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There are some similarities between cohomology of SU(2)-representationvarieties of the fundamental group of some link complements and the Khovanovhomology of the links. We start here a program to explain a possible source ofthese similarities. We introduce a symplectic manifold ${\mathscr M}$ with anaction of the braid group $B_{2n}$ preserving the symplectic structure. Theaction allows to associate a Lagrangian submanifold of ${\mathscr M}$ to everybraid. The representation variety of a link can then be described as theintersection of such Lagrangian submanifolds, given a braid presentation of thelink. We expect this to go some way in explaining the similarities mentionedabove.

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