A spanning tree model for the Heegaard Floer homology of a branched double-cover

Given a diagram of a link K in S^3, we write down a Heegaard diagram for the branched-double cover Sigma(K). The generators of the associated Heegaard Floer chain complex correspond to Kauffman states of the link diagram. Using this model we make some computations of the homology \hat{HF}(Sigma(K)) as a graded group. We also conjecture the existence of a delta-grading on \hat{HF}(Sigma(K)) analogous to the delta-grading on knot Floer and Khovanov homology.Comment: 43 pages, 20 figure

Non-simple genus minimizers in lens spaces

Given a one-dimensional homology class in a lens space, a question related to the Berge conjecture on lens space surgeries is to determine all knots realizing the minimal rational genus of all knots in this homology class. It is known that simple knots are rational genus minimizers. In this paper, we construct many non-simple genus minimizers. This negatively answers a question of Rasmussen.Comment: 16 pages, 6 figure