Braid classes and their Floer homology

Area-preserving diffeomorphisms of a 2-disc can be regarded as time-1 maps of (non-autonomous) Hamiltonian systems on $S^1 x D^2$. In this 3-dimensional setting we can think of flow-lines of the Hamilton equations as closed braids in the solid torus $S^1 x D^2$. In the spirit of positive braid classes and flat-knot types we define braid classes and use Floer’s variational approach on these spaces to define a chain-complex and the associated Floer homology invariants. This yields a Morse type theory for braided solutions of the Hamilton equations and in particular for periodic points of area-preserving diffeomorphisms of the 2-disc. The ideas presented here carry over to general two dimensional compact symplectic manifolds (M, $\omega$) (with or without boundary). In the case of the 2-disc we prove a reduction result with respect to Gar- side’s normal form for braids and conjugacy classes of braids.