Characterizing slopes for torus knots, II.

A slope p q is called a characterizing slope for a given knot K 0 ⊂ S 3 if whenever the p q -surgery on a knot K ⊂ S 3 is homeomorphic to the p q -surgery on K 0 via an orientation preserving homeomorphism, then K = K 0 . 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 T 5 , 2 . More explicitly besides all negative integer slopes there are 2 4 7 slopes which were unknown to be characterizing for T 5 , 2 , including 8 9 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 T 5 , 2 if p q > − 1 and p q ∉ { 0 , 1 , ± 1 2 , ± 1 3 }. In particular every nontrivial L -space slope of T 5 , 2 is characterizing for T 5 , 2 . More explicitly this work yields 1 2 1 new characterizing slopes for T 5 , 2 . Another interesting consequence of this work is that if a nontrivial p q -surgery on a non-torus knot in S 3 yields a manifold of finite fundamental group, then | p | > 9. [ABSTRACT FROM AUTHOR]