Dehn surgery, the fundamental group and SU(2)

TL;DR

This paper investigates the relationship between Dehn surgery on knots in the 3-sphere and the existence of non-cyclic SU(2) representations of the resulting manifold's fundamental group, establishing a threshold for the surgery coefficient.

Contribution

It demonstrates that for any non-trivial knot, the 3-manifold obtained by surgery with coefficient r ≤ 2 admits a non-cyclic SU(2) representation of its fundamental group.

Findings

Existence of non-cyclic SU(2) representations for r ≤ 2

Connection between surgery coefficient and fundamental group representations

Advancement in understanding the fundamental group of Dehn surgeries

Abstract

Let K be a non-trivial knot in the 3-sphere and let Y(r) be the 3-manifold obtained by surgery on K with surgery-coefficient a rational number r. We show that there is a homomorphism from the fundamental group of Y(r) to SU(2) with non-cyclic image if r is less than or equal to 2.