An algorithm to Legendrian realize a curve on a ribbon surface

TL;DR

This paper presents an explicit algorithm for Legendrian realization of curves on ribbon surfaces and applies it to convert open books into contact surgery diagrams, also establishing a uniqueness result.

Contribution

It introduces a new explicit algorithm for Legendrian realization on ribbon surfaces and applies it to contact topology, linking open books and surgery diagrams.

Findings

Algorithm for Legendrian realization of curves on ribbon surfaces.

Conversion method from open books with Dehn twists to contact surgery diagrams.

Uniqueness of Legendrian realizations up to Legendrian isotopy.

Abstract

We give an explicit algorithm to Legendrian realize a homologically nontrivial simple closed curve on a ribbon surface of a Legendrian graph in the standard contact structure $(\mathbb{R}^3,\xi_{\rm st})$. As an application, we obtain an algorithm that converts an abstract open book whose monodromy is written as a product of Dehn twists along homologically nontrivial curves into a contact surgery diagram for the supported contact manifold. Along the way, we also record a uniqueness statement which is implicit in earlier work but, to our knowledge, was never written in the form needed here: any two Legendrian realizations of the same curve on a ribbon surface are Legendrian isotopic, and likewise for Legendrian knots lying on pages of open books and representing the same isotopy class on the page.