Rational Seifert Surfaces in Seifert Fibered Spaces

TL;DR

This paper introduces a combinatorial approach to construct rational Seifert surfaces for links in Seifert fibered spaces, generalizing Seifert's algorithm and deriving formulas for Legendrian knot invariants.

Contribution

It establishes an equivalence between a diagram condition and the existence of rational Seifert surfaces, extending classical methods to rationally null-homologous links.

Findings

Provides a combinatorial construction of rational Seifert surfaces.

Derives explicit formulas for rational Thurston-Bennequin and rotation numbers.

Generalizes Seifert's algorithm to a broader class of knots.

Abstract

Rationally null-homologous links in Seifert fibered spaces may be represented combinatorially via labeled diagrams. We introduce an additional condition on a labeled link diagram and prove that it is equivalent to the existence of a rational Seifert surface for the link. In the case when this condition is satisfied, we generalize Seifert's algorithm to explicitly construct a rational Seifert surface for any rationally null-homologous knot. As an application of the techniques developed in the paper, we derive closed formulae for the rational Thurston-Bennequin and rotation numbers of a Legendrian knot in a contact Seifert fibered space.