  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.
