Transcript: AN ALGORITHM TO LEGENDRIAN REALIZE A CURVE ON A RIBBON SURFACE

Alex: Welcome to another episode of ResearchPod. Sam, what have you been digging into lately?

Sam: This paper by Eric Stenhede describes an algorithm to take a specific kind of loop drawn on a ribbon surface from a graph in 3D space and turn it into a Legendrian knot that fits the rules of contact geometry. The loop winds around the surface's holes in a nontrivial way.

Alex: So the paper solves how to draw these knots explicitly when simple drawings don't work?

Sam: Yes. In contact topology, contact structures on 3D spaces can be described two ways: surgery on links or open books with pages that twist along curves, like wringing a towel. The challenge is converting an open book with monodromy from Dehn twists on arbitrary loops to a surgery diagram. Previous proofs showed such Legendrian knots exist but gave no recipe to build them.

Alex: Why do simple projections fail?

Sam: A basic front projection—dropping the curve onto a plane—creates crossings at the ribbon's edge that break the Legendrian rule: the knot must stay tangent to contact planes everywhere. The algorithm breaks the curve into segments on the surface's handles, balances relative gain across them so the projection closes, and stacks pieces by prominence to set heights.

Alex: Relative gain fixes the closing. How do they compute it for each segment?

Sam: Imagine the curve split into pieces, each crossing a bridge-like 1-handle on the surface. For each, they measure the height shift in projection view—like if a path on a map climbs left or right of the bridge's center. That's relative gain, Δ(l), tracking total up-down change so the knot connects without gaps. They adjust shifts in one bridge to make the sum zero, like balancing a see-saw.

Alex: And prominence for stacking without clashes?

Sam: Prominence tracks heights at segment ends, so pieces connect at matching levels—like stacking building blocks on the right floors. For 1-handles, they copy the central line, braid copies side-by-side sorted by prominence, and shift vertically. For base 0-handles at vertices, they stack or nudge copies by prominence, like layering roads at a junction.

Alex: The front projection shows tangles and sharp turns, while the overhead Lagrangian view stays flat. Do different realizations of the same curve look the same?

Sam: The paper proves they are Legendrian isotopic—you can deform one into the other without breaking rules or self-crossing. This relies on the space of compatible contact forms being simply connected: any loop in it shrinks to a point, like pulling a rubber band tight on a balloon.

What makes that space simply connected?

AN ALGORITHM TO LEGENDRIAN REALIZE A CURVE ON A RIBBON SURFACE

ERIC STENHEDE

Apr 9, 2026·5:52·arxiv·Original Paper

Legendrian realizationRibbon surfacesContact surgery diagramsOpen book decompositionsDehn twistsLegendrian Realization

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6 min

Briefing

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.

AN ALGORITHM TO LEGENDRIAN REALIZE A CURVE ON A RIBBON SURFACE | ERIC STENHEDE | ResearchPod

AN ALGORITHM TO LEGENDRIAN REALIZE A CURVE ON A RIBBON SURFACE: Research Paper Podcast

AN ALGORITHM TO LEGENDRIAN REALIZE A CURVE ON A RIBBON SURFACE

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