A unified Casson–Lin invariant for the real forms of SL (2)

We introduce a unified framework for counting representations of knot groups into SU₂and SL₂R. Fora knot K in the 3-sphere, Lin and others showed that a Casson-style count of SU₂representations with fixed meridional holonomy recovers the signature function of K. For knots whose complement containsno closed essential surface, we show there is an analogous count for SL₂R representations. We then prove the SL₂R count is determined by the SU₂count and a single integer h (K), allowing us to show the existence of various SL₂R representations using only elementary topological hypotheses. Combined with the translation extension locus of Culler and Dunfield, we use this to prove left-order ability 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 SU₂and SL₂R comes from viewing their representations as the real points of the appropriate SL₂C character variety. While such real loci are typically highly singular at the reducible characters that are common to both SU₂and SL₂R, in the relevant situations we showhow to resolve these real algebraic sets into smooth manifolds. We construct these resolutions using the geometric transition S₂→E₂→H₂ studied from the perspective of projective geometry, and they allow us to pass between Casson–Lin counts of SU₂and SL₂R representations unimpeded.