Classical invariants of Legendrian knots in the 3-dimensional torus

Abstract

All knots in $R^3$ possess Seifert surfaces, and so the classical Thurston-Bennequin and rotation (or Maslov) invariants for Legendrian knots in a contact structure on $R^3$ can be defined. The definitions extend easily to null-homologous knots in any $3$-manifold $M$ endowed with a contact structure $\xi$. We generalize the definition of Seifert surfaces and use them to define these invariants for all Legendrian knots, including those that are not null-homologous, in a contact structure on the $3$-torus $T^3$. We show how to compute the Thurston-Bennequin and rotation invariants in a tight oriented contact structure on $T^3$ using projections.