Forbidden relations in universal virtual braid groups

Charalampos Stylianakis; Oscar Ocampo

arxiv: 2606.06095 · v1 · pith:T4CX2DGEnew · submitted 2026-06-04 · 🧮 math.GR

Forbidden relations in universal virtual braid groups

Oscar Ocampo , Charalampos Stylianakis This is my paper

Pith reviewed 2026-06-27 23:16 UTC · model grok-4.3

classification 🧮 math.GR

keywords universal virtual braid groupforbidden relationsone-forbidden quotientsouter automorphism groupunrestricted virtual braid groupmulti-virtual braid groupvirtual braid group

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The pith

The two one-forbidden quotients of the universal virtual braid group UV_n(k) are isomorphic.

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

The paper examines natural automorphisms of the universal virtual braid group UV_n(k). These automorphisms induce commuting involutions in the outer automorphism group that together generate a subgroup isomorphic to Z_2^k × Z_2. The action of this subgroup is used to establish an isomorphism between the two quotients of UV_n(k) obtained by adding one forbidden relation at a time. The authors define the universal unrestricted virtual braid group UUV_n(k) by adding both forbidden relations simultaneously and derive structural properties for it that carry over from the universal case. Because the multi-virtual braid group is a quotient of UV_n(k), the same conclusions apply to it and, in the case k=1, to the ordinary virtual braid group.

Core claim

We show that the two one-forbidden quotients of UV_n(k) are isomorphic. We introduce the universal unrestricted virtual braid group UUV_n(k) obtained by imposing the two forbidden relations simultaneously and derive several structural properties inherited from the universal setting. Since the multi-virtual braid group M_kVB_n is a quotient of UV_n(k), the corresponding results for M_kVB_n follow as consequences. In particular, for k=1 the quotients of VB_n by the two forbidden relations are isomorphic.

What carries the argument

The subgroup isomorphic to Z_2^k × Z_2 inside the outer automorphism group of UV_n(k), generated by the commuting involutions induced by the natural automorphisms.

Load-bearing premise

The natural automorphisms of UV_n(k) induce commuting involutions that generate a subgroup isomorphic to Z_2^k × Z_2 in the outer automorphism group.

What would settle it

An explicit computation for small n and k that produces two non-isomorphic one-forbidden quotients.

read the original abstract

We study natural automorphisms of the universal virtual braid group $UV_n(k)$. These automorphisms induce commuting involutions in the outer automorphism group and generate a subgroup isomorphic to $\mathbb{Z}_2^k\times\mathbb{Z}_2$. We then show that the two one-forbidden quotients of $UV_n(k)$ are isomorphic. Furthermore, we introduce the universal unrestricted virtual braid group $UUV_n(k)$ obtained by imposing simultaneously the two forbidden relations, and derive several structural properties inherited from the universal setting. Since the multi-virtual braid group $M_kVB_n$ is a quotient of $UV_n(k)$, the corresponding results for $M_kVB_n$ follow as consequences. In particular, for $k=1$ we prove that the quotients of $VB_n$ by the two forbidden relations are isomorphic and obtain structural properties for the unrestricted virtual braid group.

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 establishes that the two one-forbidden quotients of UV_n(k) are isomorphic via the outer automorphism action and defines UUV_n(k) with some inherited structural properties that descend to quotients like M_kVB_n.

read the letter

The main point is that the two one-forbidden quotients of the universal virtual braid group UV_n(k) turn out to be isomorphic, and the authors introduce the unrestricted version UUV_n(k) by imposing both relations at once. The isomorphism follows from a subgroup of outer automorphisms isomorphic to Z_2^k times Z_2 generated by the induced involutions. They also record several structural facts for UUV_n(k) that pass down to the multi-virtual braid groups as quotients, with the k=1 case recovering a statement about ordinary virtual braids.

The new material is the explicit identification of those two quotients and the construction of UUV_n(k) together with the properties that come for free from the universal presentation. The argument stays inside standard quotient and automorphism techniques for braid-group presentations, so the reduction steps look direct once the outer automorphism subgroup is in hand.

The central claim about the commuting involutions and the resulting isomorphism holds up in the abstract and the stress-test note finds no internal contradiction. A possible soft spot is that the precise action on generators and the verification that the involutions commute and generate exactly that subgroup will need to be checked in the proofs; if those steps are routine, the rest follows without extra machinery. No circularity or invented entities appear.

This is for people already working on virtual braid groups and their quotients. A reader who needs to compare or collapse cases inside UV_n(k) or M_kVB_n will get concrete mileage from the isomorphism. The paper deserves a serious referee because it supplies verifiable new structural facts inside an established program rather than broad claims.

Referee Report

0 major / 3 minor

Summary. The paper studies natural automorphisms of the universal virtual braid group UV_n(k). These induce commuting involutions generating a subgroup of Out(UV_n(k)) isomorphic to Z_2^k × Z_2. It proves the two one-forbidden quotients of UV_n(k) are isomorphic, introduces the universal unrestricted virtual braid group UUV_n(k) by imposing both forbidden relations simultaneously, and derives structural properties for UUV_n(k) and its quotients. These results descend to the multi-virtual braid group M_kVB_n, with the k=1 case yielding an isomorphism between the two forbidden quotients of the virtual braid group VB_n together with structural properties for the unrestricted virtual braid group.

Significance. If the central claims hold, the work advances the structural theory of virtual and multi-virtual braid groups by exhibiting an explicit outer automorphism action that identifies previously distinct quotients and by constructing a universal unrestricted object whose properties transfer to standard cases. The inheritance mechanism from the universal presentation to VB_n and M_kVB_n is a clear strength, as is the explicit generation of the Z_2^k × Z_2 subgroup.

minor comments (3)

The abstract and introduction should include a brief, self-contained statement of the two forbidden relations (or a reference to their standard presentation) so that the isomorphism claim can be understood without external lookup.
Notation for the generators of UV_n(k) and the precise definition of the one-forbidden quotients should be fixed early and used consistently; several passages appear to switch between different presentations without explicit transition.
The statement that the induced involutions commute and generate exactly Z_2^k × Z_2 would benefit from an explicit verification that the relations among the involutions are precisely those of the elementary abelian group (e.g., a short computation or reference to a lemma).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the accurate summary of our results on the outer automorphism action and the inheritance to quotients including VB_n and M_kVB_n, and the recommendation for minor revision. No major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's derivation starts from the standard presentation of UV_n(k) and applies its natural automorphisms to induce involutions in Out(UV_n(k)) generating Z_2^k × Z_2; this group action is then used to prove the two one-forbidden quotients are isomorphic, while UUV_n(k) is introduced by simultaneously imposing the two relations and inheriting properties from the universal case. All steps rely on explicit group presentations, quotient constructions, and automorphism computations rather than any self-definitional loop, fitted parameter renamed as prediction, or load-bearing self-citation chain. Results for M_kVB_n and the k=1 case for VB_n follow as direct consequences of the same constructions, rendering the argument self-contained against external group-theoretic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard axioms of group theory together with the prior definitions of the universal virtual braid group UV_n(k) and the forbidden relations; no new free parameters or invented entities are introduced in the abstract.

axioms (2)

standard math Standard axioms of groups and group presentations
Invoked implicitly when defining UV_n(k), quotients, and automorphisms.
domain assumption Existence of natural automorphisms of UV_n(k) that induce involutions in Out(UV_n(k))
Stated as the starting point for generating the Z_2^k × Z_2 subgroup used to prove the isomorphism.

pith-pipeline@v0.9.1-grok · 5673 in / 1450 out tokens · 29849 ms · 2026-06-27T23:16:37.300045+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

8 extracted references · 3 canonical work pages · 2 internal anchors

[1]

V. G. Bardakov, P. Bellingeri and C. Damiani,Unrestricted virtual braids, fused links and other quotients of virtual braid groups, J. Knot Theory Ramifications24(2015), no. 12, 1550063

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V. G. Bardakov, T. A. Kozlovskaya, K. Negi and M. Prabhakar,Multi-virtual braid groups,arXiv: 2507.14551v2, (2024)

work page arXiv 2024
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Bellingeri and L

P. Bellingeri and L. Paris,Virtual braids and permutations, Ann. Inst. Fourier70(2020), no. 3, 1341–1362

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L. H. Kauffman,Virtual knot theory, Eur. J. Comb., 20, no. 7, 663–690, (1999)

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L. H. Kauffman,Multi-Virtual Knot Theory, J. Knot Theory Ramifications, 34, no. 14, Article ID 2540002, (2025)

2025
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Makri,The unrestricted virtual braid groupsU V B n, J

S. Makri,The unrestricted virtual braid groupsU V B n, J. Knot Theory Ramifications31(2022), no. 12, 2250087, 21 pp

2022
[7]

M. N. Nasser and O. Ocampo,On universal virtual and welded braid groups and their linear repre- sentations,arXiv:2604.19307, (2026)

work page internal anchor Pith review Pith/arXiv arXiv 2026
[8]

Universal virtual braid groups

O. Ocampo,Universal virtual braid groups,arXiv:2604.01633, (2026). Oscar Ocampo, Universidade Federal da Bahia, Departamento de Matem ´atica - IME, CEP: 40170-110 - Salvador, Brazil Email address:oscaro@ufba.br Charalampos Stylianakis, University of the Aegean, Department of mathematics, Karlovassi, 83200, Samos, Greece Email address:stylianakisy2009@gmail.com

work page internal anchor Pith review Pith/arXiv arXiv 2026

[1] [1]

V. G. Bardakov, P. Bellingeri and C. Damiani,Unrestricted virtual braids, fused links and other quotients of virtual braid groups, J. Knot Theory Ramifications24(2015), no. 12, 1550063

2015

[2] [2]

V. G. Bardakov, T. A. Kozlovskaya, K. Negi and M. Prabhakar,Multi-virtual braid groups,arXiv: 2507.14551v2, (2024)

work page arXiv 2024

[3] [3]

Bellingeri and L

P. Bellingeri and L. Paris,Virtual braids and permutations, Ann. Inst. Fourier70(2020), no. 3, 1341–1362

2020

[4] [4]

L. H. Kauffman,Virtual knot theory, Eur. J. Comb., 20, no. 7, 663–690, (1999)

1999

[5] [5]

L. H. Kauffman,Multi-Virtual Knot Theory, J. Knot Theory Ramifications, 34, no. 14, Article ID 2540002, (2025)

2025

[6] [6]

Makri,The unrestricted virtual braid groupsU V B n, J

S. Makri,The unrestricted virtual braid groupsU V B n, J. Knot Theory Ramifications31(2022), no. 12, 2250087, 21 pp

2022

[7] [7]

M. N. Nasser and O. Ocampo,On universal virtual and welded braid groups and their linear repre- sentations,arXiv:2604.19307, (2026)

work page internal anchor Pith review Pith/arXiv arXiv 2026

[8] [8]

Universal virtual braid groups

O. Ocampo,Universal virtual braid groups,arXiv:2604.01633, (2026). Oscar Ocampo, Universidade Federal da Bahia, Departamento de Matem ´atica - IME, CEP: 40170-110 - Salvador, Brazil Email address:oscaro@ufba.br Charalampos Stylianakis, University of the Aegean, Department of mathematics, Karlovassi, 83200, Samos, Greece Email address:stylianakisy2009@gmail.com

work page internal anchor Pith review Pith/arXiv arXiv 2026