Narrow Results

We prove that every Legendrian knot in the tight contact structure of the $3$-sphere is determined by the contactomorphism type of its exterior. Moreover, by giving counterexamples we show this to be not true for Legendrian links in the tight $3$-sphere. On the way a new user-friendly formula for computing the Thurston–Bennequin invariant of a Legendrian knot in a surgery diagram is given.

Two Dehn surgeries on a knot are called purely cosmetic if their surgered manifolds are homeomorphic as oriented manifolds. Gordon conjectured that nontrivial knots in $S^3$ do not admit purely cosmetic surgeries. In this paper, we confirm this conjecture for cable knots.

A pair of surgeries on a knot is chirally cosmetic if they result in homeomorphic manifolds with opposite orientations. Using recent methods of Ichihara, Ito, and Saito, we show that, except for the (2, 5) and (2, 7)-torus knots, the genus 2 and 3 alternating odd pretzel knots do not admit any chirally cosmetic surgeries. Further, we show that for a fixed genus, at most finitely many alternating odd pretzel knots admit chirally cosmetic surgeries.

In this note, we prove that a nontrivial Kinoshita–Terasaka or Conway knot does not admit chirally cosmetic surgeries, by calculating the finite type invariant of order 3.

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