SU(2)-Cyclic Surgeries on Knots

TL;DR

This paper proves a bound on the distance between SU(2)-cyclic surgeries on knots in the 3-sphere, extending the cyclic surgery theorem using holonomy perturbations and Chern-Simons theory.

Contribution

It introduces a new bound on SU(2)-cyclic surgeries, generalizing the cyclic surgery theorem with novel holonomy perturbation techniques.

Findings

Bound on the distance of SU(2)-cyclic surgery coefficients

Extension of cyclic surgery theorem to SU(2)-cyclic case

Application of holonomy perturbations in 3-manifold topology

Abstract

A surgery on a knot in 3-sphere is called SU(2)-cyclic if it gives a manifold whose fundamental group has no non-cyclic SU(2) representations. Using holonomy perturbations on the Chern-Simons functional, we prove that the distance of two SU(2)-cyclic surgery coefficients is bounded by the sum of the absolute values of their numerators. This is an analog of Culler-Gordon-Luecke-Shalen's cyclic surgery theorem.