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arXiv · 2604.08010

An algorithm to Legendrian realize a curve on a ribbon surface

Eric Stenhede
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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 $(\mathbb{R}^3,ξ_{\rm st})$. 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.

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Eric Stenhede. 2026-04-09. An algorithm to Legendrian realize a curve on a ribbon surface. https://arxiv.org/abs/2604.08010

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