each slope \(t\in\mathbb Q\cup\{\infty\}\) in the meridian--longitude basis, let \(S^{3}_{t}(K)\) denote the result of \(t\)-Dehn filling its exterior. If \(r,s\in\mathbb Q\cup\{\infty\}\) and \(S^{3}_{r}(K)\) and \(S^{3}_{s}(K)\) are orientation-preservingly homeomorphic, then \(r=s\).
GEO-TOP-018:[BleilerHodgsonWeeks1990Cosmetic][NiWu2015Cosmetic][FuterPurcellSchleimer2025ExcludingCosmetic]
Geometry and topology

Notes

A pair of Dehn surgeries on the same knot along distinct slopes is called purely cosmetic if the resulting 33-manifolds are homeomorphic by an orientation-preserving homeomorphism. The conjecture, which dates to around 1990 and is associated with the study of cosmetic surgery on knots by Bleiler, Hodgson and Weeks [BleilerHodgsonWeeks1990Cosmetic], asserts that no nontrivial knot in S3S^3 admits such a pair: if Sr3(K)≅Ss3(K)S^3_r(K)\cong S^3_s(K) preserving orientation, then r=sr=s. Orientation-reversing ("chirally cosmetic") pairs are a different phenomenon and are not asserted to be impossible.

Heegaard Floer homology has been the main engine of progress: Ni and Wu drastically reduced the set of slope pairs that could possibly be cosmetic [NiWu2015Cosmetic]. Combined with finite-type invariants and with hyperbolic-geometric certification, computational verification now covers all knots through large crossing ranges [FuterPurcellSchleimer2025ExcludingCosmetic].

No argument yet covers all knots and all slopes simultaneously, so the conjecture remains open in general; a resolution requires closing the residual families of candidate slope pairs for arbitrary knots.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.