$SU(2)$-representations of Branched Covers

TL;DR

This paper proves the existence of irreducible $SU(2)$-representations for certain cyclic branched covers of knots in $S^3$, expanding the class of known hyperbolic integer homology spheres with such representations.

Contribution

It introduces new conditions under which cyclic branched covers admit irreducible $SU(2)$-representations, including for non-trivial prime knots with specific symmetries or tangle representations.

Findings

Irreducible $SU(2)$-representations exist for many cyclic branched covers of knots.

New infinite families of hyperbolic integer homology spheres with irreducible representations.

Applications include cases where previous criteria do not apply.

Abstract

We study the existence of irreducible $SU(2)$-representations for cyclic branched covers of knots in $S^3$. Our main result establishes that if $K$ is a non-trivial prime knot and $d$ is an integer such that $d \geq 2$ and $\Sigma_d(K)$ is an integer homology sphere, then $\pi_1(\Sigma_d(K))$ admits an irreducible $SU(2)$-representation, whenever $K$ satisfies one of two conditions: either $K$ is $2$-periodic, or $K$ can be represented as the closure of a tangle adapted to a $d\times d$ SICUP matrix. The first condition leverages a commuting trick for covering spaces to realize higher-degree branched covers as 2-fold covers, allowing us to apply recent results of Kronheimer-Mrowka and others. The second condition uses equivariant surgery descriptions and the $\nu^\sharp$ invariant from instanton Floer homology. As applications, we provide new infinite families of hyperbolic integer homology spheres admitting irreducible representations, including examples where previously known criteria fail.