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

Dunfield, Nathan M; Rasmussen, Jacob

doi:10.2140/gt.2025.29.4055

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A unified Casson–Lin invariant for the real forms of $\mathrm{SL}(2)$

Nathan M Dunfield and Jacob Rasmussen

Geometry & Topology 29 (2025) 4055–4188

DOI: 10.2140/gt.2025.29.4055

Abstract

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 $S^{2} \to 𝔼^{2} \to ℍ^{2}$ , 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.

Keywords

knot signature, character variety, Casson–Lin invariant, left-orderability

Mathematical Subject Classification

Primary: 57K10, 57K31

Secondary: 20F60, 53A20, 57K18, 57R58

References

Publication

Received: 8 January 2023

Accepted: 17 August 2024

Published: 26 November 2025

Proposed: Ciprian Manolescu

Seconded: Peter Ozsváth, Tomasz S Mrowka

Authors

Nathan M Dunfield
Department of Mathematics University of Illinois Urbana, IL United States
https://dunfield.info
Jacob Rasmussen
Department of Mathematics University of Illinois Urbana, IL United States
https://rasmusj.web.illinois.edu

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Volume 29, issue 8 (2025)