Work of Ni and Zhang has shown that, for the torus knot with , every surgery slope is a characterizing slope. We show that this can be lowered to a bound which is linear in , namely . The main technical ingredient in this improvement is to show that if is an –space bounding a sharp –manifold which is obtained by –surgery on a knot in and exceeds , then the Alexander polynomial of is uniquely determined by and . We also show that if –surgery on bounds a sharp –manifold, then bounds a sharp –manifold for all .
Duncan McCoy. "Surgeries, sharp –manifolds and the Alexander polynomial." Algebr. Geom. Topol. 21 (5) 2649 - 2676, 2021. https://doi.org/10.2140/agt.2021.21.2649