A unified Casson-Lin invariant for the real forms of SL(2)
We introduce a unified framework for counting representations of knot groups into $SU(2)$ and $SL(2, \mathbb{R})$. For a knot $K$ in the 3-sphere, Lin and others showed that a Casson-style count of $SU(2)$ representations with fixed meridional holonomy recovers the signature function of $K$. For knots whose complement contains no closed essential surface, we show there is an analogous count for $SL(2, \mathbb{R})$ representations. We then prove the $SL(2, \mathbb{R})$ count is determined by the $SU(2)$ count and a single integer $h(K)$, allowing us to show the existence of various $SL(2, \mathbb{R})$ representations using only elementary topological hypotheses. Combined with the translation extension locus of Culler-Dunfield, we use this to prove left-orderability of many 3-manifold groups obtained by cyclic branched covers and Dehn fillings on broad classes of knots. We give further applications to the existence of real parabolic representations, including a generalization of the Riley Conjecture (proved by Gordon) to alternating knots. These invariants exhibit some intriguing patterns that deserve explanation, and we include many open questions. The close connection between $SU(2)$ and $SL(2, \mathbb{R})$ comes from viewing their representations as the real points of the appropriate $SL(2, \mathbb{C})$ character variety. While such real loci are typically highly singular at the reducible characters that are common to both $SU(2)$ and $SL(2, \mathbb{R})$, in the relevant situations, we show how to resolve these real algebraic sets into smooth manifolds. We construct these resolutions using the geometric transition $S^2 \to \mathbb{E}^2 \to \mathbb{H}^2$, studied from the perspective of projective geometry, and they allow us to pass between Casson-Lin counts of $SU(2)$ and $SL(2, \mathbb{R})$ representations unimpeded.