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LEGENDRIAN WEAVES - N-GRAPH CALCULUS, FLAG MODULI and APPLICATIONS
Roger Casals,Eric Zaslow +1 more
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In this paper, the authors studied a class of Legendrian surfaces in contact five-folds by encoding their wavefronts via planar combinatorial structures, referred to these surfaces as Legendrian weaves, and to the objects as N-graphs.Abstract:
We study a class of Legendrian surfaces in contact five-folds by encoding their wavefronts via planar combinatorial structures. We refer to these surfaces as Legendrian weaves, and to the combinatorial objects as N-graphs. First, we develop a diagrammatic calculus which encodes contact geometric operations on Legendrian surfaces as multi-colored planar combinatorics. Second, we present an algebraic-geometric characterization for the moduli space of microlocal constructible sheaves associated to these Legendrian surfaces. Then we use these N-graphs and the flag moduli description of these Legendrian invariants for several new applications to contact and symplectic topology.
Applications include showing that any finite group can be realized as a subfactor of a 3-dimensional Lagrangian concordance monoid for a Legendrian surface in the 1-jet space of the two-sphere, a new construction of infinitely many exact Lagrangian fillings for Legendrian links in the standard contact three-sphere, and performing rational point counts over finite fields that distinguish Legendrian surfaces in the standard five-dimensional Darboux chart. In addition, the manuscript develops the notion of Legendrian mutation, studying microlocal monodromies and their transformations. The appendix illustrates the connection between our N-graph calculus for Lagrangian cobordisms and Elias-Khovanov-Williamson's Soergel Calculus.read more
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Kazhdan-Lusztig-Polynome und unzerlegbare Bimoduln "uber Polynomringen
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Augmentations, Fillings, and Clusters
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Book
Graph Theory
J. A. Bondy,U.S.R Murty +1 more
TL;DR: This book provides a systematic treatment of the theory of graphs without sacrificing its intuitive and aesthetic appeal, and is suitable as a textbook for advanced undergraduate and beginning graduate students in mathematics and computer science.
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Knots and Links
TL;DR: In this paper, the fundamental group of three-dimensional PL geometry Seifert surfaces Finite cyclic coverings and the torsion invariants Infinite cyclic covers and the Alexander invariant Matrix invariants 3-manifolds and surgery on links Foliations, branched covers, fibrations and so on.
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Braids, Links, and Mapping Class Groups
TL;DR: Artin's braid group has been studied extensively in the literature as discussed by the authors, where structural and algebraic properties of the braid groups of two manifolds of two different scales have been studied.
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Cluster algebras I: Foundations
Sergey Fomin,Andrei Zelevinsky +1 more
TL;DR: In this article, a new class of commutative algebras was proposed for dual canonical bases and total positivity in semisimple groups. But the study of the algebraic framework is not yet complete.