Similar to knots in S3, any knot in a lens space has a grid diagram from which one can combinatorially compute all of its knot Floer homology invariants. We give an explicit description of the generators, differentials, and rational Maslov and Alexander gradings in terms of combinatorial data on the grid diagram. Motivated by existing results for the Floer homology of knots in S3 and the similarity of the resulting combinatorics presented here, we conjecture that a certain family of knots is characterized by their Floer homology. Coupled with the work of the third author, an affirmative answer to this would prove the Berge conjecture, which catalogs the knots in S3 admitting lens space surgeries.
ASJC Scopus subject areas