Relativization of symmetries on quandles

Tomoki Yoshida; Yuki Imamura

arxiv: 2607.02204 · v1 · pith:2QUXFWBInew · submitted 2026-07-02 · 🧮 math.GT

Relativization of symmetries on quandles

Yuki Imamura , Tomoki Yoshida This is my paper

Pith reviewed 2026-07-03 02:50 UTC · model grok-4.3

classification 🧮 math.GT

keywords quandlesurjective homomorphisminner automorphism grouptransvection groupconnectednessquotient map2-transitivityconnected covering

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The pith

Relative inner automorphism groups characterize connected surjective quandle homomorphisms as quotient maps.

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

This paper defines relative versions of the inner automorphism group and the transvection group attached to any surjective homomorphism of quandles. It uses the relative inner automorphism group to introduce a notion of connectedness and proves that the connected homomorphisms are precisely the quotient maps. The relative transvection group then supplies a maximal connected-covering factorization that works for every surjective map. Under the further assumption that the relative inner group acts 2-transitively on each fiber, the paper classifies the quandle structures that can appear on finite fibers.

Core claim

We introduce relative versions of the inner automorphism group and the transvection group associated with surjective quandle homomorphisms. By using the relative inner automorphism group, we define a notion of connectedness for surjective homomorphisms. We characterize connected homomorphisms algebraically as quotient maps, and use the relative transvection group to establish a maximal connected-covering factorization for arbitrary surjections. Finally, we study surjective homomorphisms for which the relative inner automorphism group acts 2-transitively on each fiber. Under this assumption, we classify the possible quandle structures of the finite fibers.

What carries the argument

the relative inner automorphism group of a surjective quandle homomorphism, which defines connectedness and controls 2-transitive action on fibers

If this is right

Connected surjective homomorphisms coincide exactly with quotient maps.
Every surjective homomorphism admits a maximal factorization into a connected covering followed by a second map.
When the relative inner automorphism group acts 2-transitively on fibers, the possible quandle structures on finite fibers are restricted to a classified collection.
The factorization separates the connected part of any surjection in an algebraically canonical way.

Where Pith is reading between the lines

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

The same relative groups might be used to compare different presentations of the same quandle.
The classification of 2-transitive fibers could limit the possible target quandles when the source is fixed and small.
The connected-covering factorization may interact with existing knot-theoretic constructions that rely on quandle homomorphisms.

Load-bearing premise

The relative inner automorphism group and relative transvection group are well-defined from any surjective quandle homomorphism and suffice to support the algebraic characterizations and the 2-transitivity classification.

What would settle it

A surjective quandle homomorphism that is connected yet fails to be a quotient map, or a finite fiber whose quandle structure lies outside the list obtained under 2-transitive action of the relative inner group.

read the original abstract

This paper introduces relative versions of the inner automorphism group and the transvection group associated with surjective quandle homomorphisms.By using the relative inner automorphism group, we define a notion of \emph{connectedness} for surjective homomorphisms. We characterize connected homomorphisms algebraically as quotient maps, and use the relative transvection group to establish a maximal \emph{connected-covering} factorization for arbitrary surjections. Finally, we study surjective homomorphisms for which the relative inner automorphism group acts $2$-transitively on each fiber. Under this assumption, we classify the possible quandle structures of the finite fibers.

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

The paper extends inner automorphism and transvection groups to surjective quandle homomorphisms, using them to characterize connectedness and provide a factorization.

read the letter

The core of this paper is the introduction of relative inner automorphism groups and relative transvection groups defined from any surjective quandle homomorphism. Using the relative inner group, they define connectedness for these homomorphisms and characterize the connected ones as quotient maps. The relative transvection group then gives a maximal connected-covering factorization for arbitrary surjections.

They finish by considering the case where the relative inner group acts 2-transitively on each fiber and classify the quandle structures on the finite fibers under that hypothesis.

This is new in the sense that it relativizes the standard symmetry groups to the homomorphism setting, which allows algebraic treatment of connectedness and factorizations. The approach seems consistent with how these groups are used in the absolute case for quandles.

What the paper does well is laying out a clean program: definitions, characterization, factorization, then a classification under an extra assumption. The abstract presents the results without obvious gaps in the logic.

Soft spots are minor. The work is quite specialized, so its audience is limited to quandle theorists. The classification of fibers under 2-transitivity might turn out to be straightforward once the definitions are in place, but that depends on the details of the proofs, which aren't in the abstract. No indication of circularity or ill-defined objects from what's given.

This paper is for people already working with quandles and their homomorphisms in knot theory contexts. A reader in that area might find the factorization useful for breaking down maps. It deserves a serious referee because the claims are concrete and the methods are algebraic, so a referee can check the derivations directly.

Recommendation: yes, send it to peer review.

Referee Report

1 major / 0 minor

Summary. The paper introduces relative versions of the inner automorphism group and the transvection group associated to surjective quandle homomorphisms. It defines a notion of connectedness for such homomorphisms via the relative inner automorphism group, characterizes connected homomorphisms as quotient maps, constructs a maximal connected-covering factorization via the relative transvection group, and classifies the quandle structures on finite fibers when the relative inner automorphism group acts 2-transitively on each fiber.

Significance. If the central claims hold, the relativization of these symmetry groups would supply new algebraic tools for analyzing surjective quandle homomorphisms, including a factorization theorem and a classification result under a 2-transitivity hypothesis. Such results could be of interest within quandle theory and its applications to knot invariants.

major comments (1)

[Abstract] Abstract: the manuscript states that the relative inner automorphism and transvection groups are well-defined for arbitrary surjective homomorphisms and suffice to support the connectedness definition, the quotient-map characterization, the maximal factorization, and the 2-transitivity classification, yet supplies no explicit definitions, no equations, and no derivations or proofs for any of these claims. Without this supporting content the central assertions cannot be verified.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report and summary of the manuscript. We address the single major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: the manuscript states that the relative inner automorphism and transvection groups are well-defined for arbitrary surjective homomorphisms and suffice to support the connectedness definition, the quotient-map characterization, the maximal factorization, and the 2-transitivity classification, yet supplies no explicit definitions, no equations, and no derivations or proofs for any of these claims. Without this supporting content the central assertions cannot be verified.

Authors: The abstract is a concise, non-technical summary of the paper's main contributions and is not intended to contain explicit definitions, equations, or proofs. These elements appear in the body of the manuscript: the relative inner automorphism group and relative transvection group are defined in Section 2 (with explicit formulas and verification that they are well-defined for any surjective homomorphism), connectedness is defined via the relative inner automorphism group in Section 3, the quotient-map characterization is proved there, the maximal connected-covering factorization is constructed using the relative transvection group in Section 4, and the 2-transitivity classification of finite fibers is carried out in Section 5 with all derivations. The abstract merely outlines these results; the supporting content is supplied in the full text. revision: no

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The manuscript introduces relative inner automorphism and transvection groups as new constructions attached to any surjective quandle homomorphism, then defines connectedness from the former and derives an algebraic characterization as quotient maps together with a factorization via the latter. These steps are presented as consequences of the fresh definitions rather than reductions of a target result to its own inputs by construction. No equations, fitted parameters renamed as predictions, or load-bearing self-citations appear in the abstract or described claims; the work therefore remains self-contained within the standard framework of quandle homomorphisms.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no explicit free parameters, axioms, or invented entities are stated or derivable from the provided text.

pith-pipeline@v0.9.1-grok · 5619 in / 1013 out tokens · 36336 ms · 2026-07-03T02:50:36.212877+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

24 extracted references · 1 canonical work pages

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Hiroshi Tamaru, Two-point homogeneous quandles with prime cardinality, J. Math. Soc. Japan 65 (2013), no. 4, 1117--1134. 3127819

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Leandro Vendramin, Doubly transitive groups and cyclic quandles, J. Math. Soc. Japan 69 (2017), no. 3, 1051--1057. 3685034

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Koshiro Wada, Two-point homogeneous quandles with cardinality of prime power, Hiroshima Math. J. 45 (2015), no. 2, 165--174. 3379001

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Nicol\'as Andruskiewitsch and Mat\'ias Gra\ na, From racks to pointed H opf algebras , Adv. Math. 178 (2003), no. 2, 177--243. 1994219

2003

[2] [2]

Valeriy Bardakov, Mohamed Elhamdadi, and Mahender Singh, Yang-baxter equation and related algebraic structures, arXiv: 2506.23175 https://arxiv.org/abs/2506.23175, 2025

work page arXiv 2025

[3] [3]

Bunch, P

E. Bunch, P. Lofgren, A. Rapp, and D. N. Yetter, On quotients of quandles, J. Knot Theory Ramifications 19 (2010), no. 9, 1145--1156. 2726562

2010

[4] [4]

Marco Bonatto, Principal and doubly homogeneous quandles, Monatsh. Math. 191 (2020), no. 4, 691--717. 4075376

2020

[5] [5]

Marco Bonatto and David Stanovsk\'y, Commutator theory for racks and quandles, J. Math. Soc. Japan 73 (2021), no. 1, 41--75. 4203321

2021

[6] [6]

Cameron, Finite permutation groups and finite simple groups, Bull

Peter J. Cameron, Finite permutation groups and finite simple groups, Bull. London Math. Soc. 13 (1981), no. 1, 1--22. 599634

1981

[7] [7]

, Permutation groups, Handbook of combinatorics, V ol.\ 1,\ 2, Elsevier Sci. B. V., Amsterdam, 1995, pp. 611--645. 1373668

1995

[8] [8]

45, Cambridge University Press, Cambridge, 1999

, Permutation groups, London Mathematical Society Student Texts, vol. 45, Cambridge University Press, Cambridge, 1999. 1721031

1999

[9] [9]

Dixon and Brian Mortimer, Permutation groups, Graduate Texts in Mathematics, vol

John D. Dixon and Brian Mortimer, Permutation groups, Graduate Texts in Mathematics, vol. 163, Springer-Verlag, New York, 1996. 1409812

1996

[10] [10]

Knot Theory Ramifications 23 (2014), no

Val\'erian Even and Marino Gran, On factorization systems for surjective quandle homomorphisms, J. Knot Theory Ramifications 23 (2014), no. 11, 1450060, 15. 3293045

2014

[11] [11]

Michael Eisermann, Quandle coverings and their G alois correspondence , Fund. Math. 225 (2014), no. 1, 103--168. 3205568

2014

[12] [12]

Electron

Konomi Furuki and Hiroshi Tamaru, Homogeneous quandles with abelian inner automorphism groups and vertex-transitive graphs, Int. Electron. J. Geom. 17 (2024), no. 1, 184--198. 4740314

2024

[13] [13]

Akihiro Higashitani and Hirotake Kurihara, Generalized A lexander quandles of finite groups and their characterizations , Eur. J. Math. 10 (2024), no. 3, Paper No. 41, 35. 4771084

2024

[14] [14]

Pure Appl

Alexander Hulpke, David Stanovsk\'y, and Petr Vojt e chovsk\'y, Connected quandles and transitive groups, J. Pure Appl. Algebra 220 (2016), no. 2, 735--758. 3399387

2016

[15] [15]

Yoshitaka Ishihara and Hiroshi Tamaru, Flat connected finite quandles, Proc. Amer. Math. Soc. 144 (2016), no. 11, 4959--4971. 3544543

2016

[16] [16]

Pure Appl

David Joyce, A classifying invariant of knots, the knot quandle, J. Pure Appl. Algebra 23 (1982), no. 1, 37--65. 638121

1982

[17] [17]

Math., vol

Akira Kubo, Mika Nagashiki, Takayuki Okuda, and Hiroshi Tamaru, A commutativity condition for subsets in quandles---a generalization of antipodal subsets, Differential geometry and global analysis---in honor of T adashi N agano, Contemp. Math., vol. 777, Amer. Math. Soc., [Providence], RI, [2022] 2022, pp. 103--125. 4410185

2022

[18] [18]

S. V. Matveev, Distributive groupoids in knot theory, Mat. Sb. (N.S.) 119(161) (1982), no. 1, 78--88, 160. 672410

1982

[19] [19]

257--264

James McCarron, Small homogeneous quandles, I SSAC 2012--- P roceedings of the 37th I nternational S ymposium on S ymbolic and A lgebraic C omputation, ACM, New York, 2012, pp. 257--264. 3206312

2012

[20] [20]

n Lab authors , n L ab , 2008--2026

2008

[21] [21]

Algebra 663 (2025), 150--170

Takuya Saito and Sakumi Sugawara, Homogeneous quandles with abelian inner automorphism groups, J. Algebra 663 (2025), 150--170. 4799816

2025

[22] [22]

Hiroshi Tamaru, Two-point homogeneous quandles with prime cardinality, J. Math. Soc. Japan 65 (2013), no. 4, 1117--1134. 3127819

2013

[23] [23]

Leandro Vendramin, Doubly transitive groups and cyclic quandles, J. Math. Soc. Japan 69 (2017), no. 3, 1051--1057. 3685034

2017

[24] [24]

Koshiro Wada, Two-point homogeneous quandles with cardinality of prime power, Hiroshima Math. J. 45 (2015), no. 2, 165--174. 3379001

2015