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-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 S2βE2βH2, 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.Comment: 147 pages, 24 figure