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We complete the first step in a two-part program proposed by Baker, Grigsby,and the author to prove that Berge's construction of knots in the three-spherewhich admit lens space surgeries is complete. The first step, which we provehere, is to show that a knot in a lens space with a three-sphere surgery hassimple (in the sense of rank) knot Floer homology. The second (conjectured)step involves showing that, for a fixed lens space, the only knots with simpleFloer homology belong to a simple finite family. Using results of Baker, weprovide evidence for the conjectural part of the program by showing that itholds for a certain family of knots. Coupled with work of Ni, these knotsprovide the first infinite family of non-trivial knots which are characterizedby their knot Floer homology. As another application, we provide a Floerhomology proof of a theorem of Berge.

On Floer homology and the Berge conjecture on knots admitting lens space surgeries