# On Heegaard Floer and fix point Floer homologies for fibered knots

# On Heegaard Floer and fix point Floer homologies for fibered knots

Let K be a genus g fibered knot in a closed oriented 3-manifold Y. This means that the complement Y-K of K in Y fibers over

S^1 and K is the boundary of the closure of each fiber. In this talk we consider two topological invariants associated to this data. The first is the "hat" version of the knot Floer homology of K, which consists in a family \widehat{HFK}(Y,K,i) of abelian groups indexed by an integer i \in [-g,g]. The second is the "sharp" version HF^#(f) of the fix point Floer homology of any area-preserving representative f of the monodromy of the fibration of Y-K over S^1. We show that the there exists an isomorphism between \widehat{HFK}(Y,K,-g+1) and

HF^#(f) and we discuss some applications. This is a work in progress in collaboration with Paolo Ghiggini.