Characterizing slopes for torus knots, II

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.