The cosmetic surgery conjecture for non-trivial knots

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Let KK be a non-trivial knot in S3S^3. For relatively prime pairs of integers (p,q)(p,q) and (p′,q′)(p',q') with q,q′>0q,q'>0, write Sp/q3(K)S^3_{p/q}(K) and Sp′/q′3(K)S^3_{p'/q'}(K) for the corresponding Dehn surgeries.

Cosmetic surgery conjecture. If p/q≠p′/q′p/q\neq p'/q', then Sp/q3(K)S^3_{p/q}(K) and Sp′/q′3(K)S^3_{p'/q'}(K) are not orientation-preserving diffeomorphic.

The conjecture asserts that the unknot is the only knot in S3S^3 admitting distinct surgery slopes that produce orientation-preserving diffeomorphic three-manifolds. Its status is not resolved in the supplied source context.

References

Primary source

Aliakbar Daemi, Mike Miller Eismeier and Tye Lidman, “Filtered instanton homology and cosmetic surgery”, arXiv:2410.21248 (2025).

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