Symmetries and Detection of Surfaces by the Character Variety

TL;DR

This paper extends the detection of essential surfaces in 3-manifolds to 3-orbifolds using character varieties, revealing that symmetric surfaces are detected on the canonical component, with applications to symmetric double twist knots.

Contribution

It generalizes Culler-Shalen's construction to 3-orbifolds and character varieties, establishing symmetry constraints on detected slopes and providing new examples involving double twist knots.

Findings

Detected slopes on canonical components correspond to symmetric surfaces.

Symmetric double twist knots have slopes strongly detected but not on the canonical component.

Extended detection methods to include 3-orbifolds and different character varieties.

Abstract

We extend Culler and Shalen's construction of detecting essential surfaces in 3-manifolds to 3-orbifolds. We do so in the setting of the $\mathrm{SL}_2(\mathbb{C})$ character variety, and following Boyer and Zhang in the $\mathrm{PSL}_2(\mathbb{C})$ character variety as well. We show that any slope detected on a canonical component of the $\mathrm{(P)SL}_2(\mathbb{C})$ character variety of a one cusped hyperbolic 3-manifold with symmetries must be the slope of a symmetric surface. As an application, we show that for each symmetric double twist knot there are slopes which are strongly detected on the character variety but not on the canonical component.