Bounds on alternating surgery slopes

Abstract

We show that if $p∕q$–surgery on a nontrivial knot $K$ yields the branched double cover of an alternating knot, then $|p∕q|\le 4g\left(K\right)+3$. This generalises a bound for lens space surgeries first established by Rasmussen. We also show that all surgery coefficients yielding the double branched covers of alternating knots must be contained in an interval of width two and this full range can be realised only if the knot is a cable knot. The work of Greene and Gibbons shows that if $S_{p∕q}^3\left(K\right)$ bounds a sharp $4$–manifold $X$, then the intersection form of $X$ takes the form of a changemaker lattice. We extend this to show that the intersection form is determined uniquely by the knot $K$, the slope $p∕q$ and the Betti number $b_2\left(X\right)$.