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

TL;DR

The paper develops a unified framework that simultaneously counts SU(2) and SL(2,R) representations of knot groups, producing a pair of Casson–Lin-type invariants whose sum h(K) is determined by the Levine–Tristram signature and a knot-specific integer. Central to the approach is a resolution of the real loci of character varieties and a geometric transition that links SU(2) and SL(2,R) data via smooth manifolds, enabling robust intersection counts independent of the trace parameter c. The authors prove that h(K) controls the existence of SL(2,R) representations and has strong applications to left-orderability of 3-manifold groups, including large cyclic branched covers and certain Dehn surgeries; they also define an extended Lin invariant encoding parabolic SL(2,R) representations through the translation extension locus. The work blends hyperbolic-geometry-inspired configuration spaces, Kempf–Ness-type resolutions, and braid-plat techniques to provide new results and open questions connecting knot signatures, representation theory, and 3-manifold topology. It offers explicit computations for alternating, Montesinos, and torus knots and lays groundwork for further exploration of left-orderability and Seiberg–Witten-type relations in real SL(2,R) character theory.

Abstract

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.

Figures (24)

• Figure 1.7: At left in (a) is the pillowcase orbifold $X_{\mathrm{SU}_2}(\partial M_K)$ containing the image of $X_{\mathrm{SU}_2}(M_K)$ under the restriction map, where $K$ is the positive trefoil knot. Details are given in Section \ref{['sec: SLR and T2']}, but the horizontal and vertical coordinates on the pillowcase are the holonomies of $\mu$ and $\lambda$, where $\lambda$ is the Seifert longitude. The blue dots correspond to the roots of $\Delta_K(t)$ on the unit circle. At right in (b) is the corresponding picture for $X_{\mathrm{ SL}_{2} {\mathbb R}}^{\mathrm{ell}}(M_K)$.
• Figure 1.9: For the positive trefoil knot $K$, this figure shows $i^*(X_{\mathrm{SU}_2}(M_K))$ in green and $i^*(X^{\mathrm{ell}}_{\mathrm{ SL}_{2} {\mathbb R}}(M_K))$ in purple on a single pillowcase; compare Figure \ref{['Fig:CVs for trefoil']}. Excluding the locus of reducible characters, which is common to both, they come together only at the points corresponding to roots of $\Delta_K$ on the unit circle. With respect to the orientations indicated, one has $h^c_{\mathrm{SU}_2}(K) = \left\langle i^*(X^{\mathrm{irr}}_{\mathrm{SU}_2}(M_K)),\ V_c \right\rangle$ and similarly for $h^c_{\mathrm{ SL}_{2} {\mathbb R}}(K)$. Hence $h(K) = 1$ for this knot.
• Figure 1.10: At left in (a) is $X^c(S_4)$ for $c = 2 \cos \frac{2 \pi}{5}$, drawn using the equations from BenedettoGoldman1999. Topologically, $X_{\mathrm{SU}_2}(S_4)$ is a 2-sphere, whereas $X_{\mathrm{ SL}_{2} {\mathbb R}}(S_4)$ has four distinct components, each of which is a plane. The intersection $X^c_{\mathrm{SU}_2}(S_4) \cap X^c_{\mathrm{ SL}_{2} {\mathbb R}}(S_4) = X^{\mathrm{red}}(S_4)$ is just three points, indicated by small dots. The bottom component of $X_{\mathrm{ SL}_{2} {\mathbb R}}(S_4)$ corresponds to the Teichmüller space of hyperbolic structures on the orbifold with underlying space $S^2$ and four points labeled ${\mathbb Z}/5$. At right in (b) is the resolution ${\EuScript X}^c(S_n)$ where each of the three points in $X^{\mathrm{red}}(S_4)$ has been replaced by a circle.
• Figure 11.8: This figure shows our conventions for braids, including $\sigma_i$ versus $\sigma_i^{-1}$, that left-to-right in the braid word corresponds to top-to-bottom in the picture, and that the generators $s_i$ of $\pi_1(S_n)$ correspond to clockwise loops. Moreover, the induced mapping class $\varphi_\beta \in \mathrm{MCG}(D_n)$ of $\beta$ is the map that pushes curves from the bottom to top, hence $\varphi_{\alpha \beta} = \varphi_\alpha \circ \varphi_\beta$ so that $B_{n} \to \mathrm{MCG}(D_n)$ is a homomorphism rather than an antihomorphism (here functions act on the left as usual); in particular, $\varphi_{\sigma_1}(s_1) = s_2$ and $\varphi_{\sigma_1}(s_2) = s_2^{-1} s_1 s_2$.
• Figure 11.8: This figure shows our conventions for the fundamental group of the exterior ${M_K} = H_1 \cup_{\varphi_\beta} H_2$ of the plat closure $K = {{\mkern 2.5mu\widehat{\mkern-2.5mu \beta\mkern-0mu}\mkern0mu}}$. Here the canonical copy of $S_{2m}$ is the one on the boundary of $H_1$. For clarity, only half of the generators of $\pi_1(S_4)$ are drawn on each copy of $S_4$.

Theorems & Definitions (249)

• theorem 1.1
• theorem 1.2
• corollary 1.4
• theorem 1.5
• theorem 1.6
• theorem 1.7
• lemma 2.2
• proof
• remark 2.4
• lemma 2.5