Symplectic topology of SU(2)-representation varieties and link homology, I: Symplectic braid action and the first Chern class

Abstract

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 $\mathcal{M}$ with an action of the braid group $B_{2n}$ preserving the symplectic structure. The action allows to associate a Lagrangian submanifold of $\mathcal{M}$ 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. We expect this to go some way in explaining the similarities mentioned above.

78 pages, 15 figures. Version 3: Rearrangement of sections on the Chern class. A theorem on monotonicity of the extended moduli space of flat connections added. Minor additional changes

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