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arxiv: 2604.06690 · v1 · submitted 2026-04-08 · 🧮 math.GT · math.DS· math.SG

Legendrian position of veering triangulations

Chi Cheuk Tsang This is my paper

Pith reviewed 2026-05-10 17:16 UTC · model grok-4.3

classification 🧮 math.GT math.DSmath.SG
keywords veering triangulationsLegendrian arcsbicontact structuresAnosov flowspseudo-Anosov flowssteady positionhorizontal surgery
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The pith

Veering triangulations can be positioned so their edges are Legendrian arcs in a bicontact structure supporting the Anosov flow.

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

The paper proves that for veering triangulations coming from Anosov flows with orientable foliations, the edges can be made Legendrian with respect to a bicontact structure that supports the flow. It also proves that any veering triangulation can be put into steady position, where projections of edges intersect at most once transversely in the orbit space. This allows horizontal surgeries on the triangulation to be interpreted as horizontal Goodman surgeries on the flow. A sympathetic reader would care because it connects two different frameworks for studying these dynamical systems on three-manifolds.

Core claim

Given a veering triangulation corresponding to an Anosov flow with orientable stable and unstable foliations, the edges of the triangulation can be realized as Legendrian arcs with respect to a strongly adapted bicontact structure that supports the Anosov flow. Every veering triangulation can be placed in steady position, where each pair of edge projections that intersect in the orbit space only intersect once transversely.

What carries the argument

The steady position, in which intersecting edge projections cross transversely only once, which facilitates the Legendrian realization.

If this is right

  • Horizontal surgery of veering triangulations corresponds to horizontal Goodman surgery of pseudo-Anosov flows.
  • This equivalence allows combinatorial changes to the triangulation to be translated directly into modifications of the flow.
  • The Legendrian property is preserved under these operations.

Where Pith is reading between the lines

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

  • This positioning might enable the use of Legendrian knot theory techniques to study properties of the Anosov flows.
  • Steady position could simplify the analysis of intersection patterns in the orbit space for related combinatorial objects.
  • The result may inspire similar Legendrian realizations for other types of triangulations or flows.

Load-bearing premise

The veering triangulation must correspond to an Anosov flow whose stable and unstable foliations are orientable, with a strongly adapted bicontact structure available to support it.

What would settle it

A concrete veering triangulation linked to such an Anosov flow in which no placement of the edges as Legendrian arcs exists under any strongly adapted bicontact structure, or where steady position cannot be achieved.

Figures

Figures reproduced from arXiv: 2604.06690 by Chi Cheuk Tsang.

Figure 1
Figure 1. Figure 1: Left: The edges of a veering triangulation rotate towards the stable foliation as one moves along the flow. Right: The contact planes of a supporting bicontact structure (ξ+, ξ−) rotate towards the stable foliation as one moves along the flow. stable and unstable foliations of ϕ. In fact, once placed in this position, ∆ determines ϕ up to isotopic equivalence. In more detail, the edges of ∆ are determined … view at source ↗
Figure 2
Figure 2. Figure 2: Our first construction of the diagonals is to take the first￾horizontal-then-vertical arcs from the corners to an appropriate choice of a center anchor, then pull it tight [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Modifying the piecewise linear diagonals from [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The setup on the 3-torus. Symmetrically, we could make the same conjecture regarding Reeb flows of bicontact forms supporting ϕyz. The following conjecture morally combines these two versions of Conjecture 1.4. Conjecture 1.5. There exists two positive contact structures ξx, ξz and a negative contact structure ξy on Mnx,ny,nz such that • ϕxy is isotopically equivalent to a Reeb flow of ξz and is supported … view at source ↗
Figure 5
Figure 5. Figure 5: An illustration of Conjecture 1.6(III). Similarly, we conjecture that we have the following combinatorial isomorphisms: • (∆σ+ ) σ+ ∼= ∆ where both sides are taken up to horizontal surgery along A • (∆σ− ) σ− ∼= ∆ where both sides are taken up to horizontal surgery along B • ((∆σ+ ) σ− ) σ+ ∼= ((∆σ− ) σ+ ) σ− where both sides are taken up to horizontal surgery along A and B. In other words, we have a S3-to… view at source ↗
Figure 6
Figure 6. Figure 6: The stable and unstable foliations near a point on a nonsin￾gular orbit (left) and near a point on a 3-pronged singular orbit (right). A flow is transitive if it has a dense orbit. For a pseudo-Anosov flow, this is equivalent to the set of closed orbits being dense, see [BM26, Proposition 1.4.9]. All pseudo-Anosov flows in this paper will be transitive. We remark that Definition 2.2 and Definition 2.4 are … view at source ↗
Figure 7
Figure 7. Figure 7: A tetrahedron in a veering triangulation. There are no restrictions on the colors of the top and bottom edges. 3-manifold M is a decomposition of M into finitely many ideal tetrahedra glued along pairs of faces. A taut structure on an ideal triangulation is a labeling of the dihedral angles by 0 or π, such that: • Each tetrahedron has exactly two dihedral angles labeled by π, and they are opposite to each … view at source ↗
Figure 8
Figure 8. Figure 8: From left to right: an edge rectangle, a face rectangle, and a tetrahedron rectangle. Yellow dots denote elements of Ce. c [c] 1 2 3 4 [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The setting of Proposition 3.2. A face rectangle in O is a rectangle with one corner on Ce and the two opposite sides to the corner containing elements of Ce in their interior. Note that each face rectangle contains three edge subrectangles. Two of these edge rectangles have the same color, while the remaining one has the opposite color. A tetrahedron rectangle in O is a rectangle all of whose sides contai… view at source ↗
Figure 10
Figure 10. Figure 10: Analyzing arcs F ∧ Q ∩ DR. Definition 4.5 (Crossing criterion). We say that an edge candidate system {d ∧ R} satisfies the crossing criterion if the edge candidates d ∧ R are smoothly embedded, and for every R1 < R2, every intersection point between the projections of int(d ∧ R1 ) and int(d ∧ R2 ) in O is the projection of a crossing of int(d ∧ R2 ) over int(d ∧ R1 ). ♢ Proposition 4.6. Let ϕ be a pseudo-… view at source ↗
Figure 11
Figure 11. Figure 11: If R is an edge rectangle and Q is a face rectangle that intersect, then either we have Q < R or Q > R [PITH_FULL_IMAGE:figures/full_fig_p022_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: We can get rid of arcs and circles of intersection between faces by cutting and pasting along them. In the compact arc case, we claim that the endpoints of the arc must both lie below int(d ∧ R) or both lie above int(d ∧ R). Indeed, since R is an edge rectangle and Q is a face rectangle that intersect, either we have Q < R or Q > R. Without loss of generality suppose the former is true. Then also using th… view at source ↗
Figure 13
Figure 13. Figure 13: The setting of Definition 5.6. Definition 5.6 (Overlap of piecewise linear arcs). Let S be a surface equipped with a quasi-translation structure. Let p and q be two piecewise linear arcs in S, which are either both positive or both negative. Suppose p and q overlap along a (possibly degenerate) subarc r. That is, we can write p = p1 ∗v1 r ∗v2 p2 and q = q1 ∗v1 r ∗v2 q2, for (possibly degenerate) subarcs p… view at source ↗
Figure 14
Figure 14. Figure 14: Constructing the tight arc between γ1 and γ2 near w with respect to B 5.4. Canonical lifts. We continue the setting from the previous subsection. Let d be a smooth arc on O◦ . Suppose that d either has positive slope everywhere or negative slope everywhere. Consider the collection of arcs D that consists of the π1M-translates of d. Let De be the collection of preimages of arcs in D under the cover Sf◦ ∼= … view at source ↗
Figure 15
Figure 15. Figure 15: Our diagonals dR will be the union of two half-diagonals, each half-diagonal being a tight arc near near the corner that lies on the same stable leaf as the anchor, with respect to the buoys. (3) If b is a node of some diagonal dR1 that lies on some other diagonal dR2 , then it is a node of dR2 . (4) If b1 and b2 are adjacent nodes on a diagonal, then b1 and b2 do not lie in the same π1M-orbit. We provide… view at source ↗
Figure 16
Figure 16. Figure 16: Definition of strictly staircase monotone. • equivariant, i.e. g · αR = αg·R, and • staircase monotone, i.e. for every R1 < R2 that share a corner s, R(s, αR1 ) is wider and strictly shorter than R(s, αR2 ). This is the same definition as in [LMT23, Section 5.1.1]. Meanwhile, the following definition is new. A strict anchor system is a collection of one anchor αR per edge rectangle R that is • equivariant… view at source ↗
Figure 17
Figure 17. Figure 17: Defining the embedding ι : P → G. Proof. We first suppose that c is orientation-preserving. Let M, N ⊂ Ce be the two sets of corners of rectangles in P occupying the two quadrants of c. By Proposition 3.2, M and N are order isomorphic to Z under the relation m1 < m2 if R(c, m1) < R(c, m2). Moreover, identifying M, N ∼= Z, the element [c] acts by m 7→ m + m0 and n 7→ n + n0 on M and N respectively, for som… view at source ↗
Figure 18
Figure 18. Figure 18: Defining the maps r, t. Proof. Let P be the union of rectangles in P. There exists an embedding Ψ : P ,→ R 2 that maps Os/u to vertical/horizontal lines and is equivariant under the action of ⟨[c]⟩ on P and the action of ⟨[c]⟩ on R 2 with [c] · (x, y) = ( (λ −1x, λy) if c is orientation-preserving (−λ −1x, −λy) if c is orientation-reversing for some λ > 1, as explained in [LMT23, Claim 5.8]. Without loss … view at source ↗
Figure 19
Figure 19. Figure 19: The map T in the orientation-reversing case. This figure illustrates the case when r = 2. • for every R1, R2 ∈ P that share a corner s, if R1 < R2, then R(s, T(R1)) is strictly wider and strictly shorter than R(s, T(R2)). Using the fact that the image of ι lies within a strip {(m, n) ∈ Z 2 | n0 m0 m − r ≤ n ≤ n0 m0 m + r}, we can compose Ψ with a dilation (x, y) 7→ (ρx, ρy) for large ρ, so that T(R) ∈ b(R… view at source ↗
Figure 20
Figure 20. Figure 20: Left: The setting of Lemma 6.13. Right: There cannot exists subarcs p ′ ⊂ p and q ′ ⊂ q that cobound a disc. disc. Suppose otherwise, and suppose p ′ has a turn tp = (xt , yt,p). Then q ′ must contain a point (xt , yt,q) with yt,q ≤ yt,p, since otherwise q would pass through one of the slits in R\R! . See [PITH_FULL_IMAGE:figures/full_fig_p038_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: The possible configurations of type (I) and (II) pairs. • R1 is an edge rectangle, thus s2 cannot lie within R′ 1 . • R2 is an edge rectangle, thus s1 cannot lie within R′ 2 . • R1 < R2. • R1 and R2 are of the same color. • R′ 1 and R′ 2 intersect. Of the remaining 8 configurations, 4 of them are of type (I). We exhibit these in [PITH_FULL_IMAGE:figures/full_fig_p039_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Lemma 6.15 holds if we place buoys in the shaded rectangles. To argue that Q(R′ 2 ) is a rectangle, we analyze which R′ 1 can come up. If (R′ 1 , R′ 2 ) is a type (I-1) pair, then the anchor αR1 must lie in the edge rectangle R2, but not on an unstable leaf that passes through R′ 2 . Since the anchors are periodic, there are only finitely many anchors in R2. Meanwhile, for each such anchor α, there are on… view at source ↗
Figure 23
Figure 23. Figure 23: Left: The setting of Lemma 6.16. Middle: By convexity, p cannot osculate q from above. Right: There cannot exists subarcs p ′ ⊂ p and q ′ ⊂ q that cobound a disc. the common endpoints of p ′ and q ′ . Since all points of ∂S ∩ R have to lie below p ′ and above q ′ , we can consider the set R! ⊂ O◦ obtained by slitting the points of ∂S ∩ R lying below p ′ downwards, and slitting the points lying above q ′ u… view at source ↗
Figure 24
Figure 24. Figure 24: Lemma 6.17 holds if we place buoys in the shaded rectangles. Type (II-1). For each type (II-1) pair (R′ 1 , R′ 2 ), we let Q(R′ 1 , R′ 2 ) = R(hR′ 1 , αR′ 2 ) be the subrectangle of R′ 2 spanned by the hook hR′ 1 and the anchor αR′ 2 . If B contains a point b in Q(R′ 1 , R′ 2 ), then we can apply Lemma 6.16 to the rectangles R(s1, b) < R′ 2 in order to conclude. See [PITH_FULL_IMAGE:figures/full_fig_p043… view at source ↗
Figure 25
Figure 25. Figure 25: If σ, σ′ ∈ Σ are adjacent segments on dQ1 , then there is an overlap (dQ1 , dR2 , σ) if and only if there is an overlap (dQ1 , dR2 , σ′ ) for the same R2. arranged, we have a diagonal system consisting of piecewise linear arcs that intersect in points. Definition 7.3 (Peripheral, elementary overlap). We say that an overlap (dR1 , dR2 , σ) is • peripheral if σ contains an endpoint of the intersection dR1 ∩… view at source ↗
Figure 26
Figure 26. Figure 26: The setup for modifying a PLO diagonal system. σR σQ2 = σ σ ′ Q2 σ ′ R σ + Q2 σ + R dR dQ2 dQ1 dR dQ2 dQ1 [PITH_FULL_IMAGE:figures/full_fig_p047_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: Modifying a PLO diagonal system to reduce the number of bookkeeping nodes on dQ1 ∩ dQ2 . We delete from dR the union of σR and the subarc of σ ′ R from u to bR, then add back in σ + R . See [PITH_FULL_IMAGE:figures/full_fig_p047_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: Analysis for Proposition 6.1(2) if dR1 , dR2 ∈ D+. dQ1 dR1 dQ2 dR2 dQ1 dR1 dQ2 dR2 [PITH_FULL_IMAGE:figures/full_fig_p048_28.png] view at source ↗
Figure 29
Figure 29. Figure 29: Analysis for Proposition 6.1(2) if dR1 ∈ D+ but dR2 ̸∈ D+. If neither dR1 nor dR2 has a translate that lies in D+, then the modification does not involve dR1 and dR2 so Proposition 6.1(2) holds from before. Suppose dR1 has a translate that lies in D+. Up to applying the translation, we arrange it so that dR1 ∈ D+. If dR2 ∈ D+, then after modification dR1 and dR2 overlap in the same number of segments and … view at source ↗
Figure 30
Figure 30. Figure 30: Analysis for Proposition 6.1(2) if dR2 ∈ D+ but dR1 ̸∈ D+. By the elementary hypothesis, the bottom scenario cannot happen. (1) The interior of each dR is a piecewise linear arc in O◦ , under the quasi-translation structure induced from S. (2) The system {dR} satisfies the slope criterion for diagonals of the same color, i.e. for every R1 < R2 of the same color, • if R1 and R2 share a corner, then int(dR1… view at source ↗
Figure 31
Figure 31. Figure 31: Perturbing the diagonals to be smooth. many π1M◦ -orbits of diagonals in O◦ . Let de1, ..., der be a collection of one representative in each π1M◦ -orbit. Pick a large N so that Equation (7.1) holds. By Lemma 5.7, the set of π1(M◦ )[−N,N]-orbits of dei is locally finite. Thus we can choose a rectangular neighborhood νv of each non-smooth node v of dei , such that • each dei ∩ νv can be written as σ1 ∗v σ2… view at source ↗
Figure 32
Figure 32. Figure 32: Modifications in the proof of Proposition 7.10. We are now ready to modify our diagonals. Proposition 7.10. Under the hypothesis of Proposition 6.1, there exists a diagonal system {dR} satisfying the slope criterion. Proof. We first argue that up to a C 1 -small perturbation, we can assume that all misalignments have distinct projections: For each i = 1, ..., r, we consider the set of points on dei that i… view at source ↗
Figure 33
Figure 33. Figure 33: The tubular neighborhoods constructed in Construction 8.1. Note that Q0 ∼= {(t, x, y) | R/(pγZ) × R 2 \{(0, 0)} | x ∈ [0, ε], −x ≤ y ≤ x} Q1 ∼= {(t, x, y) | R/(pγZ) × R 2 \{(0, 0)} | y ∈ [0, ε], −y ≤ x ≤ y} Q2 ∼= {(t, x, y) | R/(pγZ) × R 2 \{(0, 0)} | x ∈ [−ε, 0], ε], x ≤ y ≤ −y} Q3 ∼= {(t, x, y) | R/(pγZ) × R 2 \{(0, 0)} | y ∈ [−ε, 0], y ≤ x ≤ −y} We set Q □ 0 ∼= {(t, x, y) | R/(pγZ) × R 2 \{(0, 0)} | x … view at source ↗
read the original abstract

We make a first step towards connecting the theory of veering triangulations and bicontact structures as tools for studying (pseudo-)Anosov flows: We show that given a veering triangulation corresponding to an Anosov flow with orientable stable and unstable foliations, the edges of the triangulation can be realized as Legendrian arcs with respect to a strongly adapted bicontact structure that supports the Anosov flow. Along the way, we show that every veering triangulation can be placed in `steady position', where each pair of edge projections that intersect in the orbit space only intersect once transversely. By a previous result of the author, this implies that horizontal surgery of veering triangulations correspond to horizontal Goodman surgery of pseudo-Anosov flows.

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 / 2 minor

Summary. The manuscript proves two main results connecting veering triangulations to bicontact structures and Anosov flows. Given a veering triangulation corresponding to an Anosov flow with orientable stable and unstable foliations, the edges can be realized as Legendrian arcs with respect to a strongly adapted bicontact structure supporting the flow. Separately, every veering triangulation admits a steady position in which any two edge projections that intersect in the orbit space do so exactly once and transversely. By a prior result of the author, the steady-position statement implies that horizontal surgery on veering triangulations corresponds to horizontal Goodman surgery on the associated pseudo-Anosov flows.

Significance. If the results hold, the work supplies a concrete geometric bridge between the combinatorial theory of veering triangulations and the contact-geometric theory of bicontact structures for Anosov flows. The unconditional steady-position theorem is likely to be useful on its own for controlling intersections in the orbit space. The Legendrian-realization statement is carefully scoped to the orientable-foliation case and to the existence of a strongly adapted bicontact structure, which keeps the claims falsifiable and proportionate to the tools employed.

minor comments (2)
  1. [Introduction] The introduction would benefit from a brief diagram or explicit coordinate description of the orbit space and the projection of edges, to make the definition of steady position immediately accessible to readers outside the immediate subfield.
  2. [Section 1] A short paragraph recalling the precise statement of the cited previous result on Goodman surgery would improve readability, even though the dependence is clearly flagged.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, the clear summary of our results, and the recommendation to accept. We have no major comments to address.

Circularity Check

0 steps flagged

Minor self-citation for surgery implication; central Legendrian and steady-position results are independent

full rationale

The paper directly constructs the Legendrian realization of triangulation edges with respect to a strongly adapted bicontact structure (for the scoped case of Anosov flows with orientable foliations) and the steady position for arbitrary veering triangulations. These are presented as new results without reduction to prior self-citations. The sole self-reference is the sentence 'By a previous result of the author, this implies that horizontal surgery of veering triangulations correspond to horizontal Goodman surgery of pseudo-Anosov flows,' which applies only to an additional implication and is not used to justify the main theorems. No self-definitional loops, fitted inputs renamed as predictions, ansatz smuggling, or uniqueness theorems imported from the author's prior work appear in the derivation of the primary claims. The assumptions are explicitly stated and external to the self-citation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, invented entities, or non-standard axioms are introduced; the work relies on standard definitions and existence assumptions from veering triangulation theory, Anosov flows, and contact geometry.

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Reference graph

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