Sutured Floer homology and hypergraphs

TL;DR

This paper establishes a correspondence between sutured Floer homology support and hypertrees in hypergraphs derived from link diagrams, revealing geometric structures of Floer groups and suggesting new computational methods for link invariants.

Contribution

It introduces a novel affine isomorphism between sutured Floer homology support and hypergraph hypertrees, extending to various Seifert surfaces and linking to HOMFLY polynomial coefficients.

Findings

Floer homology support corresponds to lattice points in a convex polytope.

The method applies to all homogeneous link diagrams, including alternating and positive.

Potential to compute HOMFLY polynomial coefficients from Floer homology.

Abstract

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 support of the sutured Floer homology of the sutured manifold complementary to F is affine isomorphic to the set of lattice points given as hypertrees in a certain hypergraph that is naturally associated to the diagram. This implies that the Floer groups in question are supported in a set of Spin^c structures that are the 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, together with work in progress of the second author and others, our correspondence suggests 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.