9,794 research outputs found
On the sheaf-theoretic SL(2, C) Casson–Lin invariant
No full textWe prove that the (τ-weighted, sheaf-theoretic) SL(2, C) Casson–Lin invariant introduced by Manolescu and the first author is generically independent of the parameter τ and additive under connected sums of knots in integral homology 3-spheres. This addresses two questions asked by Manolescu and the first author. Our arguments involve a mix of topology and algebraic geometry, and rely crucially on the fact that the SL(2, C) Casson–Lin invariant admits an alternative interpretation via the theory of Behrend functions.</p
Comparative studies on the enantioseparation of hydrobenzoin and structurally related compounds by capillary zone electrophoresis with sulfated beta-cyclodextrin as the chiral selector in the presence and absence of borate complexation
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A unified Casson-Lin invariant for the real forms of SL(2)
Full text linkWe introduce a unified framework for counting representations of knot groups into and . For a knot in the 3-sphere, Lin and others showed that a Casson-style count of representations with fixed meridional holonomy recovers the signature function of . For knots whose complement contains no closed essential surface, we show there is an analogous count for representations. We then prove the count is determined by the count and a single integer , allowing us to show the existence of various 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 and comes from viewing their representations as the real points of the appropriate character variety. While such real loci are typically highly singular at the reducible characters that are common to both and , in the relevant situations, we show how to resolve these real algebraic sets into smooth manifolds. We construct these resolutions using the geometric transition , studied from the perspective of projective geometry, and they allow us to pass between Casson-Lin counts of and representations unimpeded.148 pages, 24 figures; v2: incorporates referee\u27s comments; to appear in Geometry and Topolog
Enantioseparations of hydrobenzoin and structurally related compounds in capillary zone electrophoresis using heptakis(2,3-dihydroxy-6-O-sulfo)-beta-cyclodextrin as chiral selector and enantiomer migration reversal of hydrobenzoin with a dual cyclode
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Enantioseparation of benzoins and enantiomer migration reversal of hydrobenzoin in capillary zone electrophoresis with dual cyclodextrin systems and borate complexation
No full textA procedure to minimize lower lid retraction during large inferior rectus recession in Graves ophthalmopathy.
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