Bounds on alternating surgery slopes

TL;DR

This paper establishes bounds on surgery slopes that produce double branched covers of alternating knots, showing these slopes are limited and related to the knot's genus, with implications for the intersection forms of bounding 4-manifolds.

Contribution

It generalizes Rasmussen's lens space surgery bounds to alternating knots and links, and characterizes the intersection forms of associated 4-manifolds based on knot and surgery data.

Findings

Bound on |p/q| in terms of knot genus: |p/q| ≤ 4g(K)+3.

Surgery coefficients for alternating double covers lie within an interval of width two.

Intersection form of bounding 4-manifold is uniquely determined by knot, slope, and Betti number.

Abstract

We show that if $p/q$-surgery on a nontrivial knot $K$ yields the branched double cover of an alternating knot or link, then $|p/q|\leq 4g(K)+3$. This generalises a bound for lens space surgeries first established by Rasmussen. We also show that all surgery coefficients yielding the double branched cover of an alternating knot or link must be contained in an interval of width two and this full range can be realised only if $K$ is a cable knot. The work of Greene and Gibbons shows that if $S^3_{p/q}(K)$ 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 $b_2(X)$.