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

We introduce a unified framework for counting representations of knot groups into SU 2 and SL 2 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 R representations.We then prove the SL 2 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 R representations using only elementary topological hypotheses.Combined with the translation extension locus of Culler and 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 R comes from viewing their representations as the real points of the appropriate SL 2 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 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 !E 2 !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 R representations unimpeded.