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We extend Culler and Shalen's construction of detecting essential surfaces in3-manifolds to 3-orbifolds. We do so in the setting of the$\mathrm{SL}_2(\mathbb{C})$ character variety, and following Boyer and Zhang inthe $\mathrm{PSL}_2(\mathbb{C})$ character variety as well. We show that anyslope 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 bethe slope of a symmetric surface. As an application, we show that for eachsymmetric double twist knot there are slopes which are strongly detected on thecharacter variety but not on the canonical component.

Symmetries and Detection of Surfaces by the Character Variety