Rational linking and contact geometry

TL;DR

This paper extends classical invariants and inequalities to Legendrian and transverse knots in rationally null-homologous types, providing new classifications and corrections to previous definitions in contact geometry.

Contribution

It generalizes key invariants and inequalities for rational knots, classifies when bounds are sharp, and corrects earlier definitions of rational self-linking number.

Findings

Generalized self-linking number, Thurston-Bennequin invariant, and rotation number for rational knots.

Proved a version of Bennequin's inequality for these knots.

Classified when the Bennequin bound is sharp for fibered knot types.

Abstract

In the note we study Legendrian and transverse knots in rationally null-homologous knot types. In particular we generalize the standard definitions of self-linking number, Thurston-Bennequin invariant and rotation number. We then prove a version of Bennequin's inequality for these knots and classify precisely when the Bennequin bound is sharp for fibered knot types. Finally we study rational unknots and show they are weakly Legendrian and transversely simple. This version of the paper corrects the definition of rational self-linking number in the previous and published version of the paper. With this correction all the main results of the paper remain true as originally stated.