Characterizing slopes for torus knots, II

Abstract

A slope p/q is called a characterizing slope for a given knot K₀ ⊂ S³ if whenever the p/q-surgery on a knot K ⊂ S³ is homeomorphic to the p/q-surgery on K₀ via an orientation preserving homeomorphism, then K = K₀. In a previous paper, we showed that, outside a certain finite set of slopes, only the negative integers could possibly be non-characterizing slopes for the torus knot T₅,₂. More explicitly besides all negative integer slopes there are 247 slopes which were unknown to be characterizing for T₅,₂, including 89 nontrivial L-space slopes. Applying recent work of Baldwin–Hu–Sivek, we improve our result by showing that a nontrivial slope p/q is a characterizing slope for T₅,₂ if p/q > −1 and p/q ∉ {0, 1, ±1/2, ±1/3}. In particular every nontrivial L-space slope of T₅,₂ is characterizing for T₅,₂. More explicitly this work yields 121 new characterizing slopes for T₅,₂. Another interesting consequence of this work is that if a nontrivial p/q-surgery on a non-torus knot in S³ yields a manifold of finite fundamental group, then |p| > 9.

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