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Mutation Sequences along Weaves and Amalgamation of Braid Varieties

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Yuma Mizuno

Algebraic Geometry · Submitted 5 Sep 2026 · 20 references

Pith's verdict: A solid structural result with a clear gap: the inductive sweep-mutation engine delegates the B2 and G2 checks to unperformed computations, so the main theorem is conditional on those checks holding up.

T0 · AI review

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arXiv ID2609.05833 (version 1)
Pith ID4YVBFLY4, Pith's permanent record of this paper and its review
CitationYuma Mizuno. "Mutation Sequences along Weaves and Amalgamation of Braid Varieties." arXiv:2609.05833 (2026). https://pith.science/paper/4YVBFLY4
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The paper

The paper claims

A quasi-cluster isomorphism links double Bott-Samelson strata to products of braid varieties.

The reading

This paper establishes that for positive braids p and q with Demazure products u and v, the open stratum Conf(p,q)_{u,v} of a double Bott-Samelson cell carries a cluster structure obtained by freezing a subset of mutable vertices from the cluster structure on the whole cell. Moreover, the natural splitting isomorphism Conf(p,q)_{u,v} → Conf(u,v) × X(p) × X($q^{{op}}$) is a quasi-cluster isomorphism, so the cluster variables on the three factors pull back to cluster variables on the stratum. The proof constructs a sweep mutation sequence along a double Demazure weave, using one mutation per trivalent vertex, which terminates at an amalgamation of an extension of the double-word quiver with the weave quiver. The result implies, in the half-decorated case, the conjectured quasi-cluster property of Gorsky–Kim–Scroggin–Simental splicing maps.

What carries the argument

The sweep mutation sequence along a double Demazure weave: an inductively defined sequence of quiver mutations that tracks how the quiver changes as one reduces the double word via braid moves and contractions. At each contraction, a single mutation is performed at the middle string vertex, and that vertex is relabeled as a new frozen vertex (a trivalent vertex of the weave). After processing the entire weave, the quiver becomes an amalgamation of the terminal double-word quiver Q(δ) and the weave quiver Q(W), with the trivalent vertices providing the bridge.

Extended reading notes

Core claim

Theorem 1.1 (Corollary 5.24 and Theorem 5.25) proves that for positive braids p,q with Demazure products u,v, there exists a seed Σ on Conf(p,q) and a subset S of mutable vertices such that (1) the frozen seed describes the coordinate ring of Conf(p,q)_{u,v}, (2) the freezing is a cluster localization, and (3) the splitting isomorphism Conf(p,q)_{u,v} → Conf(u,v) × X(p) × X($q^{{op}}$) is a quasi-cluster isomorphism. The seed is built from a sweep mutation sequence along a double Demazure weave, one mutation per trivalent vertex, ending at an amalgamation of an extension of the double word quiver with the weave quiver.

Load-bearing premise

The inductive step of the sweep mutation sequence (Lemma 4.25) relies on verifying the exchange-matrix identity (55) for types B2 and G2 by finite computation 'left to the reader'; if those computations contain an error, the main mutation sequence may not terminate at the claimed quiver. Additionally, the trivalent identity of [CGGLSS25, Definition5.8, Lemma5.9] is imported without reproof.

Editorial extensions

If this is right

  • The cluster structure on Conf(p,q)_{u,v} is explicitly described as a cluster localization of the known cluster structure on Conf(p,q).
  • The splitting map provides a quasi-cluster isomorphism between the stratum and the product Conf(u,v) × X(p) × X(q^{op}), showing that cluster variables on the three factors map to cluster variables on the stratum.
  • In the half-decorated setting, the splicing maps Ψ_{r,w_0} and Ψ_{N,e} of Gorsky–Kim–Scroggin–Simental are proved to be quasi-cluster isomorphisms, confirming their conjecture.
  • The sweep mutation sequence provides an explicit combinatorial recipe to compute the cluster variables on the stratum from those on the whole cell.

Reading between the lines

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

  • The sweep mutation sequence construction likely generalizes to other truncations of double Bott-Samelson cells beyond endpoint strata, providing cluster localizations for many open subvarieities.
  • The explicit mutation sequence could be used to compute cluster variables for braid varieties from known formulas on double Bott-Samelson cells, offering a new computational tool.
  • The relation to local systems on a tetrahedron suggests that sweep mutation sequences along weaves of type ΔΔ → Δ realize flips of ideal triangulations in higher Teichmüller theory for all Lie types.
Original abstract

Let $p,q$ be positive braids with Demazure products $u,v$. The endpoint stratum $\mathrm{Conf}(p,q)_{u,v}$ of the double Bott-Samelson cell splits as $\mathrm{Conf}(u,v) \times X(p) \times X(q^{\mathrm{op}})$, a double Bruhat cell times two braid varieties. We prove that the stratum's cluster structure is given by a cluster localization of the one on $\mathrm{Conf}(p,q)$, and that the splitting map is a quasi-cluster isomorphism. This comes from a mutation sequence along a double Demazure weave, one mutation per trivalent vertex, ending at an amalgamation of an extension of the double word quiver with the weave quiver. In the half-decorated case, this proves the conjecture of Gorsky-Kim-Scroggin-Simental for the splicing map $X(p) \times X(\Delta\Delta) \to X(p\Delta)$, where $\Delta$ is a reduced positive braid for $w_0$ and $\mathrm{dem}(p) = w_0$.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,559 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.19 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.63 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

Let $p,q$ be positive braids with Demazure products $u,v$. The endpoint stratum $\mathrm{Conf}(p,q)_{u,v}$ of the double Bott-Samelson cell splits as $\mathrm{Conf}(u,v) \times X(p) \times X(q^{\mathrm{op}})$, a double Bruhat cell times two braid varieties. We prove that the stratum's cluster structure is given by a cluster localization of the one on $\mathrm{Conf}(p,q)$, and that the splitting map is a quasi-cluster isomorphism. This comes from a mutation sequence along a double Demazure weave, one mutation per trivalent vertex, ending at an amalgamation of an extension of the double word quiver with the weave quiver. In the half-decorated case, this proves the conjecture of Gorsky-Kim-Scroggin-Simental for the splicing map $X(p) \times X(\Delta\Delta) \to X(p\Delta)$, where $\Delta$ is a reduced positive braid for $w_0$ and $\mathrm{dem}(p) = w_0$.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Let $p,q$ be positive braids with Demazure products $u,v$. background 1.00 0.07 0.02 unclear 0.65
s1 The endpoint stratum $\mathrm{Conf}(p,q)_{u,v}$ of the double Bott-Samelson cell splits as $\mathrm{Conf}(u,v) \times X(p) \times X(q^{\mathrm{op}})$, a double Bruhat cell times two braid varieties. result 0.91 0.44 0.17 asserted 1.00
s2 We prove that the stratum's cluster structure is given by a cluster localization of the one on $\mathrm{Conf}(p,q)$, and that the splitting map is a quasi-cluster isomorphism. result 0.90 0.95 0.57 asserted 1.00
s3 This comes from a mutation sequence along a double Demazure weave, one mutation per trivalent vertex, ending at an amalgamation of an extension of the double word quiver with the weave quiver. method 0.96 0.87 0.21 asserted 0.99
s4 In the half-decorated case, this proves the conjecture of Gorsky-Kim-Scroggin-Simental for the splicing map $X(p) \times X(\Delta\Delta) \to X(p\Delta)$, where $\Delta$ is a reduced positive braid for $w_0$ and $\mathrm{dem}(p) = w_0$. result 0.50 0.84 0.18 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
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not examined
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  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. The sweep mutation sequence along the type
Figure 2. Continuation of Figure 1
Figure 3. An A3 weave W : ∆∆ → ∆, where ∆ = 121321, drawn as the tetrahedral projection and the string diagram. initial Q(∆∆) final Qsweep(W) sweep µW ∆∆ −→ ∆
Figure 4. The sweep mutation sequence for W : 121321 121321 → 121321. 1.4 Relation with local systems on a tetrahedron The N-decomposition subdivides each ideal tetrahedron of a triangulated 3-manifold into N+1 3  octahedra labeled by (a, b, c, d) ∈ N 4 with a + b + c + d = N − 2. It gives coordinates on framed flat PGLN (C)-local systems. Each octahedron carries shape parameters, which satisfy the higher￾rank form of Thurston’s gluing equations [GGZ15; DGG16]. Also, each octahedron contributes one Ptolemy relation for the Ptolemy coordinates [GTZ15]. A flip of an ideal triangulation is a…
Figure 5. A C3 weave W : ∆∆ → ∆, where ∆ = 123123123, drawn as the tetrahedral projection. Declaration of AI use. The author used GPT 5.6 Sol in preparing this paper: to help refine the mathematical formulation, to draw the TikZ figures, and to help write the text in English. The author takes full responsibility for the contents of this paper. Acknowledgments. This project originated in discussions with Tsukasa Ishibashi on the local systems on a tetrahedron. The author also thanks him for his many comments on preliminary versions of this paper. The author was supported by the Irish Research…
Figure 6. Flag configurations for the double Bott–Samelson cell and its two half-decorated versions.
Figure 7. Elementary triangle quivers. The displayed
Figure 8. The string expression for β = 1122112 and its associated quiver. Dotted edges represent exchange-matrix entries equal to 1/2. 4.2 Demazure weaves and double weaves Following [CGGLSS25], we review the construction of a quiver associated with a Demazure weave. It defines a cluster chart on a braid variety. Below, in order to compare it with the cluster chart on a double Bott–Samelson cell, we introduce double Demazure weaves, which extend Demazure weaves. As shown in Proposition 4.14, however, the resulting quiver decomposes into its plus and minus parts. For distinct i, j ∈ I, set mij
Figure 9. Elementary weaves and shuffles. A Demazure weave can be regarded as a double Demazure weave by attaching a plus tag to every edge. Suppose that a double weave W : β → η, double words L, R, and an elementary double weave σ : b → b ′ satisfy η = L b R. Then W′ := (idL ⊗σ ⊗ idR) ◦ W : β −→ L b′ R (31) is called the one-step extension of W by σ. Below, we abbreviate double Demazure weave to double weave. To define the boundary pairing of cycles while separating the two tag components, we use the lattices with componentwise pairings X∗ (H 2 ) := X∗ (H) ⊕ X∗ (H), X∗(H 2 ) := X∗(H) ⊕…
Figure 10. An example of the plus–minus separation of a double weave appearing in Proposi
Figure 11. Extension of the quiver Q(δ) by the terminal weights of Lusztig cycles. In this figure, the weave is assumed to be simply laced and to have only plus tags. 23
Figure 12. An extension datum for Q(1122112) in type A2. Here T = {σ, τ}, aσ = (0, 1, 0, 0, 0, 0, 0), and aτ = (0, 1, 0, 1, 0, 1, 0). The two additional vertices are frozen, and ε Q στ = 0. We put a u-variable on each horizontal edge and a monomial Mℓ below each string vertex ℓ. These functions are defined in Section 5.1. Q(i +i +) i [0, 1] [1, 2] [2, ∞] k µk k 7→ σ µσ
Figure 13. The cluster transformation associated with a contraction. The top row is the case
Figure 14. The extended cluster transformation µσ : Q → Q′ associated with the type A2 braid move σ : iji → jij. The components a ∨ τ = (a1, a2, a3) and aτ ′ = (b1, b2, b3) change according to the tropical Lusztig map, and the change in the edge between τ, τ ′ is given by the local intersection term (40). 26

Figures reproduced from arXiv:2609.05833 by the authors.

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