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 \(p/q \geq \frac{30}{67}(r^2-1)(s^2-1)\) is a characterizing slope. In this paper, we show that this can be lowered to a bound which is linear in \(rs\), namely, \(p/q\geq \frac{43}{4}(rs-r-s)\). 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 \(p/q\)-surgery on a knot \(K\) in \(S^3\) and \(p/q\) exceeds \(4g(K)+4\), 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 \(S^3_{p'/q'}(K)\) bounds a sharp 4-manifold for all \(p'/q'\geq p/q\).