Variants of Coxeter quandles associated with Pin groups

Yuichi Kabaya

arxiv: 2606.02023 · v2 · pith:56KGBNJ2new · submitted 2026-06-01 · 🧮 math.GT · math.GR

Variants of Coxeter quandles associated with Pin groups

Yuichi Kabaya This is my paper

Pith reviewed 2026-06-28 11:53 UTC · model grok-4.3

classification 🧮 math.GT math.GR

keywords Coxeter quandlesPin groupsAndruskiewitsch-Graña quandlerotational D_n quandleinner automorphism groupsconjugation quandlesreflection groups

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The pith

The Andruskiewitsch-Graña quandle is realized as a conjugation quandle in a Pin group, while the rotational D_n quandle arises from right-angle rotations under conjugation in the D_n Coxeter group.

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

The paper shows that the Andruskiewitsch-Graña quandle, originally defined on roots using negatives of reflections, coincides with conjugation inside a Pin group. It introduces the rotational D_n quandle as the set of certain right-angle rotations in the Coxeter group of type D_n, equipped with the conjugation operation. The inner automorphism groups of both families are computed in detail. These groups are observed to be quite similar. The constructions therefore supply explicit group-theoretic models for two variants of Coxeter quandles.

Core claim

The Andruskiewitsch-Graña quandle is realized as a conjugation quandle in a Pin group. The rotational D_n quandle is the set of some right angle rotations in the Coxeter group of type D_n with binary operation given by conjugation. Their inner automorphism groups are determined and observed to be quite similar.

What carries the argument

Conjugation inside a Pin group applied to the set of roots, and conjugation inside the D_n Coxeter group applied to the set of right-angle rotations.

If this is right

The explicit inner automorphism groups give complete descriptions of the symmetries of each quandle.
The observed similarity between the two inner automorphism groups points to parallel algebraic structure in the two families.
Both quandles can now be studied through the geometry and representation theory of Pin groups and Coxeter groups of type D_n.
The realizations embed the quandles as substructures inside well-studied reflection groups.

Where Pith is reading between the lines

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

The same conjugation technique might produce analogous quandles for other Coxeter types whose root systems admit suitable subsets closed under the operation.
The similarity of inner automorphism groups raises the question whether the two quandles are isomorphic or related by a functor that preserves symmetry.
These models could supply new ways to compute quandle cohomology or colorings by transferring calculations to the ambient linear groups.

Load-bearing premise

The chosen subsets of roots and right-angle rotations remain closed under conjugation and satisfy all three quandle axioms without extra relations forced by the ambient groups.

What would settle it

An explicit pair of elements from one of the subsets whose conjugation operation violates quandle idempotence, or an automorphism of the quandle not generated by the inner automorphisms already computed.

read the original abstract

We study two families of quandles arising from Coxeter quandles. One is the quandle defined by Andruskiewitsch-Gra\~{n}a, which is the set of roots with binary operation defined by using the negatives of reflections. We observe that this is realized as a conjugation quandle in a Pin group. The other, which we call a rotational $D_n$ quandle, is the set of some right angle rotations in the Coxeter group of type $D_n$ with binary operation given by conjugation. We determine their inner automorphism groups, and observe that they are quite similar.

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.

Desk Editor's Note private letter to a colleague

Paper gives explicit Pin-group realization of the Andruskiewitsch-Graña quandle and a new rotational D_n quandle, plus inner automorphism computations for both.

read the letter

The paper supplies two concrete quandle constructions and works out their inner automorphism groups. One realizes the Andruskiewitsch-Graña quandle as a conjugation quandle inside a Pin group; the other, called the rotational D_n quandle, takes right-angle rotations in the D_n Coxeter group and uses conjugation as the operation. Both inner automorphism groups are computed directly and noted to be similar.

These are the actual new pieces: the specific Pin-group embedding and the rotational D_n definition do not appear in the cited prior work. The computations themselves are the main output, and they look like standard direct verification once the sets are fixed.

The soft spot is that the abstract leaves the closure and quandle-axiom checks implicit. In this literature those steps are routine, but without seeing the explicit verification it is impossible to judge whether the Pin-group case introduces any extra relations or whether the similarity between the two automorphism groups is only superficial. The paper does not claim deeper structural reasons for the similarity, so the observation stays at the level of explicit calculation.

This is a specialized note aimed at readers already working with quandles coming from reflection groups or knot theory. It adds usable examples rather than a new general theory. The claims are modest and the methods are standard, so the work is coherent on its own terms.

I would send it to a referee who knows the quandle literature; the explicit constructions are worth checking even if the broader impact stays inside the subfield.

Referee Report

0 major / 3 minor

Summary. The manuscript studies two families of quandles arising from Coxeter quandles. The first realizes the Andruskiewitsch-Graña quandle (roots with operation via negatives of reflections) as a conjugation quandle inside a Pin group. The second defines the rotational D_n quandle on a subset of right-angle rotations in the type-D_n Coxeter group, again using conjugation as the operation. The inner automorphism groups of both quandles are computed explicitly, and the authors observe that these groups are quite similar.

Significance. If the realizations and computations are correct, the work supplies explicit group-theoretic models for these quandles, which may simplify the calculation of quandle invariants or the study of their automorphism groups in knot-theoretic contexts. The explicit determination of inner automorphism groups is a concrete contribution that strengthens the algebraic understanding of these structures.

minor comments (3)

The abstract states that the inner automorphism groups 'are quite similar' without indicating the precise sense of similarity (e.g., isomorphism, common subgroup structure, or order comparison). A brief clarifying sentence would strengthen the claim.
The manuscript should explicitly verify closure of the chosen subsets (roots for the first family, right-angle rotations for the second) under the defined conjugation operations and confirm that the resulting structures satisfy the three quandle axioms; these verifications are central to the constructions but are only alluded to in the abstract.
Notation for the binary operation (especially the use of negatives of reflections) should be introduced with a short displayed equation in the introduction to improve readability for readers unfamiliar with the Andruskiewitsch-Graña construction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and significance assessment of the manuscript, as well as the recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; constructions are explicit and self-contained

full rationale

The paper presents two explicit algebraic constructions: realizing the Andruskiewitsch-Graña quandle as a conjugation quandle inside a Pin group, and defining the rotational D_n quandle on right-angle rotations in the Coxeter group of type D_n with conjugation operation. It then computes their inner automorphism groups. These steps consist of direct verification of closure under the operations, satisfaction of quandle axioms, and group computations. No equations reduce a claimed prediction or result to a fitted parameter or self-referential definition by construction. No load-bearing self-citations or uniqueness theorems imported from prior author work are invoked to force the central claims. The derivation chain is independent of the target results and relies on standard verification in quandle theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, invented entities, or non-standard axioms; constructions rest on background facts from Coxeter group theory and quandle axioms.

axioms (1)

standard math Standard axioms of Coxeter groups and Pin groups as reflection and double-cover structures
The quandle operations are defined using reflections and conjugations whose properties are taken from established group theory.

pith-pipeline@v0.9.1-grok · 5618 in / 1269 out tokens · 29551 ms · 2026-06-28T11:53:44.391092+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

9 extracted references

[1]

Toshiyuki Akita, The adjoint group of a Coxeter quandle, Kyoto J. Math. 60 (2020), no. 4, 1245–-1260

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[2]

Andruskiewitsch and M

N. Andruskiewitsch and M. Gra\ n a, From racks to pointed Hopf algebras, Adv. Math. 178 (2003), no. 2, 177–-243

2003
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Clark, M

W. Clark, M. Elhamdadi, X. Hou, M. Saito and T. Yeatman, Connected quandles associated with pointed abelian groups, Pac. J. Math. 264 (1) (2013) 31–-60

2013
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Eisermann, Homological characterization of the unknot, J

M. Eisermann, Homological characterization of the unknot, J. Pure Appl. Algebra 177 (2003), no. 2, 131–-157

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James Humphreys, Reflection groups and Coxeter groups, Cambridge Stud. Adv. Math., 29
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Lawson and M

H. Lawson and M. Michelsohn, Spin geometry, Princeton Math. Ser., 38 (1989)

1989
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Vendramin, Rig, a GAP package for racks, quandles and Nichols algebras, Available at https://github.com/gap-packages/rig/

L. Vendramin, Rig, a GAP package for racks, quandles and Nichols algebras, Available at https://github.com/gap-packages/rig/
[9]

Knot Theory Ramifications 21 (2012), no

Vendramin, On the classification of quandles of low order, J. Knot Theory Ramifications 21 (2012), no. 9, 1250088, 10 pp

2012

[1] [1]

Toshiyuki Akita, The adjoint group of a Coxeter quandle, Kyoto J. Math. 60 (2020), no. 4, 1245–-1260

2020

[2] [2]

Andruskiewitsch and M

N. Andruskiewitsch and M. Gra\ n a, From racks to pointed Hopf algebras, Adv. Math. 178 (2003), no. 2, 177–-243

2003

[3] [3]

Clark, M

W. Clark, M. Elhamdadi, X. Hou, M. Saito and T. Yeatman, Connected quandles associated with pointed abelian groups, Pac. J. Math. 264 (1) (2013) 31–-60

2013

[4] [4]

Eisermann, Homological characterization of the unknot, J

M. Eisermann, Homological characterization of the unknot, J. Pure Appl. Algebra 177 (2003), no. 2, 131–-157

2003

[5] [5]

(https://www.gap-system.org)

The GAP Group, GAP -- Groups, Algorithms, and Programming, Version 4.15.1; 2025. (https://www.gap-system.org)

2025

[6] [6]

James Humphreys, Reflection groups and Coxeter groups, Cambridge Stud. Adv. Math., 29

[7] [7]

Lawson and M

H. Lawson and M. Michelsohn, Spin geometry, Princeton Math. Ser., 38 (1989)

1989

[8] [8]

Vendramin, Rig, a GAP package for racks, quandles and Nichols algebras, Available at https://github.com/gap-packages/rig/

L. Vendramin, Rig, a GAP package for racks, quandles and Nichols algebras, Available at https://github.com/gap-packages/rig/

[9] [9]

Knot Theory Ramifications 21 (2012), no

Vendramin, On the classification of quandles of low order, J. Knot Theory Ramifications 21 (2012), no. 9, 1250088, 10 pp

2012