Tight contact structures on Seifert surface complements

TL;DR

This paper explores the classification of tight contact structures on Seifert surface complements of special alternating links, revealing connections to knot invariants and Floer homology.

Contribution

It introduces a method to enumerate tight contact structures using Honda's approach, linking their count to the Alexander polynomial's leading coefficient.

Findings

Number of tight contact structures equals Alexander polynomial's leading coefficient.

Euler classes correspond to hypertrees in a hypergraph.

Contact invariants form a basis for sutured Floer homology.

Abstract

We consider complements of standard Seifert surfaces of special alternating links. On these handlebodies, we use Honda's method to enumerate those tight contact structures whose dividing sets are isotopic to the link, and find their number to be the leading coefficient of the Alexander polynomial. The Euler classes of the contact structures are identified with hypertrees in a certain hypergraph. Using earlier work, this establishes a connection between contact topology and the Homfly polynomial. We also show that the contact invariants of our tight contact structures form a basis for sutured Floer homology. Finally, we relate our methods and results to Kauffman's formal knot theory.