Cosmetic surgery in L–space homology spheres
نویسندگان
چکیده
In particular, let K be a framed knot in a closed oriented three-manifold Y . For a rational number r , let Yr .K/ be the manifold obtained by Dehn surgery along K with slope r . Two surgeries along K with distinct slopes r and r 0 are called equivalent if there exists an orientation-preserving homeomorphism of the complement of K taking one slope to the other; and they are called truly cosmetic if there exists an orientation-preserving homeomorphism between Yr .K/ and Yr 0.K/. When K D U is the unknot in S , there are truly cosmetic surgeries: S p=q .U / Š S p=pCq .U /, and S p=q1 Š S p=q2 .U / when q1q2 1 .mod p/: While p=q and p=.pCq/ are equivalent slopes, p=q1 and p=q2 as above are usually not. By contrast, there are no known truly cosmetic surgeries on a nontrivial knot. Indeed, it is one of the outstanding problems conjectured in Kirby’s problem list, Problem 1.81(1):
منابع مشابه
Lens space surgeries and L-space homology spheres
We describe necessary and sufficient conditions for a knot in an L-space to have an L-space homology sphere surgery. We use these conditions to reformulate a conjecture of Berge about which knots in S admit lens space surgeries.
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