Characterizing slopes for torus knots

Abstract

A slope p/q is called a characterizing slope for a given knot K_0 in S^3 if whenever the p/q–surgery on a knot K in S^3 is homeomorphic to the p/q–surgery on K_0 via an orientation preserving homeomorphism, then K D K0 . In this paper we try to find characterizing slopes for torus knots T_(r,s). We show that any slope p/q which is larger than the number 30(r^(2)-1)(s^(2)-1)/67 is a characterizing slope for T_(r,s). The proof uses Heegaard Floer homology and Agol–Lackenby’s 6–theorem. In the case of T(5,2), we obtain more specific information about its set of characterizing slopes by applying further Heegaard Floer homology techniques.