Characterizing slopes for torus knots, II

Abstract

A slope [Formula: see text] is called a characterizing slope for a given knot [Formula: see text] if whenever the [Formula: see text]-surgery on a knot [Formula: see text] is homeomorphic to the [Formula: see text]-surgery on [Formula: see text] via an orientation preserving homeomorphism, then [Formula: see text]. 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 [Formula: see text]. More explicitly besides all negative integer slopes there are [Formula: see text] slopes which were unknown to be characterizing for [Formula: see text], including [Formula: see text] nontrivial [Formula: see text]-space slopes. Applying recent work of Baldwin–Hu–Sivek, we improve our result by showing that a nontrivial slope [Formula: see text] is a characterizing slope for [Formula: see text] if [Formula: see text] and [Formula: see text]. In particular every nontrivial [Formula: see text]-space slope of [Formula: see text] is characterizing for [Formula: see text]. More explicitly this work yields [Formula: see text] new characterizing slopes for [Formula: see text]. Another interesting consequence of this work is that if a nontrivial [Formula: see text]-surgery on a non-torus knot in [Formula: see text] yields a manifold of finite fundamental group, then [Formula: see text].