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

##### Authors
Jacobsson, Magnus
Rubinsztein, Ryszard L.
##### Description
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 ${\mathscr M}$ with an action of the braid group $B_{2n}$ preserving the symplectic structure. The action allows to associate a Lagrangian submanifold of ${\mathscr M}$ 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.
Comment: 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
##### Keywords
Mathematics - Geometric Topology, 57R17, 57M27