pith. sign in

arxiv: 2504.12201 · v2 · submitted 2025-04-16 · 🧮 math.AT

A combinatorial genesis of the right-angled relations in Artin's classical braid groups

Omar Alvarado-Gardu\~no , Jes\'us Gonz\'alez , Matthew Kahle This is my paper

Pith reviewed 2026-05-22 20:04 UTC · model grok-4.3

classification 🧮 math.AT
keywords braid groupsright-angled Artin groupsconfiguration spacesunit squaresLusternik-Schnirelmann categorytopological complexity
0
0 comments X

The pith

A right-angled Artin group presentation for the braid group B_n arises from the configuration space of unit squares when the rectangle has width two.

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

The paper constructs an explicit right-angled Artin group presentation for the fundamental group B_n(p×q) of the space UC(n,p×q) of n non-overlapping unit squares in a p by q rectangle. This space recovers the homotopy type of the classical unlabeled point configuration space in the plane whenever the smaller side is at least n. When the smaller side equals two the new presentation coincides with Artin's classical braid group presentation after the Artin-Tits relations are deleted. The match supplies the Lusternik-Schnirelmann category of these aspherical spaces together with all their k-sequential topological complexities in both the classical and distributional settings.

Core claim

The authors describe a right-angled Artin group presentation for B_n(p×q) in the cases where UC(n,p×q) is known to be aspherical. When min{p,q}=2 this presentation agrees with Artin's classical presentation for B_n after the Artin-Tits relations are removed, which directly yields the Lusternik-Schnirelmann category of the corresponding spaces UC(n,p×q) and all their k-sequential topological complexities.

What carries the argument

The right-angled Artin group presentation for the fundamental group B_n(p×q) of the unit-square configuration space UC(n,p×q).

If this is right

  • The Lusternik-Schnirelmann category of UC(n,2×q) equals the value already known for the classical configuration space.
  • All k-sequential topological complexities of UC(n,2×q) are determined in both the Rudyak classical sense and the Dranishnikov distributional sense.
  • The (p,q)-approximations B_n(p×q) supply combinatorial models that isolate the right-angled relations inside the classical braid group presentation.

Where Pith is reading between the lines

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

  • The separation of right-angled relations from Artin-Tits relations may allow similar combinatorial presentations to be extracted from other discrete models of configuration spaces.
  • The same method could be tested on configuration spaces of other shapes or in higher dimensions to see whether right-angled Artin presentations appear more generally.

Load-bearing premise

The configuration space UC(n,p×q) is aspherical in the cases where the right-angled presentation is given.

What would settle it

A direct computation showing that the fundamental group of UC(n,2×q) is not the right-angled Artin group described in the presentation would disprove the claimed agreement with the classical braid group.

Figures

Figures reproduced from arXiv: 2504.12201 by Jes\'us Gonz\'alez, Matthew Kahle, Omar Alvarado-Gardu\~no.

Figure 1
Figure 1. Figure 1: The simple collapse for a W-pair and the corresponding step in the reduction process Here, as depicted in [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Left: The d-cell c = {e1, . . . , ed} ∪ Uc determines and is determined by the cell φ(c) = {e1, . . . , ed} ∪ Wc. Here Uc = {u1, . . . , ur−d} as in the proof, and Uc, Wc and the end points of the edges ei partition V . Right: The face of c obtained after replacing ed by u corresponds to the face of φ(c) obtained after replacing ed by w. 8 [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Grid Γ6,4 (left) and its maximal tree (right) 3.3 Farley-Sabalka gradient field Fix positive integers p and q. In this work we focus attention on the graph Γp,q given by the restriction of the integer grid Z × Z to the rectangle [1, p] × [1, q]. The left hand-side in [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: A typical step in the construction of reduced forms [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Wp,q,n-critical 2-cell in Case 1 (left) and the resulting relation (right) Case 2. Let c be a Wp,q,n-critical 2-cell coming from UConf(Γp,q, n) with deleted edge ingredients ei = [i, j] and ei ′ = [i ′ , j′ ], say with i < i′ . Assume now j ′ > j > i′ and consider the corresponding five non-negative integers r, s, t, u, v depicted on the left hand-side of [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: that add up to n − 2 and describe the distribution of blocked vertices in c. This time the resulting relation in Bn(p × q) is εi(r, s+t+ 1, u+v)· εi ′(r +s, t+u+ 1, v)· εi(r, s + t, u + v + 1)· εi ′(r + s + 1, t + u, v). (8) · · · · · · • • • • i ′ i j j ′ r t s u v i j i ′ j ′ ei ei ′ (r + s + 1, t + u, v) ′ (r + s, t + u + 1, v) ei(r, s + t, u + v + 1) ei(r, s + t + 1, u + v) [PITH_FULL_IMAGE:figures/fu… view at source ↗
Figure 7
Figure 7. Figure 7: Wp,q,n-critical 2-cell in Case 3 (left) and the resulting relation (right) Case 4. Let c = {c1, . . . , cn} be a Wp,q,n-critical 2-cell outside UConf(Γp,q, n), say with square ingredient c1 determined by the four vertices i, i+ 1, j −1, j. This time we only need the three non-negative integers r, s, t depicted in [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Wp,q,n-critical 2-cell outside UConf(Γp,q, n) in Case 4 Relations of the first type above are the only ones relevant for Theorem 1.3. Proof of Theorem 1.3. Proposition 3.3 suggests the following efficient notation for genera￾tors of B2p−2(p × 2). For an edge e = [i, j], i < j, and a vertex h different from i and j, let e[h] stand for the generator coming from the 1-cell of UX(2p − 2, p × 2) consisting of e… view at source ↗
Figure 9
Figure 9. Figure 9: Generators and relations in B2p−2(p × 2) in view of Corollary 5.3. The resulting presentation can then be simplified by using (11) to get rid of ci ′ = b1ai ′b1 for 1 < i′ ≤ p − 1. This yields the presentation with generators b1, ai and bi for 2 ≤ i ≤ p − 1, subject to the relations biai ′bib1ai ′b1 for 2 ≤ i < i′ ≤ p − 1. Then the rules ai 7→ xi for 2 ≤ i ≤ p − 1 ai ←[ xi bi 7→ y1yi for 2 ≤ i ≤ p − 1 bib1… view at source ↗
read the original abstract

The configuration space $\text{UC}(n,p\times q)$ of $n$ unlabelled non-overlapping unit squares in a $p\times q$ rectangle is known to recover the homotopy type of the classical configuration space of $n$ unlabelled points in the plane, provided $\min\{p,q\}\geq n$. Thus the fundamental group $B_n(p\times q)$ of $\text{UC}(n,p\times q)$ yields a $(p,q)$-approximation of Artin's classical braid group $B_n$. We describe a right-angled Artin group presentation for $B_n(p\times q)$ in cases where $\text{UC}(n,p\times q)$ is known to be aspherical. When $\min\{p,q\}=2$, our presentation agrees with Artin's classical presentation for $B_n$ removing the Artin-Tits relations. This allows us to deduce the value of the Lusternik-Schnirelmann category of the corresponding aspherical spaces $\text{UC}(n,p\times q)$, as well as the values of all their $k$-sequential topological complexities, both in the classical (Rudyak et al.) and distributional (Dransihnikov et al.) contexts.

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

1 major / 0 minor

Summary. The manuscript claims to derive a right-angled Artin group presentation for the fundamental group B_n(p×q) of the configuration space UC(n,p×q) of n unlabelled unit squares in a p×q rectangle, specifically in cases where UC(n,p×q) is known to be aspherical. It further asserts that when min{p,q}=2 this presentation agrees with Artin's classical presentation for B_n after removing the Artin-Tits relations, and uses the presentation to deduce the Lusternik-Schnirelmann category of UC(n,p×q) together with all its k-sequential topological complexities (in both classical and distributional senses).

Significance. If the asphericity hypotheses are secured, the work supplies an explicit combinatorial route to relations in approximations of braid groups and yields concrete values for LS category and sequential topological complexity on these spaces, extending prior results on configuration spaces and right-angled Artin groups. The claimed agreement with Artin's presentation (minus Artin-Tits relations) when min{p,q}=2 constitutes a verifiable strength that could be checked directly against the classical generators and relations.

major comments (1)
  1. [Abstract] Abstract (final paragraph): the deduction of LS category and k-sequential topological complexities is stated to follow from the right-angled Artin presentation, yet this step presupposes that UC(n,p×q) is aspherical (hence a K(π,1)) when min{p,q}=2. The only cited homotopy equivalence to the classical configuration space requires min{p,q}≥n, which fails for n>2; no separate argument, citation, or computation establishing vanishing of π_k for k≥2 is supplied in this regime, rendering the assumption load-bearing for the principal applications.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment of significance, and constructive major comment. We address the point directly below and will revise the manuscript to strengthen the presentation of the asphericity hypothesis.

read point-by-point responses

  1. Referee: [Abstract] Abstract (final paragraph): the deduction of LS category and k-sequential topological complexities is stated to follow from the right-angled Artin presentation, yet this step presupposes that UC(n,p×q) is aspherical (hence a K(π,1)) when min{p,q}=2. The only cited homotopy equivalence to the classical configuration space requires min{p,q}≥n, which fails for n>2; no separate argument, citation, or computation establishing vanishing of π_k for k≥2 is supplied in this regime, rendering the assumption load-bearing for the principal applications.

    Authors: We agree that the manuscript should make the supporting references for asphericity explicit when min{p,q}=2. The text already restricts the right-angled Artin presentation and the subsequent LS-category / sequential-complexity deductions to those cases where asphericity of UC(n,p×q) is known; the combinatorial agreement with Artin’s presentation (minus Artin-Tits relations) is independent of the homotopy-type claim. Nevertheless, the abstract and the applications section rely on this hypothesis without citing the specific sources that establish vanishing of higher homotopy groups for the width-2 regime. We will add the appropriate citations (or a short explanatory paragraph if a direct reference is unavailable) to clarify that the K(π,1) property holds independently of the min{p,q}≥n equivalence. This revision removes the load-bearing gap identified by the referee. revision: yes

Circularity Check

0 steps flagged

No significant circularity; combinatorial derivation is self-contained

full rationale

The paper derives the right-angled Artin presentation combinatorially from the UC(n,p×q) model and states the LS category and sequential complexity deductions only for cases where asphericity is already known from prior external results (explicitly conditioned on min{p,q}≥n for homotopy equivalence to classical configuration spaces). No step reduces a claimed prediction or central result to a fitted parameter, self-definition, or load-bearing self-citation chain; the min{p,q}=2 agreement is presented as an output of the combinatorial construction rather than an input, and external citations (Rudyak, Dranishnikov) supply the invariant computations without circular dependence on the present work.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on two key prior results: the homotopy equivalence of the square configuration space to the classical point configuration space when the rectangle is sufficiently wide, and the asphericity of UC(n,p×q) in the cases considered. No free parameters or invented entities are introduced.

axioms (2)
  • domain assumption UC(n,p×q) recovers the homotopy type of the classical configuration space of n unlabelled points in the plane when min{p,q}≥n.
    Stated as known in the abstract and used to identify B_n(p×q) as an approximation to B_n.
  • domain assumption UC(n,p×q) is aspherical in the cases where the right-angled Artin group presentation is described.
    Invoked to justify applying the presentation and to compute the topological invariants.

pith-pipeline@v0.9.0 · 5760 in / 1459 out tokens · 63069 ms · 2026-05-22T20:04:57.308329+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages · 1 internal anchor

  1. [1]

    ProQuest LLC, Ann Arbor, MI, 2000

    Aaron David Abrams.Configuration spaces and braid groups of graphs. ProQuest LLC, Ann Arbor, MI, 2000. Thesis (Ph.D.)–University of California, Berkeley

    work page 2000

  2. [2]

    Right-angled Artin groups, polyhedral products and the TC-generating function.Proc

    Jorge Aguilar-Guzm´ an, Jes´ us Gonz´ alez, and John Oprea. Right-angled Artin groups, polyhedral products and the TC-generating function.Proc. Roy. Soc. Edinburgh Sect. A, 152(3):649–673, 2022

    work page 2022

  3. [3]

    Homology of configuration spaces of hard squares in a rectangle.Algebr

    Hannah Alpert, Ulrich Bauer, Matthew Kahle, Robert MacPherson, and Kelly Spendlove. Homology of configuration spaces of hard squares in a rectangle.Algebr. Geom. Topol., 23(6):2593–2626, 2023

    work page 2023

  4. [4]

    Configuration spaces of disks in an infinite strip.J

    Hannah Alpert, Matthew Kahle, and Robert MacPherson. Configuration spaces of disks in an infinite strip.J. Appl. Comput. Topol., 5(3):357–390, 2021. 30

    work page 2021

  5. [5]

    Square-section braid groups and Higman-Neumann-Neumann extensions.arXiv.2510.17707

    Omar Alvarado-Gardu˜ no and Jes´ us Gonz´ alez. Square-section braid groups and Higman-Neumann-Neumann extensions.arXiv.2510.17707

    work page arXiv

  6. [6]

    Theorie der Z¨ opfe.Abh

    Emil Artin. Theorie der Z¨ opfe.Abh. Math. Sem. Univ. Hamburg, 4(1):47–72, 1925

    work page 1925

  7. [7]

    Rudyak, and Dai Tamaki

    Ibai Basabe, Jes´ us Gonz´ alez, Yuli B. Rudyak, and Dai Tamaki. Higher topological complexity and its symmetrization.Algebr. Geom. Topol., 14(4):2103–2124, 2014

    work page 2014

  8. [8]

    Ameri- can Mathematical Society, Providence, RI, 2003

    Octavian Cornea, Gregory Lupton, John Oprea, and Daniel Tanr´ e.Lusternik- Schnirelmann category, volume 103 ofMathematical Surveys and Monographs. Ameri- can Mathematical Society, Providence, RI, 2003

    work page 2003

  9. [9]

    On the complexity of parametrized motion planning algorithms.arXiv.2508.17629

    Navnath Daundkar and Ekansh Jauhari. On the complexity of parametrized motion planning algorithms.arXiv.2508.17629

    work page internal anchor Pith review arXiv

  10. [10]

    Higher topological complexity of aspherical spaces

    Michael Farber and John Oprea. Higher topological complexity of aspherical spaces. Topology Appl., 258:142–160, 2019

    work page 2019

  11. [11]

    Discrete Morse theory and graph braid groups

    Daniel Farley and Lucas Sabalka. Discrete Morse theory and graph braid groups. Algebr. Geom. Topol., 5:1075–1109, 2005

    work page 2005

  12. [12]

    Rational homotopy of the polyhedral product functor

    Yves F´ elix and Daniel Tanr´ e. Rational homotopy of the polyhedral product functor. Proc. Amer. Math. Soc., 137(3):891–898, 2009

    work page 2009

  13. [13]

    A discrete Morse theory for cell complexes

    Robin Forman. A discrete Morse theory for cell complexes. InGeometry, topology, & physics, volume IV ofConf. Proc. Lecture Notes Geom. Topology, pages 112–125. Int. Press, Cambridge, MA, 1995

    work page 1995

  14. [14]

    Higher topological com- plexity of subcomplexes of products of spheres and related polyhedral product spaces

    Jes´ us Gonz´ alez, B´ arbara Guti´ errez, and Sergey Yuzvinsky. Higher topological com- plexity of subcomplexes of products of spheres and related polyhedral product spaces. Topol. Methods Nonlinear Anal., 48(2):419–451, 2016

    work page 2016

  15. [15]

    A. Hurwitz. ¨Uber Riemann’sche Fl¨ achen mit gegebenen Verzweigungspunkten.Math. Ann, 39:1–61, 1891

    work page

  16. [16]

    On sequential versions of distributional topological complexity.Topol- ogy Appl., 363:Paper No

    Ekansh Jauhari. On sequential versions of distributional topological complexity.Topol- ogy Appl., 363:Paper No. 109271, 28, 2025

    work page 2025

  17. [17]

    Graph braid groups and right- angled Artin groups.Trans

    Jee Hyoun Kim, Ki Hyoung Ko, and Hyo Won Park. Graph braid groups and right- angled Artin groups.Trans. Amer. Math. Soc., 364(1):309–360, 2012

    work page 2012

  18. [18]

    Geometric aspects of configuration spaces of thick particles in a rect- angle.Unpublished

    Leonid Plachta. Geometric aspects of configuration spaces of thick particles in a rect- angle.Unpublished

    work page

  19. [19]

    Abrams’s stable equivalence for graph braid groups

    Paul Prue and Travis Scrimshaw. Abrams’s stable equivalence for graph braid groups. Topology Appl., 178:136–145, 2014

    work page 2014

  20. [20]

    Yuli B. Rudyak. On higher analogs of topological complexity.Topology Appl., 157(5):916–920, 2010

    work page 2010

  21. [21]

    The sequential (distributional) topological complexity of the ordered configuration space of disks in a strip.arXiv:2412.19943

    Nicholas Wawrykow. The sequential (distributional) topological complexity of the ordered configuration space of disks in a strip.arXiv:2412.19943

    work page arXiv

  22. [22]

    The topological complexity of the ordered configuration space of disks in a strip.Proc

    Nicholas Wawrykow. The topological complexity of the ordered configuration space of disks in a strip.Proc. Amer. Math. Soc. Ser. B, 11:638–652, 2024. 31

    work page 2024