Mathematical Research Letters
Volume 19 (2012)
Sutured Floer homology and hypergraphs
Pages: 1309 – 1328
By applying Seifert’s algorithm to a special alternating diagram of a link $L$, one obtains a Seifert surface $F$ of $L$. We show that the set of Spin$^c$ structures that support the sutured Floer homology of the sutured manifold complementary to $F$ is affine isomorphic to the set of hypertrees in a certain hypergraph that is naturally associated to the diagram. This implies that the support in question is the set of integer lattice points of a convex polytope. This property has an immediate extension to Seifert surfaces arising from homogeneous link diagrams (including all alternating and positive diagrams).
In another direction, our results and work in progress of the second author with Murakami and Postnikov suggest a method for computing the “top” coefficients of the HOMFLY polynomial of a special alternating link from the sutured Floer homology of a Seifert surface complement for a certain dual link.