Exceptional surgeries on alternating knots

All Exceptional Surgeries on Alternating Knots Are Integral Surgeries

We show that all exceptional surgeries on hyperbolic alternating knots in the 3-sphere are integral surgeries.

The Classification of Exceptional Dehn Surgeries on 2-bridge Knots

A nontrivial Dehn surgery on a hyperbolic knot K in S is exceptional if the resulting manifold is either reducible, or toroidal, or a Seifert fibered manifold whose orbifold is a sphere with at most three exceptional fibers, called a small Seifert fibered space. Thus an exceptional Dehn surgery is non-hyperbolic, and using a version of Thurston’s orbifold theorem proved by Boileau and Porti [BP...

Exceptional Surgery on Knots

Let M be an irreducible, compact, connected, orientable 3-manifold whose boundary is a torus. We show that if M is hyperbolic, then it admits at most six finite/cyclic fillings of maximal distance 5. Further, the distance of a finite/cyclic filling to a cyclic filling is at most 2. If M has a nonboundary-parallel, incompressible torus and is not a generalized 1-iterated torus knot complement, t...

Alternating Knots

Cosmetic surgeries on genus one knots

Let Y be a closed oriented three-manifold and K be a framed knot in Y . For a rational number r , let Yr(K) be the resulting manifold obtained by Dehn surgery along K with slope r with respect to the given framing. For a knot K in S3 , we will always use the Seifert framing. Two surgeries along K with distinct slopes r and s are called cosmetic if Yr(K) and Ys(K) are homeomorphic. The two surge...