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arxiv: 2604.08010 · v1 · submitted 2026-04-09 · 🧮 math.GT · math.SG

An algorithm to Legendrian realize a curve on a ribbon surface

Eric Stenhede This is my paper

Pith reviewed 2026-05-10 17:47 UTC · model grok-4.3

classification 🧮 math.GT math.SG
keywords Legendrian realizationribbon surfaceLegendrian graphopen bookcontact surgeryDehn twistcontact manifoldstandard contact structure
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The pith

An explicit algorithm turns any homologically nontrivial simple closed curve on a ribbon surface into a Legendrian knot in standard contact R^3.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes an explicit algorithm that produces a Legendrian realization of any homologically nontrivial simple closed curve lying on the ribbon surface of a Legendrian graph in the standard contact structure on R^3. A sympathetic reader would care because this turns abstract topological data on a surface into concrete Legendrian knots that can be studied or manipulated in contact geometry. The same algorithm converts an abstract open book whose monodromy consists of Dehn twists along such curves into an explicit contact surgery diagram for the supported contact manifold. The paper also records that any two Legendrian realizations of the same curve on a ribbon surface are Legendrian isotopic, and the same holds for knots on open book pages that are isotopic on the page.

Core claim

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 (R^3, ξ_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. 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.

What carries the argument

The explicit algorithm that constructs a Legendrian realization from a given homologically nontrivial curve on the ribbon surface of a Legendrian graph, together with the uniqueness statement that all such realizations are Legendrian isotopic.

If this is right

  • Any abstract open book with monodromy given by Dehn twists along homologically nontrivial curves can be converted algorithmically into a contact surgery diagram.
  • Legendrian knots on pages of open books that represent the same isotopy class on the page are Legendrian isotopic.
  • Legendrian realizations of curves on ribbon surfaces are unique up to Legendrian isotopy whenever the homology condition holds.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The uniqueness result implies that the Legendrian isotopy class of such a knot is completely determined by its isotopy class on the ribbon surface or page.
  • The algorithms make it possible in principle to compute contact-geometric invariants directly from combinatorial data on surfaces or open books.
  • The method may extend to producing Legendrian representatives for other classes of curves or surfaces, though the paper restricts to the homologically nontrivial case.

Load-bearing premise

The input curve on the ribbon surface must be homologically nontrivial.

What would settle it

A concrete homologically nontrivial simple closed curve on the ribbon surface of some Legendrian graph in (R^3, ξ_st) for which no Legendrian realization exists, or two non-isotopic Legendrian realizations of the same curve.

Figures

Figures reproduced from arXiv: 2604.08010 by Eric Stenhede.

Figure 1.1
Figure 1.1. Figure 1.1: Left: the front projection of a ribbon surface Σ. The crossing indicated by an arrow is one of the crossings that makes a naive Legendrian realization impossible. Middle: a homologically nontrivial curve a on Σ. Right: the Legendrian realization of a produced by the algorithm. Very briefly, the algorithm works as follows. Starting from a Legendrian graph G ⊂ (R 3 , ξst), we use the handle decomposition o… view at source ↗
Figure 2.1
Figure 2.1. Figure 2.1: Left: coordinates on R 3 . Center: the front projection. Right: the La￾grangian projection. If ⋆ is a Legendrian object in (R 3 , ξst), we denote its front projection by ⋆F and its Lagrangian projection by ⋆L. Let r = (rx, ry, rz): I → (R 3 , ξst) be a Legendrian immersion. Its front projection is rF = (ry, rz). A point of the image of a generic front projection has one of the local models shown in [PIT… view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: Local models for points in the front projection of a generic Legendrian graph. In particular, if r is a closed Legendrian curve, then I rL x dy = 0. Moreover, r is embedded if and only if every loop in rL (except, in the closed case, the full loop itself) encloses a nonzero oriented area [PITH_FULL_IMAGE:figures/full_fig_p004_2_2.png] view at source ↗
Figure 2.3
Figure 2.3. Figure 2.3: Lagrangian projections corresponding to the front projections in Fig￾ure 2.2. The y–coordinate agrees with that of the front projection, and the x–coordinate parametrizes the vertical direction (top to bottom is positive). 2.3. Convex surfaces and Legendrian realization. Given an oriented surface S in a contact manifold (M, ξ), we denote by Sξ its characteristic foliation. We say that S is convex if ther… view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: Left: a Legendrian graph in (R 3 , ξst). Right: a ribbon surface of the Legendrian graph. Every Legendrian graph G admits a ribbon surface: one can take a sufficiently small “thickening” of G in the direction of the contact structure. Any such ribbon is automatically convex, since it is transverse to a Reeb vector field. However, we can say more about this. Lemma 3.2. Let Σ be a ribbon surface. Then Σ is… view at source ↗
Figure 3.2
Figure 3.2. Figure 3.2: The handle decomposition of the ribbon from [PITH_FULL_IMAGE:figures/full_fig_p007_3_2.png] view at source ↗
Figure 3.3
Figure 3.3. Figure 3.3: Left: the characteristic foliation on a 1–handle of a generic ribbon, obtained by concatenating pieces like those at the top-left and bottom-left. Right: the characteristic foliation on a 0–handle. Here the vertex has valency 4, but the general case is analogous. 3.0.1. Ribbon surfaces in the standard contact structure. We now specialize to (R 3 , ξst). Given the front projection of a generic Legendrian … view at source ↗
Figure 3.4
Figure 3.4. Figure 3.4: Top row: front projection of the ribbon near a non-vertical edge, a crossing, cusps, and a vertex. Bottom row: corresponding Lagrangian projections. If the vertex in [PITH_FULL_IMAGE:figures/full_fig_p008_3_4.png] view at source ↗
Figure 3.5
Figure 3.5. Figure 3.5: (f) (g) [PITH_FULL_IMAGE:figures/full_fig_p008_3_5.png] view at source ↗
Figure 3.6
Figure 3.6. Figure 3.6: shows an example of this construction applied to a Legendrian graph G [PITH_FULL_IMAGE:figures/full_fig_p009_3_6.png] view at source ↗
Figure 3.7
Figure 3.7. Figure 3.7: Local simplifications of the ribbon via elliptic–hyperbolic cancellation. Horizontal and vertical reflections are also allowed (with crossings and shading adjusted). A global example is given in [PITH_FULL_IMAGE:figures/full_fig_p009_3_7.png] view at source ↗
Figure 3.8
Figure 3.8. Figure 3.8: Simplifications of the ribbon in [PITH_FULL_IMAGE:figures/full_fig_p009_3_8.png] view at source ↗
Figure 5.1
Figure 5.1. Figure 5.1: Left: a portion of a Legendrian graph G containing a vertex v and an edge. Middle: a portion of a ribbon surface subdivided into handles. Right: the Legendrian core e of the 1–handle and the Legendrian skeleton of the 0–handle. • Choose a 1–handle of S whose cocore intersects a an odd number of times (this exists by Lemma 3.4). Denote this distinguished 1–handle by H⋆. • Choose an orientation for the cor… view at source ↗
Figure 5.2
Figure 5.2. Figure 5.2: An abstract 1–handle, with oriented core e. In all subsequent pictures of abstract 1–handles we adopt this convention. For example, if a 1–handle H looks as on the left of [PITH_FULL_IMAGE:figures/full_fig_p018_5_2.png] view at source ↗
Figure 5.3
Figure 5.3. Figure 5.3: Left: a 1–handle H with a green and an orange segment and Legendrian core e. Right: the corresponding segments on the abstract handle of [PITH_FULL_IMAGE:figures/full_fig_p018_5_3.png] view at source ↗
Figure 5.4
Figure 5.4. Figure 5.4: Left: a 1–handle of S and the segment of a it contains after Step 1. Right: an example of how a may look on a 0–handle after Step 1. We now: • choose an orientation of a; [PITH_FULL_IMAGE:figures/full_fig_p018_5_4.png] view at source ↗
Figure 5.5
Figure 5.5. Figure 5.5: Top: a 1–handle with an odd number of segments of a. Bottom: a 1–handle with an even number of segments. The values ∆e are indicated. We then define the relative gain ∆(lji ) by ∆(lji ) := δji ∆( e lji ), where δji := ( 1, if the orientation of lji agrees with the orientation of the core e of H, −1, otherwise. For the segments in [PITH_FULL_IMAGE:figures/full_fig_p019_5_5.png] view at source ↗
Figure 5.6
Figure 5.6. Figure 5.6: The relative gains (displacement indices) of the segments from [PITH_FULL_IMAGE:figures/full_fig_p019_5_6.png] view at source ↗
Figure 5.7
Figure 5.7. Figure 5.7: Left: The front projection of the handle H and three segments on it (one being the Legendrian core of the handle). Right: The Legendrian realization of these three segments. Denote by e the Legendrian core of the handle H (the central blue segment in [PITH_FULL_IMAGE:figures/full_fig_p020_5_7.png] view at source ↗
Figure 5.8
Figure 5.8. Figure 5.8: Left: a 1–handle H with six segments of a. Right: the Legendrian core e = G ∩ H. Remove from e a small open subsegment disjoint from cusps and crossings. Denote the two resulting components by e + and e −, see [PITH_FULL_IMAGE:figures/full_fig_p021_5_8.png] view at source ↗
Figure 5.9
Figure 5.9. Figure 5.9: Left: the two components e + and e − of e after removing a small open interval. Right: k vertical translates of e + ⊔ e −. p + ∗ l∗ p − ∗ [PITH_FULL_IMAGE:figures/full_fig_p022_5_9.png] view at source ↗
Figure 5.10
Figure 5.10. Figure 5.10: A segment l∗ in H with endpoints p ± ∗ on the sides of e ±. Label the translates of e ± by e ±(1), . . . , e ±(k) as follows: the copy labeled e ±(l) lies higher in z than the copy e ±(m) if • P(p ± l ) > P(p ± m), or • P(p ± l ) = P(p ± m) but P(p ∓ l ) > P(p ∓ m). For each i = 1, . . . , k, connect e +(i) to e −(i) by a straight segment in the front projection. The resulting Legendrian arc is declared… view at source ↗
Figure 5.11
Figure 5.11. Figure 5.11: shows a concrete choice of prominences for the segments in [PITH_FULL_IMAGE:figures/full_fig_p022_5_11.png] view at source ↗
Figure 5.12
Figure 5.12. Figure 5.12: Legendrian realization of the segments of [PITH_FULL_IMAGE:figures/full_fig_p023_5_12.png] view at source ↗
Figure 5.13
Figure 5.13. Figure 5.13: A 0–handle H and its Legendrian skeleton. The Legendrian realization of a segment l∗ in H is obtained by concatenating two of the Legendrian segments e1, . . . , en, possibly translated in the ∂z direction and slightly perturbed. More precisely, if l∗ is the segment of a that joins a 1–handle H1 to a 1–handle H2, then its Legendrian realization joins the corresponding pieces of the edges of G passing th… view at source ↗
Figure 5.14
Figure 5.14. Figure 5.14: Left: a 0–handle H with several segments of a. Right: their Legendrian realizations when all prominences are distinct. The dots indicate the relative vertical position. In the case where P(ljl ) = P(ljm), we again draw the Legendrian realization of these segments as the concatenation of two of the Legendrian segments of the Legendrian skeleton of H, but we perturb them slightly to avoid overlaps. The re… view at source ↗
Figure 5.15
Figure 5.15. Figure 5.15: Left: a 0–handle with several segments of a of equal prominence. Right: their Legendrian realizations after a small perturbation. G R H⋆ l1 → [PITH_FULL_IMAGE:figures/full_fig_p024_5_15.png] view at source ↗
Figure 5.16
Figure 5.16. Figure 5.16: Left: the Legendrian graph G. Middle: a ribbon surface R of G and an oriented, homologically nontrivial simple closed curve a on R (in green). Right: the handle decomposition of R and the curve a subdivided into segments as in Step 1. of R (three 0–handles in blue, and six 1–handles in yellow), together with the curve a subdivided into segments as prescribed after Step 1. In particular, [PITH_FULL_IMAG… view at source ↗
Figure 5.17
Figure 5.17. Figure 5.17: From left to right: the Legendrian realizations of the segments l1 through l4 [PITH_FULL_IMAGE:figures/full_fig_p025_5_17.png] view at source ↗
Figure 5.18
Figure 5.18. Figure 5.18: From left to right: the Legendrian realizations of the segments l5 through l8 [PITH_FULL_IMAGE:figures/full_fig_p025_5_18.png] view at source ↗
Figure 5.19
Figure 5.19. Figure 5.19: From left to right: the Legendrian realizations of the segments l9 through l12. but one may run out of space in the front projection if a new segment needs to be drawn between two previously drawn segments. Denote by G(s) the graph G translated vertically by an amount s. We draw G(0), G(−ε), G(−2ε), . . . , G(−5ε). These are the light blue graphs in Figures 5.17–5.20. Consider first the Legendrian reali… view at source ↗
Figure 5.20
Figure 5.20. Figure 5.20: From left to right: the Legendrian realizations of the segments l13 through l16. first picture in [PITH_FULL_IMAGE:figures/full_fig_p026_5_20.png] view at source ↗
Figure 5.21
Figure 5.21. Figure 5.21: The Legendrian realization of the curve a from [PITH_FULL_IMAGE:figures/full_fig_p026_5_21.png] view at source ↗
Figure 5.22
Figure 5.22. Figure 5.22: Ribbon surfaces that are not adapted to the braid in [PITH_FULL_IMAGE:figures/full_fig_p027_5_22.png] view at source ↗
Figure 5.23
Figure 5.23. Figure 5.23: Left: a 1–handle and its characteristic foliation, with a single hyperbolic point. Right: the segments of a arranged along the leaves, except near the unstable separatrix. Top pictures: front projections; bottom pictures: schematic versions. This clearly implies (1). By pushing the segments of a closer to the central one we can make all |Di | as small as desired. Moreover, segments that are further away… view at source ↗
Figure 5.24
Figure 5.24. Figure 5.24: Segments of a in the distinguished handle H⋆ after the additional isotopy used to implement the modified gains. This proves (3). □ [PITH_FULL_IMAGE:figures/full_fig_p028_5_24.png] view at source ↗
Figure 5.25
Figure 5.25. Figure 5.25: Legendrian realization li of a segment li in a 1–handle. The lower surface is Σ, the others are vertical translates. For example, if we take the segments in [PITH_FULL_IMAGE:figures/full_fig_p029_5_25.png] view at source ↗
Figure 5.26
Figure 5.26. Figure 5.26: The Legendrian realization of the segments in [PITH_FULL_IMAGE:figures/full_fig_p029_5_26.png] view at source ↗
Figure 5.27
Figure 5.27. Figure 5.27: Left: a segment li in a 0–handle, isotoped into the desired position (purple). Right: comparison between the leaves of the characteristic foliation (blue) and the Legendrian realization of li (purple). The bump near the vertex corrects the mismatch in z at the endpoints. After this correction, we have hi(0) = hi(1) for all segments in 0–handles. □ 5.4.5. Gluing the segments together. Lemma 5.14. Let li … view at source ↗
Figure 5.28
Figure 5.28. Figure 5.28: Left: a section of the original ribbon Σ: green points represent a, pink points represent L ′ . Right: a small perturbation of Σ that contains L ′ . Because the quantities ε1 and ε0 in the construction of the hi can be chosen arbitrarily small by isotoping a closer to the cores of the handles, this perturbation can be made arbitrarily C 0–small [PITH_FULL_IMAGE:figures/full_fig_p030_5_28.png] view at source ↗
Figure 5.29
Figure 5.29. Figure 5.29: shows schematically how bringing a closer to the core reduces the vertical displacement between a and L ′ [PITH_FULL_IMAGE:figures/full_fig_p031_5_29.png] view at source ↗
Figure 5.30
Figure 5.30. Figure 5.30: (a) or [PITH_FULL_IMAGE:figures/full_fig_p031_5_30.png] view at source ↗
Figure 5.31
Figure 5.31. Figure 5.31: Left: the Legendrian realization of li and lj in the situation of Fig￾ure 5.30(a). Right: the Legendrian realization of li and lj in the situation of Fig￾ure 5.30(b). (a) (b) li lj li lj [PITH_FULL_IMAGE:figures/full_fig_p032_5_31.png] view at source ↗
Figure 5.32
Figure 5.32. Figure 5.32: (a) The Legendrian segments li and lj are initially embedded in the same ribbon surface at height hi(0) = hj (0). The 1–handle of this ribbon surface is depicted. (b) The Legendrian segments li and lj are eventually embedded in the same ribbon surface at height hi(1) = hj (1). Again, the 1–handle of this ribbon surface is depicted. If hi(1) = hj (1), then we have hi(0) > hj (0), and it follows that the … view at source ↗
Figure 5.33
Figure 5.33. Figure 5.33: (a) The Legendrian segments li and lj are initially embedded in the same ribbon surface at height hi(0) = hj (0). The 1–handle of this ribbon surface is depicted. (b) The Legendrian segments li and lj cross. The last remaining case is hi(1) < hj (1) and hi(0) > hj (0). In this case, the relative position of the two segments in H is as in [PITH_FULL_IMAGE:figures/full_fig_p032_5_33.png] view at source ↗
Figure 5.34
Figure 5.34. Figure 5.34: (a) Segments of a in a 0–handle. (b) Legendrian realizations when the prominences are distinct: higher prominence means higher z–level. (c) Legendrian realizations when the prominences agree: all lie on the same level and are separated by a small perturbation. This proves all assertions in Proposition 5.8 and hence completes the justification of the algorithm. □ 6. Applications: from open books to conta… view at source ↗
Figure 6.1
Figure 6.1. Figure 6.1: A B [PITH_FULL_IMAGE:figures/full_fig_p033_6_1.png] view at source ↗
Figure 6.2
Figure 6.2. Figure 6.2: A Legendrian graph obtained by concatenating pieces of type A and B. Its ribbon has genus equal to the number of A–pieces and boundary components equal to one plus the number of B–pieces. The ribbons of these graphs are pages of open books supporting (S 3 , ξst) (because of Theorem 2.5). Thus we have explicit models of open books for (S 3 , ξst) with any prescribed topology of the page. Proof of Corollar… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript gives an explicit algorithm to Legendrian realize a homologically nontrivial simple closed curve on the ribbon surface of a Legendrian graph in (R^3, xi_st). It applies the algorithm to produce contact surgery diagrams from abstract open books whose monodromy is a product of Dehn twists along such curves, and records a uniqueness statement that any two Legendrian realizations of the same curve on a ribbon surface (or of the same isotopy class on an open-book page) are Legendrian isotopic.

Significance. The explicit, constructive nature of the algorithm, built from standard contact-geometric operations (Legendrian isotopy, stabilization, and framing adjustments), supplies a practical tool for realizing curves on surfaces while respecting the Thurston-Bennequin inequality via the homological nontriviality hypothesis. The uniqueness result formalizes an implicit fact from earlier literature in a form directly usable for open-book applications. These features strengthen the manuscript's utility for constructing and comparing contact structures via surgery diagrams and open books.

minor comments (3)
  1. The algorithm is presented as a sequence of moves; adding a short pseudocode block or numbered checklist in the main algorithm section would make the steps easier to follow and verify.
  2. The uniqueness statement is described as implicit in prior work; a brief sentence recalling the precise earlier reference (e.g., the relevant theorem number) would clarify what is being recorded versus newly proved.
  3. Figure captions for the illustrative diagrams of the realization process could explicitly label the successive stabilizations and isotopies to match the textual steps.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, including recognition of the explicit algorithm's practical value and the utility of the uniqueness result for open-book applications. We appreciate the recommendation for minor revision.

Circularity Check

0 steps flagged

No significant circularity; explicit algorithmic construction is self-contained

full rationale

The paper presents an explicit algorithm for Legendrian realization of a homologically nontrivial simple closed curve on a ribbon surface, built from standard contact-geometric operations (Legendrian isotopy, stabilization, and surface framing adjustments) that are fully described without reduction to fitted parameters or self-referential definitions. The homological nontriviality condition is stated upfront as an assumption to ensure compatibility with the Thurston-Bennequin inequality. The uniqueness statement is recorded as implicit in prior work but is not used as a load-bearing self-citation chain for the main algorithm; it is presented separately for the needed form. No predictions, ansatzes, or renamings reduce by construction to the inputs. The result is a constructive procedure, self-contained against external contact-geometric benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the standard axioms of contact geometry in R^3 and the definition of ribbon surfaces and Legendrian graphs; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption The standard contact structure on R^3 satisfies the usual contact condition and admits Legendrian realizations of suitable curves.
    Invoked implicitly when stating Legendrian realization is possible.

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