Rational Seifert surfaces in Seifert fibered spaces

Joan E. Licata and Joshua M. Sabloff

Vol. 258 (2012), No. 1, 199–221

DOI: 10.2140/pjm.2012.258.199

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 link. As an application of the techniques developed in the paper, we derive closed formulae for the rational Thurston–Bennequin and rotation numbers of a rationally null-homologous Legendrian knot in a contact Seifert fibered space.

Keywords

Seifert surface, Seifert fibered space, Legendrian knot

Mathematical Subject Classification 2010

Primary: 57M27

Secondary: 57R17

Milestones

Received: 10 August 2011

Accepted: 16 April 2012

Published: 21 July 2012