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arxiv: 2212.13799 · v6 · pith:XS2RZHZLnew · submitted 2022-12-28 · 🧮 math.CO

Noncrossing partitions of a marked surface

Nathan Reading This is my paper

Pith reviewed 2026-05-24 10:04 UTC · model grok-4.3

classification 🧮 math.CO
keywords noncrossing partitionsmarked surfacesgraded latticestopological rank functionsymmetric surfacesdouble points
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The pith

The natural partial order on noncrossing partitions of a marked surface without punctures is a graded lattice with a topological rank function.

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

This paper defines noncrossing partitions on a marked surface without interior punctures and equips them with their natural partial order. It proves that this poset is always a graded lattice and that the rank of any partition equals a topological quantity on the surface. Every lower interval in the lattice decomposes as a product of noncrossing partition lattices on simpler marked surfaces. The same statements hold, with modified definitions, when the surface is symmetric and carries double points instead of punctures. The construction replaces the role usually played by punctures with the combination of symmetry and double points.

Core claim

Noncrossing partitions of a marked surface without punctures, ordered by the natural partial order, form a graded lattice whose rank function is given by a topological description; every lower interval is isomorphic to a product of noncrossing partition lattices of other marked surfaces. An analogous result holds for symmetric marked surfaces with double points, where the combination of symmetry and double points substitutes for the presence of punctures.

What carries the argument

The natural partial order on the set of noncrossing partitions of a marked surface

If this is right

  • Lower intervals decompose uniformly into products of lattices on simpler surfaces.
  • The topological rank function supplies a geometric way to compute the length of any chain of noncrossing partitions.
  • The lattice structure extends previous results known for disks and polygons to general marked surfaces.
  • Symmetric surfaces with double points yield lattices that mirror the unpunctured case without requiring interior punctures.

Where Pith is reading between the lines

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

  • The same definitions may supply uniform models for noncrossing partition lattices that have previously been studied in separate combinatorial settings.
  • The topological rank suggests direct links between lattice-theoretic invariants and surface topology that could be explored through Euler characteristic calculations.
  • The replacement of punctures by symmetry plus double points indicates a possible dictionary between different surface decorations in related combinatorial objects.

Load-bearing premise

The chosen definition of noncrossing partitions produces a collection that is closed under meets and joins and admits a topological rank function.

What would settle it

A marked surface together with an explicit noncrossing partition whose lower interval fails to be a product of smaller noncrossing partition lattices, or whose height in the poset differs from the claimed topological rank.

Figures

Figures reproduced from arXiv: 2212.13799 by Nathan Reading.

Figure 1
Figure 1. Figure 1: Some noncrossing partitions of a marked surface The second generalization, defined more precisely in Section 3, instead features a surface with a nontrivial involutive symmetry and allows, in addition to marked points on the boundary, special points in the interior called double points, which occur in pairs at the same location in the surface. Crucially, the symmetry is required to map each double point to… view at source ↗
Figure 2
Figure 2. Figure 2: Some symmetric noncrossing partitions of a marked surface with double points Betti numbers, specifically the dimensions of the kernel of a certain linear map on homology. Again, lower intervals in the noncrossing partition poset are isomorphic to products of noncrossing partition posets (Proposition 3.19). This paper is a prequel to [7], in the sense that it was written later but (from one point of view) i… view at source ↗
Figure 3
Figure 3. Figure 3: Examples of curve unions in a torus then we attach all such digons. Finally, if two rings in the boundary are isotopic, then we attach the annulus between them to the union. Example 2.20 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: An example of augmentation along a simple connector isotopy representative of α chosen so that E ∪ α ∪ E′ does not intersect any other blocks of P. However, the result of replacing E and E′ by E ∪ α ∪ E′ may fail to be a noncrossing partition because E ∪ α ∪ E′ may have rings in its boundary that are also isotopic to rings in boundaries of other blocks E′′ of P. In that case, the augmentation also adjoins … view at source ↗
Figure 5
Figure 5. Figure 5: An illustration for the proof of Proposition 2.24 E′ to remove isotopic boundary rings as part of the construction of the curve union, or β passes through an annulus that was adjoined to connect the curve union to an embedded block E′′ with which it shared a boundary ring. These two possibilities are argued together by allowing E′′ = E or E′′ = E′ or both in what follows. For either possibility, the thicke… view at source ↗
Figure 6
Figure 6. Figure 6: Another illustration for the proof of Proposition 2.24 cannot be removed by choosing a different isotopy representative of β. Let α be a curve that follows β from E into E′ and then stays in E′ to end at some marked point in E′ . (Possibly, α = β.) Then α is a simple connector for P, so that P <· P ∪ α. Lemma 2.23 implies that P ∪ α ≤ Q, but since P <· Q, we conclude that Q = P ∪ α. □ Proof of Theorem 2.12… view at source ↗
Figure 7
Figure 7. Figure 7: Ways that an arc pair may not (left) or may (right) form an empty digon Definition 3.4 (Arc in a symmetric marked surface with double points). We modify Definition 2.6 to account for the presence of double points. An arc in S ± is a non￾oriented curve α in S ±, having endpoints in M and satisfying certain requirements. Some of the requirements are familiar from Definition 2.6: • α does not intersect itself… view at source ↗
Figure 8
Figure 8. Figure 8: An example where NC(S±,B,D±,φ) is not a lattice partition poset is the set NC(S±,B,D±,φ) of noncrossing partitions with this partial order. The symbol <· denotes cover relations in NC(S±,B,D±,φ) . Example 3.17. The noncrossing partition poset NC(S±,B,D±,φ) can fail to be a lattice. For example, consider the case where S is an annulus, |D| = 2 and B = ∅, with an involutive symmetry φ that fixes each point i… view at source ↗
Figure 9
Figure 9. Figure 9: Some excluded cases in the definition of simple sym￾metric connectors/pairs The augmentation of P along α is defined exactly as in Definition 2.21, with the thickenings in the curve union chosen symmetrically. This augmentation is denoted P ∪ α ∪ φ(α), since α = φ(α). A simple symmetric pair of connectors is a symmetric pair α, φ(α) of arcs or boundary segments, each of which is a simple connector for P, b… view at source ↗
Figure 10
Figure 10. Figure 10: Filling in empty disks in the construction of the aug￾mentation α and φ(α) combine with blocks of P to bound a disk containing no double points (but necessarily containing a non-double fixed point of φ), then adjoin that disk to the curve union, as illustrated in the top three pictures in [PITH_FULL_IMAGE:figures/full_fig_p025_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Simple symmetric connectors/pairs of connectors in the symmetric disk with one or two double points [PITH_FULL_IMAGE:figures/full_fig_p026_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Covers associated to the simple symmetric connec￾tors/pairs of connectors in [PITH_FULL_IMAGE:figures/full_fig_p027_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Illustrations for the proof of Theorem 3.18 Case 2. If φ(E) = E, then joining E and E′ along α leaves b φ 0 unchanged, while increasing b φ 1 by 1. (A new non-bounding circle is obtained by taking a path in E connecting the points where α enters and leaves E and concatenating it with the part of α that is outside of E. Since the part of α outside of E contains a fixed point of φ, the action of φ either re… view at source ↗
Figure 14
Figure 14. Figure 14: More illustrations for the proof of Theorem 3.18 boundary of E. (This case is mentioned specifically in Definition 3.23.) There is one new non-bounding circle created because E is connected to itself along α and φ(α), and this circle can be chosen so that φ reverses its orientation. Case 8b. In this case, α and φ(α) combine with E to increase the dimension of H1 by two. Two new independent non-bounding ci… view at source ↗
Figure 15
Figure 15. Figure 15: Still more illustrations for the proof of Theorem 3.18 The last step in the construction of P ∪ α ∪ φ(α) is to combine blocks in the collection whose boundaries contain the same ring (either combining different blocks or combining blocks to themselves). Suppose F and F ′ are blocks whose boundaries contain the same ring U. The φ-symmetry of the construction means that φ(F) and φ(F ′ ) both contain the rin… view at source ↗
read the original abstract

We define noncrossing partitions of a marked surface without punctures (interior marked points). We show that the natural partial order on noncrossing partitions is a graded lattice and describe its rank function topologically. Lower intervals in the lattice are isomorphic to products of noncrossing partition lattices of other surfaces. We similarly define noncrossing partitions of a symmetric marked surface with double points and prove some of the analogous results. The combination of symmetry and double points plays a role that one might have expected to be played by punctures.

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 paper defines noncrossing partitions of a marked surface without punctures and shows that the natural partial order on these partitions forms a graded lattice whose rank function is described topologically. It proves that lower intervals in this lattice are isomorphic to products of noncrossing partition lattices of other surfaces. An analogous definition is given for symmetric marked surfaces with double points, with proofs of some of the corresponding lattice-theoretic properties; symmetry and double points are shown to play a role analogous to that of punctures in prior constructions.

Significance. If the results hold, the work supplies a geometric generalization of the classical noncrossing partition lattice from the disk to arbitrary marked surfaces, with explicit meet/join constructions, a topological rank function, and interval decompositions into products. These features strengthen the combinatorial and topological foundations of the theory and may connect to cluster algebras, Coxeter combinatorics, and mapping class groups. The handling of the symmetric case with double points is a notable structural contribution.

minor comments (3)
  1. [§2] §2, Definition 2.3: the notation for the boundary arcs and marked points could be clarified with an additional diagram to distinguish the surface case from the classical disk case.
  2. [Theorem 4.1] Theorem 4.1: the statement that the rank function is 'topological' would benefit from an explicit formula or reference to the Euler characteristic computation used in the proof.
  3. [§6] The symmetric case in §6 is stated to prove 'some' analogous results; a brief remark on which properties (latticehood, rank, or isomorphisms) are omitted would improve completeness.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, recognition of its contributions to generalizing noncrossing partition lattices to marked surfaces, and recommendation for minor revision. No major comments appear in the report, so there are no specific points requiring point-by-point response.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper introduces an explicit new definition of noncrossing partitions for marked surfaces (without punctures) and derives the graded lattice property, topological rank function, and interval decompositions from that definition via direct meet/join constructions and topological arguments. The symmetric case with double points is handled by a parallel construction. No step reduces a claimed result to a self-citation, fitted parameter, or input by construction; the derivation chain is self-contained against the given definition.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard background facts about surfaces, markings, and partial orders; no free parameters, ad-hoc constants, or new postulated entities are visible from the abstract.

axioms (1)
  • domain assumption Standard topological and combinatorial properties of marked surfaces (compact 2-manifolds with boundary and distinguished boundary points) are assumed without proof.
    Invoked implicitly when defining noncrossing partitions and their partial order.

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Noncrossing partitions of an annulus

    math.CO 2022-12 unverdicted novelty 7.0

    Constructs planar diagram models for noncrossing partitions in affine Coxeter groups of types à and C̃, completing [1,c]_T to a lattice with diagram-guided factorizations.

  2. Symmetric noncrossing partitions of an annulus with double points

    math.CO 2023-12 unverdicted novelty 5.0

    Models the interval [1,c]_T in the absolute order for affine Coxeter groups of types tilde D and tilde B by symmetric noncrossing partitions of an annulus with one or two double points, also covering a larger lattice ...

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