Classical invariants of Legendrian knots in the 3-dimensional torus

Abstract

All knots in R3 possess Seifert surfaces, and so the classical Thurston–Bennequin and rotation (or Maslov) invariants for Legendrian knots in a contact structure on R3 can be defined. The definitions extend easily to null-homologous knots in any 3-manifold M endowed with a contact structure ξ. 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 T3. We show how to compute the Thurston–Bennequin and rotation invariants in a tight oriented contact structure on T3 using projections.