Noncharacterizing slopes for hyperbolic knots

A nontrivial slope r on a knot K in S³ is called a characterizing slope if whenever the result of r-surgery on a knot K’ is orientation-preservingly homeomorphic to the result of r-surgery on K, then K’ is isotopic to K. Ni and Zhang ask: for any hyperbolic knot K, is a slope r=p/q with |p|+|q| sufficiently large a characterizing slope? In this article, we prove that if we can take an unknot c so that (0,0)-surgery on K∪c results in S³ and c is not a meridian of K, then K has infinitely many noncharacterizing slopes. As the simplest known example, the hyperbolic, two-bridge knot 8₆ has no integral characterizing slopes. This answers the above question in the negative. We also prove that any L-space knot never admits such an unknot c.