Surgeries, sharp 4–manifolds and the Alexander polynomial

Abstract

Work of Ni and Zhang has shown that, for the torus knot $T_{r,s}$ with , every surgery slope $p∕q\ge \frac{30}{67}\left(r^2-1\right)\left(s^2-1\right)$ is a characterizing slope. We show that this can be lowered to a bound which is linear in $rs$, namely $p∕q\ge \frac{43}{4}\left(rs-r-s\right)$. The main technical ingredient in this improvement is to show that if $Y$ is an $L$–space bounding a sharp $4$–manifold which is obtained by $p∕q$–surgery on a knot $K$ in $S^3$ and $p∕q$ exceeds $4g\left(K\right)+4$, then the Alexander polynomial of $K$ is uniquely determined by $Y$ and $p∕q$. We also show that if $p∕q$–surgery on $K$ bounds a sharp $4$–manifold, then $S_{p^{\prime }∕q^{\prime }}^3\left(K\right)$ bounds a sharp $4$–manifold for all $p^{\prime }∕q^{\prime }\ge p∕q$.