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Symplectic topology of SU(2)-representation varieties and link homology, I: Symplectic braid action and the first Chern class
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics. (Algebra, Geometry and Logic)
(English)Manuscript (preprint) (Other academic)
Abstract [en]

There are some similarities between cohomology of SU(2)-representation varieties of the fundamental group of some link complements and the Khovanov homology of the links. We start here a program to explain a possible source of these similarities. We introduce a symplectic manifold M with an action of the braid group B(2n) preserving the symplectic structure. The action allows to associate a Lagrangian submanifold of M to every braid. The representation variety of a link can then be described as the intersection of such Lagrangian submanifolds, given a braid presentation of the link. In a sequel to this paper we shall refine representation varieties of links using this description. We expect this to go some way in explaining the similarities mentioned above.

Keyword [en]
Topology of SU(2)-representation varieties, symplectic structure invariant under a braid group action, its Lagrangian submanifolds, its almost complex structure.
National Category
Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-105809OAI: oai:DiVA.org:uu-105809DiVA: diva2:222546
Projects
Topological quandles and link homology
Available from: 2009-06-09 Created: 2009-06-09 Last updated: 2010-01-14

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