![]() ![]() Generalizing rational tangles, a split tangle is a rationaltangle with links possibly tied into either of the strands. The invariant HFT in fact detectsrational tangles. Eachcomponent is one of two types: rational (which is the immersed curve invariant of a rationaltangle) or irrational. Furthermore, a structure theorem has beenestablished for what the individual components of multicurves can look like. This multicurve is a geometric realization of abordered sutured Heegaard Floer invariant called the peculiar module of T, and the LagrangianFloer homology of two such multicurves describes the link Floer homology of a suitable tanglesum of the two corresponding tangles. ![]() Following a construction ofthe third author, we associate a decorated immersed multicurve HFT( T ) in the four-puncturedsphere to a four-ended (i.e. We now describe the rough strategy for the proof of this theorem. As an immediate corollary, we obtain:Ĭonway mutation preserves L-space knots. This answers affirmatively the conjecture posed by the first and second authors in. It is essential if the corresponding four-punctured sphere is incompressible in the knotexterior.The main purpose of this article is to prove: Main Theorem.Ī knot in S with a non-trivial L-space surgery admits no essential Conwaysphere. ![]() Recall that a Conway sphere is a two-sphere intersecting the knot transversely in fourpoints. It is natural to ask whether the existence ofcertain essential surfaces in the complement of a knot can obstruct non-trivial surgeries yieldingL-spaces. Both properties are inherentlystatements about surfaces in the knot exterior. Ozsváth and Szabó established a structuretheorem for the knot Floer homology of such knots, which allows one to show that theyare fibered and prime. While Floer theory has become a very effective modern tool for an-swering questions about Dehn surgery, a geometric characterization of knots admitting L-spacesurgeries remains a difficult outstanding problem. This class includes, for example, all three-manifolds with finite fundamental group. L-spaces are closed, oriented three-manifolds with the simplest possible Heegaard Floerhomology. We consider the problem of whether Dehn surgery along a knot K in S produces an L-space. We prove that L-space knots do not have essential Conway spheres with the tech-nology of peculiar modules, a Floer theoretic invariant for tangles. This is joint work with Bach Nguyen (Temple University) and Matt Lee (University of Illinois at Chicago).LL-SPACE KNOTS HAVE NO ESSENTIAL CONWAY SPHERES We will conclude with a discussion of the Yangian, its relation to the quantum loop algebra, and some recent work concerning its global Weyl modules. Along the way, the utility of highest–weight representations, and of the (local and global) Weyl Modules, in all of these settings will be described. In this introductory talk, we proceed by example from the classical structure and representation theory of the special linear algebra in dimensions 2 and 3, to that of the corresponding Loop algebras and quantum groups. Particular examples of highest–weight representations of certain infinite–dimensional Lie algebras called the Weyl modules (for loop and quantum algebras) were introduced by Chari and Pressley in 2000. Highest-weight representations play a prominent role in the representation theory of Lie algebras and quantum groups. Highest-weight representations and global Weyl modules: from classical Lie algebras to YangiansĪuthors: Prasad Senesi*, Bach Nguyen and Matt Lee ![]()
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