Surgeries, sharp 4-manifolds and the Alexander polynomial

Work of Ni and Zhang has shown that for the torus knot $T_{r,s}$ with $r>s>1$ every surgery slope $p\mathrm{/}q\ge \frac{30}{67}(r^2-1)(s^2-1)$ is a characterizing slope. In this paper, we show that this can be lowered to a bound which is linear in $rs$, namely, $p\mathrm{/}q\ge \frac{43}{4}(rs-r-s)$. 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 $p\mathrm{/}q$-surgery on a knot $K$ in $S^3$ and $p\mathrm{/}q$ exceeds $4g(K)+4$, then the Alexander polynomial of $K$ is uniquely determined by $Y$ and $p\mathrm{/}q$. We also show that if $p\mathrm{/}q$-surgery on $K$ bounds a sharp 4-manifold, then $S_{p^{\mathrm{\prime }}\mathrm{/}{q}^{\mathrm{\prime }}}^3(K)$ bounds a sharp 4-manifold for all $p^{\mathrm{\prime }}\mathrm{/}{q}^{\mathrm{\prime }}\ge p\mathrm{/}q$.