Small Heegaard genus and SU(2)

TL;DR

This paper investigates the existence of non-abelian SU(2) and SO(3) representations of the fundamental group for certain 3-manifolds with genus 2, and characterizes trefoil knots via irreducible SU(2) representations.

Contribution

It establishes conditions under which 3-manifolds of genus 2 admit non-abelian SU(2) or SO(3) representations and characterizes trefoil knots through unique irreducible SU(2) representations.

Findings

Existence of non-abelian SU(2) representations for manifolds with specific first homology orders.

Existence of non-abelian SO(3) representations for manifolds with first homology of order 2.

Characterization of trefoil knots via unique conjugacy classes of irreducible SU(2) representations.

Abstract

Let $Y$ be a closed, orientable 3-manifold with Heegaard genus 2. We prove that if $H_1(Y;\mathbb{Z})$ has order $1$, $3$, or $5$, then there is a representation $\pi_1(Y) \to \mathrm{SU}(2)$ with non-abelian image. Similarly, if $H_1(Y;\mathbb{Z})$ has order $2$ then we find a non-abelian representation $\pi_1(Y) \to \mathrm{SO}(3)$. We also prove that a knot $K$ in $S^3$ is a trefoil if and only if there is a unique conjugacy class of irreducible representations $\pi_1(S^3\setminus K) \to \mathrm{SU}(2)$ sending a fixed meridian to $\left(\begin{smallmatrix}i&0\\0&-i\end{smallmatrix}\right)$.