Classical invariants of Legendrian knots in the 3-dimensional torus

TL;DR

This paper extends the classical invariants of Legendrian knots, specifically Thurston-Bennequin and rotation numbers, to all Legendrian knots in the 3-torus, including non-null-homologous ones, and provides methods for their computation.

Contribution

It generalizes the definition of Seifert surfaces and invariants to all Legendrian knots in the 3-torus, including those not null-homologous, and offers computational techniques.

Findings

Invariants are defined for all Legendrian knots in T^3.

Methods for computing invariants in tight contact structures on T^3.

Extension of classical invariants to non-null-homologous knots.

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.