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Let K \subset Y be a knot in a three manifold which admits a longitude-framedsurgery such that the surgered manifold has first Betti number greater thanthat of Y. We find a formula which computes the twisted Floer homology of thesurgered manifold, in terms of twisted knot Floer homology. Using this, wecompute the twisted Heegaard Floer homology \underline{HF}^+ of the mappingtorus of a diffeomorphism of a closed Riemann surface whose mapping class isperiodic, giving an almost complete description of the structure of thesegroups. When the mapping class is nontrivial, we find in particular that in the"second-to-highest" Spin^c structure, this is isomorphic to a free module (overa certain ring) whose rank is equal to the Lefschetz number of thediffeomorphism. After taking a tensor product with Z/2Z, this agrees preciselywith the symplectic Floer homology of the diffeomorphism, as calculated byGautschi.

On the twisted Floer homology of mapping tori of periodic diffeomorphisms