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Nodal Tangles

T0 review · 4 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Any two toric moment maps on a closed symplectic four-manifold are connected by a nodal tangle, resolving Symington's conjecture and yielding applications to displacement energy and Lagrangian knots.

desk verdict The nodal tangle framework is new and promising, but Proposition 3.17 contains an explicit local check that fails, leaving Theorem A unproven as written. read the letter →

arxiv 2506.23754 v4 pith:FBBP7CHE submitted 2025-06-30 math.SG

Joel Schmitz This is my paper · ORCID
classification math.SG MSC 53D3553D1253D20
keywords almosttoricfibrationsnodaltanglesSymingtonconjecturemomentmapsdisplacementenergyLagrangiantorusknotsPoincarérecurrenceintegralaffinesurfaces
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Nodal tangles are one-parameter families of nodal slides in the base of an almost toric fibration. The paper's central result is that any two toric moment maps on the same closed symplectic four-manifold are connected by such a tangle, so the corresponding toric fibrations are homotopic through almost toric fibrations; this proves Symington's conjecture for closed toric four-manifolds. The same machinery gives a canonical form for Delzant polygons, which is used to construct Hamiltonian diffeomorphisms violating Lagrangian Poincaré recurrence in every compact non-monotone toric four-manifold, to compute the displacement energy of most toric fibres, and to give an elementary recipe for infinite families of Lagrangian torus knots.

What carries the argument

Central is the nodal integral affine surface: a surface with isolated 'nodes' whose complement carries an integral affine structure with monodromy, together with charts whose cuts are directed weighted graphs. A nodal tangle is a one-parameter family of such surfaces in which nodes slide along their eigenlines and can split, keeping the affine structure away from the sliding locus fixed; its transition map is a piecewise integral affine map built from half-shears. The proof of the main theorem uses a canonical form (Proposition 3.17) obtained by filling the caustic of a weakly Delzant polygon with cuts and parking nodes, so that the height function becomes integral affine outside an $\varepsilon$-hat; the possible hats are classified by type, and the canonical type is shown to be a symplectic invariant. Lifting a tangle gives a path of fibrations, and the transition map governs how Lagrangian invariant germs change along the tangle.

What would settle it

An independent computation of the displacement energy of a toric fibre $\mu^{-1}(x)$ with $x \in \Delta \setminus \mathcal{K}_\Delta$ and $\mathcal{F}_\Delta(x) < \tfrac12 \sup \mathcal{F}_\Delta$ that yields a value different from $\mathcal{F}_\Delta(x)$ would refute Theorem C; alternatively, a weakly Delzant polygon $\Delta$ for which some trimmed domain $\Delta_t$ or some vertex or edge of its caustic fails to have the local affine normal forms quoted from [MS23] would break the canonical form and with it Theorem A.

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Extended reading notes

Core claim

The paper introduces nodal tangles as deformations of nodal integral affine surfaces and proves three translation theorems: a nodal surface determines an almost toric fibration, a nodal tangle lifts to a path of almost toric fibrations, and invariant germs transform by the tangle's transition map. The main theorem (Theorem A, Corollary 3.26) states that for two toric moment maps $\mu_i \colon X \to \Delta_i$ on a closed four-dimensional symplectic manifold, the bases $\Delta_0$ and $\Delta_1$ are connected by a nodal tangle; consequently there is a continuous path of almost toric fibrations $\pi_t \colon X \to B_t$ from one base to the other. The proof deforms each $\Delta_i$ into a canonical form depending only on $X$, classified by hat class, maximum height, and heights of parked nodes. Applications include Theorem B (non-recurrence on a set of almost full measure in every compact non-monotone toric four-manifold), Theorem C ($e(\mu^{-1}(x)) = \mathcal{F}_\Delta(x)$ outside the caustic when $\mathcal{F}_\Delta(x) < \frac12 \sup \mathcal{F}_\Delta$), and Theorem D (alternatingly sliding two nodes through a point yields infinitely many pairwise non-symplectomorphic Lagrangian tori when $k_{\mathfrak{a}}k_{\mathfrak{b}} \det(v_{\mathfrak{a}}, v_{\mathfrak{b}})^2 \ge 4$ and an affine invariant germ exists).

Load-bearing premise

The load-bearing premise is that the distance-to-the-boundary function of each polygonal domain in the admissible class has exactly the local shapes asserted by the tropical-geometry input, near every edge and vertex of its caustic; if those local shapes admit exceptions, the nodal slides cannot be arranged to flatten the height function, and the proofs of Theorems A, B, and C collapse.

Editorial extensions

If this is right

  • On any closed toric symplectic four-manifold, the space of toric fibrations is connected through almost toric fibrations, since any two moment polytopes have the same canonical form and the tangle between them lifts to a path.
  • In every compact non-monotone toric four-manifold, Lagrangian Poincaré recurrence fails on a set of fibres of almost full measure, with the rotation amounts governed by irrational ratios of a length function to height.
  • For toric fibres outside the caustic with height below half the maximum, the displacement energy is exactly the height to the boundary, giving a computable invariant for distinguishing Lagrangian tori.
  • Alternatingly sliding two nodes with $k_{\mathfrak a}k_{\mathfrak b}\det(v_{\mathfrak a},v_{\mathfrak b})^2 \ge 4$ through a common point produces infinitely many pairwise non-symplectomorphic Lagrangian tori whenever a non-constant affine invariant germ is available.
  • An elementary Farey-tree sliding procedure realizes Lagrangian pinwheels of every coprime type in monotone $\mathbb{C}P^2 \# n \overline{\mathbb{C}P}^2$ for $5 \le n \le 8$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the canonical-form strategy extends from Delzant polygons to all almost toric bases of a fixed rational surface (the paper's Question 5.4), the connectivity proved here would become a special case of a much broader statement: the entire base space of a rational symplectic four-manifold would be connected by nodal tangles.
  • The entangling-node transition maps in Section 4.2 are governed by the same recurrence as rank-2 cluster algebras; a natural testable extension is that the infinite families of Theorem D persist for any affine invariant germ, not only displacement energy, and that the accumulation points of tangling points (Question 5.7) carry infinitely many almost-Hamiltonian-isotopic, non-symplectomorphic tori.
  • The author's Remark 3.33 suggests the half-maximum condition in Theorem C is an artifact of the proof; a concrete programme is to continue the probe through the $\varepsilon$-hat to establish $e(\mu^{-1}(x)) = \mathcal{F}_\Delta(x)$ for all fibres outside the caustic.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 3 minor

Summary. The paper introduces nodal tangles, one-parameter families of nodal integral affine surfaces, and proves translation theorems that lift such tangles to paths of almost toric fibrations and track symplectic invariant germs. The main results are: Theorem A, resolving Symington's conjecture for closed toric four-manifolds by showing that any two toric moment polygons are connected by a nodal tangle; Theorem B, constructing Lagrangian Poincaré non-recurrence in compact non-monotone toric four-manifolds; Theorem C, computing displacement energies of toric fibres outside a one-dimensional caustic; and Theorem D, producing infinite families of almost Hamiltonian isotopic Lagrangian torus knots. The paper also gives an elementary construction of Lagrangian pinwheels in del Pezzo surfaces.

Significance. If the results are correct, this is a substantial contribution. Theorem A resolves a twenty-year-old conjecture, the displacement-energy formula is explicit and parameter-free, and the Lagrangian-knot construction gives a systematic combinatorial recipe with concrete examples. The framework of nodal tangles is well motivated, and the translation theorems are clearly useful. The paper is organized and includes worked examples. However, the central canonical-form construction in Proposition 3.17 contains an incorrect local verification, and the bijection in Theorem 3.25 is asserted rather than proved; since Theorem A and its applications depend on these points, the main results are not yet fully supported.

major comments (4)
  1. [§3.1, Proposition 3.17, Eq. (2)] The local verification in the proof of Proposition 3.17 is incorrect as written. In the case n=0, Theorem 3.13 gives F_Δ(a)=min{x,x+y,y} after translation, and the proof sets R_0=h_{(1,1)}∘h_{(1,0)} with h_v as in Eq. (1). Direct computation contradicts the displayed identity F_Δ(R_0^{-1}a)=F_1(x)+⟨(1,0),a⟩. For a=(1,0), R_0^{-1}(1,0)=(1,0) and F_Δ(1,0)=0, whereas the right-hand side is 1; for a=(0,1), R_0^{-1}(0,1)=(3,2) and F_Δ(3,2)=2, whereas the right-hand side is 0. The same failure occurs for the opposite half-shear convention. Since Proposition 3.17 is the basis for the canonical form and hence for Theorems A, B, and C, a correct rectifying map or an additional half-shear must be supplied and the local identity checked.
  2. [§3.2, Theorem 3.25] The proof of the bijection α:𝒞→ℛ ends with “It is easily checked that β∘α=id and α∘β=id” without carrying out the check. This is load-bearing: the uniqueness of the canonical type, and therefore Theorem A, depends on the inverse formulas in the second table. At minimum the verification should be written out, especially for hats of types B–E, where the formulas involve case distinctions and the ordering of the α_i.
  3. [§3.3, proof of Theorem B] The formula for the length function g(h) is stated without derivation. The construction of the Hamiltonian diffeomorphism requires that for a point x at height h the orbit under ψ is a translation of length 2(M-h) on a level set whose total length is g(h); the irrationality condition g(h)/(2(M-h))∉ℚ is what produces non-recurrence. The reader cannot verify this key step without a proof of the stated length formula, including the dependence on the hat class and the contributions min{h-α_i,0} of parked nodes.
  4. [§3.1, Proposition 3.17, proof] The inductive “filling” argument for parking nodes is described only informally (“we see inductively”). In particular, the claim that when a node reaches a vertex where a previous node slid off, Theorem 3.13 forces its multiplicity to be 1 and keeps it on the caustic, should be stated as a precise induction using the local normal forms. This matters because the cut graph of φ_1 is used to define the canonical form and to justify that ℱ_1 is integral affine away from the ε-hat.
minor comments (3)
  1. [§2.3, Theorem 2.26] The final sentence of the uniqueness statement has broken punctuation: “a fibred symplectomorphism, .” should be cleaned up, and the diagrammatic sentence beginning “commutes, that is” should be rephrased.
  2. [§3.1, before Eq. (2)] The notation h_{n+1(1,0)} is ambiguous; it should be typeset as h_{(1,0)}^{n+1} or otherwise explicitly defined, since the subscript currently looks like a vector with a coefficient.
  3. [§2.4, proof of Lemma 2.33] In the proof, “Let a,b∈I and x∈B∖π_B(𝔑)” should be “let x_a∈B_a∖π_B(𝔑)” to match the notation used in the statement and in Corollary 2.34.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central proofs are self-contained given external tropical-geometry and symplectic inputs, and the only self-citations are contextual.

full rationale

The derivation is not circular. The canonical-form construction (Prop. 3.17) and the canonical-type classification (Thm. 3.25) rest on the external tropical-geometry theorems of Mikhalkin-Shkolnikov, restated as Thms. 3.10 and 3.13; these are independent inputs rather than outputs of this paper. The translation theorems 2.26, 2.29, and Cor. 2.34 are explicitly reformulations of [Sym03], [Eva23], and [BHS24]; the last shares an author, but it is used only for standard versal-deformation and flux formalism, not to assert any of the paper's conclusions. The displacement-energy equality in Cor. 3.32 combines an external lower bound [Bre23, Prop. 3.2] with an upper bound from an explicitly constructed probe in Thm. 3.29; the probe length is read off from the base geometry, not chosen to force the value, so the equality is not a fit in disguise. Theorem B's Hamiltonian diffeomorphism is built from an explicit nodal tangle and an integral-affine identification, and the only self-citation [Sch24] is contextual: the proof gives a different construction. The skeptic's objection to Eq. (2) in Prop. 3.17 is a local-computation or correctness concern, not a circularity, since a wrong rectifying map would make the proof incomplete rather than making the theorem equivalent to its inputs. No free parameter is fitted to a target result, and no load-bearing claim is justified solely by the author's own prior work. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central results rest on a stack of prior theorems from symplectic and tropical geometry; the genuinely new combinatorial machinery (nodal tangles, canonical form) is built on these. No free parameters are fitted; all numerical data are derived from the moment polytope.

assumptions (7)
  • standard math Existence and essential uniqueness of almost toric fibrations over nodal integral affine surfaces (Theorem 2.26).
    Imported from [Sym03, Theorem 5.2] and [Eva23, Proof of Theorem 8.5]; used to give a fibration over each base and to identify symplectic manifolds via the same base.
  • standard math Every nodal tangle lifts to a one-parameter family of almost toric fibrations (Theorem 2.29).
    Consequence of [Eva23, Proof of Theorem 8.10] and [Sym03, Theorem 6.5]; this is the bridge from base combinatorics to symplectic paths.
  • standard math For weakly Delzant polygonal domains, the trimmed domains Δ_t are weakly Delzant and the height function F_Δ has the local affine normal forms in Theorem 3.13.
    Restates [MS23, Theorems 26 and 46]; underpins the nodal slides in Proposition 3.17 that produce the canonical form.
  • standard math Symplectic forms on rational surfaces are classified by symplectic reduced vectors (Karshon-Kessler).
    Used in Theorem 3.25 to map canonical types to toric reduced vectors and to reconstruct the underlying symplectic manifold.
  • standard math Lower bound e(μ^{-1}(x)) ≥ F_Δ(x) for toric fibres (Lemma 3.31).
    From [Bre23, Proposition 3.4]; combined with the probe upper bound to get the exact displacement energy in Corollary 3.32.
  • standard math The symplectic mapping class group of a rational surface is finite.
    From [LLW22]; used in the proof of Theorem B to turn the symplectomorphism ψ into a Hamiltonian diffeomorphism ψ^m.
  • standard math Local model and monodromy of focus-focus singularities in almost toric fibrations.
    Background from [Zun97] and [Sym03, Section 4.4]; used to define nodal charts, monodromy pairs, and eigenlines.

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Pith. "Pith review of Nodal Tangles." pith.science (2026). https://pith.science/paper/FBBP7CHE

@misc{pith2026250623754,
  author       = {Pith},
  title        = {Pith review of: Nodal Tangles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FBBP7CHE}},
  note         = {Machine review of arXiv:2506.23754}
}
read the original abstract

We study piecewise linear knot diagrams in the base of almost toric fibrations of symplectic four-manifolds. These diagrams translate to deformations of the almost toric fibration. We give several applications to symplectic topology, among them a proof of a conjecture by Symington, simpler counterexamples to Lagrangian Poincar\'e recurrence in dimension four, the calculation of the displacement energy for many fibres of toric moment maps, and an elementary recipe for building and distinguishing Lagrangian torus knots.

Figures

Figures reproduced from arXiv: 2506.23754 by the authors.

Figure 1
Figure 1. The behaviour of 𝜓 on the base space for 𝑆 2 × 𝑆2 . Nodes are represented by arrowheads, the emerging lines are branch cuts. Level sets of ℱ are in grey. The shaded region in the middle is ℱ −1((𝑀 − 𝜀, 𝑀 ]). 1.2 Lagrangian Poincaré non-recurrence In [Sch24] we constructed a counterexample to Lagrangian Poincaré recurrence in dimension four using almost toric fibrations. That is, we constructed a Hamiltonian diffeomo… view at source ↗
Figure 2
Figure 2. The green area marks the fibres displaceable via Theorem C, the red curve marks the caustic, the excluded one-dimensional subset. where 𝜆𝑖 ∈ ℤ2 are primitive vectors. Define ℱΔ ∶ Δ → ℝ≥0 , ℱΔ(𝑥) ≔ min{⟨𝜆𝑖 , 𝑥⟩ + 𝑐𝑖 } , where we take the minimum over all the half-planes defining an edge of Δ. Hence 𝜕Δ = ℱ −1 Δ (0) and ℱΔ gives a notion of distance to the boundary of Δ. 2 The canonical form constructed in Section 3.1 … view at source ↗
Figure 3
Figure 3. A topological picture of a nodal tangle used to produce Lagrangian knots. Here we slide two nodes 𝔞, 𝔟 alternatingly over 𝑥0 . 1.4 Lagrangian knots The study of Lagrangian knots in a symplectic manifold (𝑋 , 𝜔) involves classifying embedded Lagrangians in 𝑋 up to either Lagrangian isotopy, symplectomorphism or Hamiltonian diffeomorphism, see [PS24] for a survey. In this paper we are concerned with the classification… view at source ↗
Figures from the paper (30 more)
Figure 4
Figure 4. Figure 4: Two nodes 𝔞, 𝔟, marked by arrowheads, whose eigenlines pass through 𝑥0 . The eigenlines are parallel to the primitive integer vectors 𝑣𝔞 , 𝑣𝔟 (the direction of the arrow tips). The level sets of the invariant ℐ are in grey. Remark 1.5. The assumption that [ℐ ]𝑥0 is aff…
Figure 5
Figure 5. Figure 5: The rectifying map in the case of Remark 2.8 [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: Four different nodal charts of ℝ≥0 × ℝ. Each adjacent pair is related by a half-shear. On the right is the integral affine chart idℝ≥0×ℝ. This makes 𝐺 into a directed weighted graph, with direction on 𝓁 given by 𝑣 and weight given by 𝑘. The pair (im 𝜑, 𝐺) lets one reco…
Figure 7
Figure 7. Figure 7: A nodal chart diagram of a neighbourhood of a node 𝔞 of multiplicity 𝑘. 11 [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]
Figure 8
Figure 8. Figure 8: A nodal tangle. Horizontal slices are nodal chart diagrams. On the right are the top and bottom nodal chart diagrams. In the image on the left, the nodes 𝔑 are given by the thick coloured curves. 2.2 Nodal tangles A nodal slide as in [Sym03, §6.1] can be described as t…
Figure 9
Figure 9. Figure 9: Building a tangle. Moving on, we can slide 𝔟 along its eigendirection, past the point 𝑥. As our nodal chart 𝜑1 has a cut through 𝑥, the straight line segment in 𝐵 along which we slide 𝔟 appears broken in im 𝜑1 at the point 𝑥. Using the recipe in Remark 2.19, we find th…
Figure 10
Figure 10. Figure 10: A more complicated tangle of nodes. The small hooks at 𝑥 and 𝑦 illustrate the node reversing its sliding direction, not an actual lateral movement. Let 𝐵 × [0, 1] be a nodal tangle, and 𝜑0 ∶ 𝐵0 ⊂ 𝑈0 → ℝ2 a simple nodal chart. Perturb 𝔑 (disregarding the integral affin…
Figure 11
Figure 11. Figure 11: Modifying an invariant germ by a simple nodal tangle. We start with a nodal chart 𝜑0 which is affine around 𝑥0 for 𝐵0 , then we modify 𝐵0 by a nodal tangle to obtain 𝐵1 and with 𝑥1 = 𝜏 1 0 𝑥0 , first giving us a nodal chart 𝜑1 = 𝜑0 ∘ 𝜏 0 1 . We then compose 𝜑1 with th…
Figure 12
Figure 12. Figure 12: Level sets of ℱΔ in dark grey and the caustic 𝒦Δ in red. Let Δ ⊂ ℝ2 be a weakly Delzant polygonal domain, and write Δ = ⋂𝐻𝜆,𝑐∈ℋ 𝐻𝜆,𝑐, where ℋ is a minimal set of half-planes. Let ℱΔ ∶ ℝ2 → ℝ ∪ {−∞} 𝑝 ↦ inf 𝐻𝜆,𝑐∈ℋ {⟨𝜆, 𝑝⟩ + 𝑐} , be the height function of Δ. We can writ…
Figure 13
Figure 13. Figure 13: Level sets of ℱΔ near a vertex of 𝒦Δ outside of Δ𝑀 . 𝓁 𝐽 𝜆1 − 𝐽 𝜆2 𝜆1 𝜆2 [PITH_FULL_IMAGE:figures/full_fig_p024_13.png]
Figure 14
Figure 14. Figure 14: Smoothing an 𝐴𝑛 corner. Theorem 3.13 ([MS23, Theorems 26, 46]). Let Δ be weakly Delzant and 𝑀 = sup ℱΔ. If 𝓁 is an edge of 𝒦Δ in Δ ∖ Δ𝑀 of weight 𝑛 + 1, 𝑥 is in the interior of 𝓁 and 𝑎 ∈ ℝ2 small enough, then, after an integral affine transformation, we can write ℱΔ(𝑥…
Figure 15
Figure 15. Figure 15: a): smoothing a weakly Delzant polygon Δ. b): filling the caustic 𝒦Δ with cuts. Nodes of multiplicity 2 have a double arrow head. Nodes parked in b) and their cuts are drawn in blue. 𝑛 + 1 1 [PITH_FULL_IMAGE:figures/full_fig_p026_15.png]
Figure 16
Figure 16. Figure 16: Filling 𝒦Δ with cuts near a vertex of 𝒦Δ. 26 [PITH_FULL_IMAGE:figures/full_fig_p026_16.png]
Figure 17
Figure 17. Figure 17: Constructing an “open” nodal chart. Parked nodes are drawn in blue. 1 2 3 4 5 6 7 [PITH_FULL_IMAGE:figures/full_fig_p028_17.png]
Figure 18
Figure 18. Figure 18: Possible caustics near sup ℱΔ for Delzant polygonal domains. 𝒦Δ in red, level sets of ℱΔ in grey. 28 [PITH_FULL_IMAGE:figures/full_fig_p028_18.png]
Figure 19
Figure 19. Figure 19: All possible 𝜀-hats up to nodal tangle. change of nodal chart change of nodal chart [PITH_FULL_IMAGE:figures/full_fig_p029_19.png]
Figure 20
Figure 20. Figure 20: Nodal tangle transforming part of an 𝜀-hat of positive width. Definition 3.20. In this case we call 𝐵1 as constructed in Proposition 3.17 the canonical form of Δ. The 𝜀-hat of 𝐵1 is given by the nodal integral affine surface 𝐻𝜀 = {𝑥 ∈ 𝐵1 | ℱ1 (𝑥) > 𝑀 − 𝜀} . The hat wi…
Figure 21
Figure 21. Figure 21: Constructing visible tropical spheres for hats of type E (top) and B (bottom). From left to right: in a toric model we first mark which spheres we would like to blow down in green. Then we perform nodal trades to obtain the middle diagram where the marked spheres are …
Figure 22
Figure 22. Figure 22: Two nodal charts for the same canonical type. may choose the nodes in the hat 𝐻 to have at least height 𝑀 − 𝜀. The level sets ℱ −1(ℎ) with ℎ ∈ [0, 𝑀 − 𝜀) are closed straight lines in 𝐵0 . We have 𝑀 = max ℱ , and we call Δ𝑀 ≔ ℱ −1(𝑀 ) the ridge of 𝐵0 . The integral aff…
Figure 23
Figure 23. Figure 23: Nodal tangle transforming part of an 𝜀-hat of positive width. On the left the nodal tangle for an end of Δ𝑀 with two incoming nodes, on the right the one for an end with three incoming nodes. The top row shows the nodal tangle in a nodal chart as in [PITH_FULL_IMAGE:…
Figure 24
Figure 24. Figure 24: Modifying invariant germs by sliding over one node. Corollary 3.32 with 𝑥 not being in the caustic. Using Lemma 2.36 we may identify the invariant germ [ℐ ]𝜋 −1 0 (𝑥0 ) with [ℐ ]𝑥0 , and similarly for 𝑥1 . Using Corollary 2.34 we can write [ℐ ]𝑥1 |𝑈 ∖𝓁 = [ℐ ]𝑥0 |𝑈 ∖𝓁 …
Figure 25
Figure 25. Figure 25: Entangling two nodes let (𝐵𝑛+2, 𝔑𝑛+2) be the nodal integral affine surface obtained from (𝐵𝑛+1, 𝔑𝑛+1) by sliding 𝔟 through 𝑥 once. We obtain a nodal tangle (𝐵 × ℝ≥0, 𝔑). See [PITH_FULL_IMAGE:figures/full_fig_p040_25.png]
Figure 26
Figure 26. Figure 26: The rectifying map 𝑅. On the left the domain of 𝑅 and on the right its image. The domain is divided into four closed cones 1 , 2 , 3 , 4 on which 𝑅 acts linearly by the matrices − id, −𝐴, −𝐵𝐴, −𝐵 respectively. The rays dividing these regions form the cut graph of 𝑅 (a…
Figure 27
Figure 27. Figure 27: Nodal tangle diagram for the case 𝑎𝑏 = 1 at 𝑡 = 5. Here the rectifying map at 𝑥 is integral linear. 𝑡 = 2𝑛 𝑣2𝑛−1 𝑣2𝑛 … −𝑣1 𝑣2𝑛 𝑣0 𝑣2𝑛−1 id (𝐵𝐴) 1−𝑛𝐴(𝐵𝐴) 𝑛−1 𝐵𝐴 𝐵 [PITH_FULL_IMAGE:figures/full_fig_p042_27.png]
Figure 28
Figure 28. Figure 28: The rectifying map 𝑅2𝑛 have 𝑣𝑛 = { 𝑣0 + 𝑛(𝑣1 − 𝑣0 ) if 𝑎 = 2 1 √𝑎 2−4 ((𝜇𝑛 − 𝜇−𝑛)𝑣1 − (𝜇𝑛−1 − 𝜇1−𝑛)𝑣0) if 𝑎 > 2 . To prove Proposition 4.1 we use the following definition and lemma. Definition 4.4. A vector 𝑢 ≠ 0 is an eigenvector of a piecewise linear map 𝑇 if 𝑇 𝑢 = …
Figure 29
Figure 29. Figure 29: Some Lagrangian knots in 𝑆 2 × 𝑆2 ; we only picture the interior of the moment polytope. Grey dots correspond to points in 𝒟′ . Proof. Condition 1. puts us in the case 𝑎𝑏 ≥ 4 in Proposition 4.1. Let 𝜑0 be an integral affine chart at 𝑥0 , and let 𝜑𝑛 ≔ 𝜑0 ∘ 𝜏 0 𝑛 and 𝑥𝑡…
Figure 30
Figure 30. Figure 30: Constructing Vianna tori via nodal tangles where 𝑝̂3 = 3𝑝1𝑝2−𝑝3 is the Markov number given by mutation at 𝑝3 . This equation follows from the fact that 1 𝑝1 (𝑝1 , 𝑞1 ), 1 𝑝2 (𝑝2 , 𝑞2 ), 1 𝑝̂3 (−𝑝3 , −𝑞3 ) form the corners of a Vianna triangle, see [Eva23, Appendix I] …
Figure 31
Figure 31. Figure 31: The first four rows of the Farey tree. 𝑥 𝔞1 𝔞2 𝔞3 a) b) c) d) [PITH_FULL_IMAGE:figures/full_fig_p048_31.png]
Figure 32
Figure 32. Figure 32: Stepping through the Farey tree with nodal slides. The double arrow heads mark that the nodes are of multiplicity two. Definition 4.10. A Farey pair (𝑣0 , 𝑣1 ) is a positive basis of ℤ 2 , i.e. det(𝑣0 , 𝑣1 ) = 1. Its children are the Farey pairs (𝑣0 , 𝑣0 + 𝑣1 ) and (𝑣…
Figure 33
Figure 33. Figure 33: Nodal chart diagrams for monotone ℂ𝑃2 #𝑛ℂ𝑃2 with 𝑛 ≥ 5 showing the desired nodal configuration. This configuration of nodes can be found in the monotone ℂ𝑃2 #𝑛ℂ𝑃2 with 𝑛 ≥ 5, as illustrated in [PITH_FULL_IMAGE:figures/full_fig_p049_33.png]

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  1. More counterexamples to Lagrangian Poincar\'e recurrence in dimension four

    math.SG 2025-07 conditional novelty 5.0 of 10

    Every non-monotone toric symplectic four-manifold admits a Hamiltonian diffeomorphism and a Lagrangian torus that never intersects its own image under iteration.

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