[2209.03382] Unified Casson-Lin invariant of the real form of SL(2)

[Submitted on 7 Sep 2022]

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Overview: Introduce a unified framework for counting knot group representations into $SU(2)$ and $SL(2, \mathbb{R})$. For the three-sphere knot $K$, Lin and others have shown that his Casson-style counting of the $SU(2)$ representation with a fixed meridian holonomy recovers the signature function of $K$. rice field. We show that for knots whose complements do not contain closed intrinsic surfaces, the $SL(2, \mathbb{R})$ representation has similar counts. We then prove that the count of $SL(2, \mathbb{R})$ is determined by the count of $SU(2)$ and the single integer $h(K)$, and the various $SL ( 2, \mathbb{R})$ expressions use only basic topological hypotheses.

In combination with the Culler-Dunfield translational extension locus, we use this to prove the left-order possibility of many 3-manifold groups obtained by Dehn filling of a wide class of circular branch covers and knots. We show further applications to the existence of real parabolic representations, including alternating knot generalizations of the Riley conjecture (proven by Gordon). These invariants exhibit some interesting patterns worthy of explanation and contain many open questions.

The close relationship between $SU(2)$ and $SL(2, \mathbb{R})$ makes their representations the proper $SL(2, \mathbb{C})$ It comes from looking at it as a point. Usually such real loci are highly singular with reducible characters common to both $SU(2)$ and $SL(2, \mathbb{R})$, but in relevant situations these We show how to solve the real algebra of . Set to a smooth manifold. We construct these resolutions using the geometric transition $S^2 \to \mathbb{E}^2 \to \mathbb{H}^2$ studied in terms of projective geometry, giving Casson-Lin allow it to pass through. Counting of $SU(2)$ and $SL(2, \mathbb{R})$ expressions is not prevented.