# A bound for rational Thurston-Bennequin invariants

## TL;DR

This paper introduces a rational tau invariant for knots in contact 3-manifolds, providing bounds on Legendrian knot invariants and computable in certain cases, advancing the understanding of knot invariants in contact topology.

## Contribution

The paper defines a new rational tau invariant for rationally null-homologous knots in contact 3-manifolds, extending the toolkit for studying Legendrian knots.

## Key findings

- The rational tau invariant bounds the sum of rational Thurston-Bennequin and rotation numbers.
- In Floer simple knots in L-spaces, the invariant can be explicitly computed using correction terms.
- The invariant offers new insights into the structure of Legendrian knot invariants.

## Abstract

In this paper, we introduce a rational $\tau$ invariant for rationally null-homologous knots in contact 3-manifolds with nontrivial Ozsv\'{a}th-Szab\'{o} contact invariants. Such an invariant is an upper bound for the sum of rational Thurston-Bennequin invariant and the rational rotation number of the Legendrian representatives of the knot. In the special case of Floer simple knots in L-spaces, we can compute the rational $\tau$ invariants by correction terms.