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arxiv: 2312.17331 · v4 · pith:R3OZ2HW3new · submitted 2023-12-28 · 🧮 math.CO

Symmetric noncrossing partitions of an annulus with double points

Nathan Reading This is my paper

Pith reviewed 2026-05-25 08:52 UTC · model grok-4.3

classification 🧮 math.CO
keywords noncrossing partitionsaffine Coxeter groupsabsolute orderannulusdouble pointstilde Dtilde BMcCammond-Sulway lattice
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The pith

Symmetric noncrossing partitions of an annulus with double points model the absolute order interval in affine Coxeter groups of types tilde D and tilde B.

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

The paper establishes a model for the interval from the identity to a Coxeter element in the absolute order of affine Coxeter groups in types tilde D and tilde B. Symmetric noncrossing partitions drawn on an annulus with one or two double points represent the elements and their order relations. The same diagrams capture the larger McCammond-Sulway lattice in type tilde B and almost so in type tilde D. This supplies a concrete combinatorial representation for the structure of these intervals in infinite Coxeter groups.

Core claim

For affine Coxeter groups of affine types tilde D and tilde B, the interval [1,c]_T in the absolute order is modeled by symmetric noncrossing partitions of an annulus with one or two double points. In type tilde B (and almost in type tilde D), the diagrams also model the larger lattice defined by McCammond and Sulway.

What carries the argument

symmetric noncrossing partitions of an annulus with one or two double points, which encode the elements, covering relations, and partial order of the interval [1,c]_T

Load-bearing premise

The specific diagrams of symmetric noncrossing partitions on the annulus with double points correctly encode the covering relations and partial order of the interval [1,c]_T without additional verification steps.

What would settle it

A specific pair of elements in [1,c]_T whose covering relation fails to correspond to an allowed local change between their annulus partition diagrams would show the model does not hold.

Figures

Figures reproduced from arXiv: 2312.17331 by Nathan Reading.

Figure 1
Figure 1. Figure 1: Left: The symmetric annulus with two double points. Center: Inner, outer and double points for c = s3s6s2s0s1s5s7s4. Right: The date line. doubled, we let the map ϕ also swap + and − at each double point. The left picture in [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Symmetric noncrossing partitions of an annulus with two double points that starts inside E, leaves E, intersects no other block of P, then enters E′ and stays there until it ends there. We allow E = E′ , but if so, since κ is not in curve(P), it does not have a representative that is entirely contained in E. A simple symmetric pair of connectors for P is a symmetric pair κ, ϕ(κ) of simple connectors for P,… view at source ↗
Figure 3
Figure 3. Figure 3: The interval [1, c2c3]T ∪F . Cycles (a b) in the figure should be read as ((a b))2n [PITH_FULL_IMAGE:figures/full_fig_p040_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The non-lattice [1, c2c3]T ∪L As in Lemma 3.1, source-sink moves change the partition of {±1, . . . , ±(n − 1)} into outer, inner, and double points. Every Coxeter element can be obtained from our special choice of Coxeter element by a sequence of source-sink moves. Every source-sink move is conjugation by an element of S, and this conjugation makes the same rearrangements of the cycle notation of elements… view at source ↗
Figure 5
Figure 5. Figure 5: Left: Not a symmetric pair of arcs; Right: A symmetric pair of arcs • α does not combine with a boundary segment to bound a digon in B. • Either α = ϕ(α) or α and ϕ(α) don’t intersect, except possibly at endpoints. • α and ϕ(α) do not combine to form a digon in B unless that digon contains a double point. However, we do not rule out α and ϕ(α) if, at a vertex of the digon, the two edges incident to that po… view at source ↗
Figure 6
Figure 6. Figure 6: Symmetric noncrossing partitions of an annulus with one double point Theorem 7.9. The poset NCgB c of symmetric noncrossing partitions of an annulus with n − 2 marked points on each boundary and one pair of double points is graded, with rank function given by n − 1 minus the number of symmetric pairs of distinct non-annular blocks plus the number of symmetric annular blocks. As in type D, one may go up by … view at source ↗
read the original abstract

For affine Coxeter groups of affine types $\tilde D$ and $\tilde B$, we model the interval $[1,c]_T$ in the absolute order by symmetric noncrossing partitions of an annulus with one or two double points. In type $\tilde B$ (and \emph{almost} in type $\tilde D$), the diagrams also model the larger lattice defined by McCammond and Sulway.

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 paper claims that for affine Coxeter groups of affine types tilde D and tilde B, the interval [1,c]_T in the absolute order is modeled by symmetric noncrossing partitions of an annulus with one or two double points. In type tilde B (and almost in type tilde D), the diagrams also model the larger lattice defined by McCammond and Sulway.

Significance. If the result holds, the work supplies explicit bijections, covering-relation checks, and order-isomorphism proofs between the symmetric noncrossing partitions (with one or two double points) and the intervals [1,c]_T, together with the McCammond-Sulway lattice in type tilde B. These parameter-free constructions with direct verification of the absolute order provide a concrete combinatorial model that extends noncrossing partition techniques to affine types.

minor comments (2)
  1. [Abstract] Abstract: the qualifier 'almost in type tilde D' is imprecise; state explicitly which covering relations or lattice properties fail to hold in the tilde D case.
  2. The manuscript would benefit from a short table summarizing the number of double points required for each type and each modeled poset.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the paper and for recommending minor revision. The report lists no major comments, so there are no specific points requiring point-by-point rebuttal or revision at this stage.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The manuscript supplies explicit bijections, covering-relation checks, and order-isomorphism proofs between the symmetric noncrossing partitions with one or two double points and the intervals [1,c]_T in affine types tilde B and tilde D, as well as the McCammond-Sulway lattice in type tilde B. These are direct, parameter-free constructions with verification that the diagrams respect the absolute order, rendering the modeling self-contained against external benchmarks without any reduction to self-definitions, fitted inputs, or self-citation chains.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5573 in / 1171 out tokens · 52268 ms · 2026-05-25T08:52:55.846447+00:00 · methodology

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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. Noncrossing partitions of a marked surface

    math.CO 2022-12 unverdicted novelty 7.0

    Defines noncrossing partitions of marked surfaces without punctures, proves the poset is a graded lattice with topological rank function, and shows lower intervals factor as products of smaller such lattices; analogou...

Reference graph

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13 extracted references · 13 canonical work pages · cited by 2 Pith papers · 2 internal anchors

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