# Rotation numbers and the Euler class in open books

## TL;DR

This paper develops methods to compute Legendrian knot invariants, such as rotation number and Euler class, using front and Lagrangian projections in contact manifolds supported by open books, with explicit formulas and applications.

## Contribution

It introduces new techniques for calculating Legendrian invariants from front projections in open book decompositions, including formulas for intersection numbers and the Euler class.

## Key findings

- Computed rotation numbers from front projections.
- Derived explicit formulas for intersection numbers.
- Calculated the Euler class of the contact structure.

## Abstract

This paper introduces techniques for computing a variety of numerical invariants associated to a Legendrian knot in a contact manifold presented by an open book with a Morse structure. Such a Legendrian knot admits a front projection to the boundary of a regular neighborhood of the binding. From this front projection, we compute the rotation number for any null-homologous Legendrian knot as a count of oriented cusps and linking or intersection numbers; in the case that the manifold has non-trivial second homology, we can recover the rotation number with respect to a Seifert surface in any homology class. We also provide explicit formulas for computing the necessary intersection numbers from the front projection, and we compute the Euler class of the contact structure supported by the open book. Finally, we introduce a notion of Lagrangian projection and compute the classical invariants of a null-homologous Legendrian knot from its projection to a fixed page.