SU(2)-cyclic surgeries and the pillowcase

TL;DR

This paper investigates knots in three-dimensional space with infinitely many special surgeries called $SU(2)$-cyclic surgeries, revealing their bounded nature, boundary slope properties, and connections to instanton L-space surgeries.

Contribution

It proves that knots with infinitely many $SU(2)$-cyclic surgeries have a bounded set of slopes with a unique rational limit point, and are prime with infinitely many instanton L-space surgeries.

Findings

The set of $SU(2)$-cyclic slopes is bounded and has a unique rational limit point.

Knots with such surgeries are prime.

These knots have infinitely many instanton L-space surgeries.

Abstract

We study knots in $S^3$ with infinitely many $SU(2)$-cyclic surgeries, which are Dehn surgeries such that every representation of the resulting fundamental group into $SU(2)$ has cyclic image. We show that for every such nontrivial knot $K$, its set of $SU(2)$-cyclic slopes is bounded and has a unique limit point, which is both a rational number and a boundary slope for $K$. We also show that such knots are prime and have infinitely many instanton L-space surgeries. Our methods include the application of holonomy perturbation techniques to instanton knot homology, using a strengthening of recent work by the second author.