Floer homology, the HOMFLY polynomial, and combinatorics

All oriented links $L$ have special diagrams. Based on such a diagram we construct a sutured handlebody $M$ which embeds in the branched double cover of $L$. From the sutured Floer homology of $M$ we recover the Alexander polynomial $\Delta$ of $L$ via a simple forgetful map. More surprisingly, in cases when the diagram is also positive (so that $L$ is a special alternating link), $\mathrm{SFH}(M)$ can be used to compute those coefficients of the HOMFLY polynomial of $L$ whose sum is the leading coefficient of $\Delta$. To extract this information algebraically, we develop the notion of the interior polynomial of a bipartite graph. The talk involves joint results with A. Juh\'asz, H. Murakami, A. Postnikov, J. Rasmussen, and D. Thurston.