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

Magnus Jacobsson; Ryszard L. Rubinsztein

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arXiv · 0806.2902

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

Magnus Jacobsson·Ryszard L. Rubinsztein

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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 ${\mathscr M}$ 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.

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Magnus Jacobsson, Ryszard L. Rubinsztein. 2009-06-19. Symplectic topology of SU(2)-representation varieties and link homology, I: Symplectic braid action and the first Chern class. https://arxiv.org/abs/0806.2902

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