pith. sign in

arxiv: 2207.13885 · v2 · submitted 2022-07-28 · 🧮 math.GR · math.GT

On virtual singular braid groups

Oscar Ocampo This is my paper

Pith reviewed 2026-05-24 11:45 UTC · model grok-4.3

classification 🧮 math.GR math.GT
keywords virtual singular braid groupgroup homomorphismssymmetric groupsemi-direct productbraid groupsvirtual braidssingular braidsgroup presentations
0
0 comments X

The pith

Virtual singular braid groups VSG_n admit semi-direct product decompositions from their homomorphisms to the symmetric group S_n.

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

The paper determines all group homomorphisms from VSG_n to S_n up to conjugation and uses them to obtain semi-direct product decompositions of VSG_n. It also defines numerical invariants via exponent sums in words from VSG_n and identifies the kernels of the maps to abelian groups. For the case n=2 the kernels receive explicit presentations. The work further shows that some relations cannot hold in VSG_n and constructs natural quotients, including welded and unrestricted versions, that satisfy analogous structural results.

Core claim

All group homomorphisms from the virtual singular braid group VSG_n to the symmetric group S_n are determined up to conjugation, yielding corresponding semi-direct product decompositions of VSG_n. Numerical invariants arise from exponent sums, their kernels to abelian groups are described explicitly, and for n=2 the kernels receive concrete presentations. Certain relations are shown to be forbidden in VSG_n, and welded and unrestricted quotients are introduced with parallel decomposition results.

What carries the argument

The homomorphisms from VSG_n to S_n, which produce the semi-direct product decompositions of the virtual singular braid group.

If this is right

  • VSG_n splits as a semi-direct product for each such homomorphism to S_n.
  • The kernels of the exponent-sum maps to abelian groups are explicitly described.
  • For n=2 the kernels of the maps to S_2 receive explicit group presentations.
  • Welded and unrestricted quotients of VSG_n inherit the same homomorphism classification and decomposition property.

Where Pith is reading between the lines

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

  • The classification may allow systematic computation of representations of VSG_n that factor through S_n.
  • The forbidden relations could be used to distinguish VSG_n from other braid-like groups in presentations.
  • The numerical invariants might descend to invariants of virtual singular links via the quotient maps.

Load-bearing premise

The relations that define VSG_n as a common generalization of singular and virtual braid groups are consistent and complete enough for the listed homomorphisms and decompositions to hold.

What would settle it

Discovery of a homomorphism from VSG_n to S_n that is not conjugate to any of the classified ones.

Figures

Figures reproduced from arXiv: 2207.13885 by Oscar Ocampo.

Figure 1
Figure 1. Figure 1: Classical crossings, for i = 1, . . . , n − 1. The singular braid monoid with n strands, denoted by SBn, was introduced independ￾ently by Baez [2] and Birman [8]. It was shown by Fenn, Keyman and Rourke [15] that the singular braid monoid embeds in a group called the singular braid group and denoted by SGn. The singular braid monoid has been shown to be a useful object to consider, for example, in the cont… view at source ↗
Figure 2
Figure 2. Figure 2: Singular and virtual crossings, for i = 1, . . . , n − 1. (SR1) τiτj = τj τi for |i − j| ≥ 2. (MR1) τiσj = σj τi for |i − j| ≥ 2. (MR2) τiσi = σiτi for i = 1, 2, . . . , n − 1. (MR3) σiσi+1τi = τi+1σiσi+1, for i = 1, 2, . . . , n − 2. (MR4) σi+1σiτi+1 = τiσi+1σi , for i = 1, 2, . . . , n − 2. Virtual singular braids are similar to classical braids, in addition to having classical crossings, they also have … view at source ↗
Figure 3
Figure 3. Figure 3: Classical and extended Reidemeister move, for i = 1, . . . , n − 1. In [10, Definition 3] was defined an abstract group called the virtual singular braid group such that the virtual singular braid monoid V SBn (defined in [9]) embeds in it. We shall use the presentation as stated in [10, Pages 5-6]. We note that the relations of the groups Bn, V Bn and SGn given in their respective presentations (see Defin… view at source ↗
Figure 4
Figure 4. Figure 4: Two point relations, for i = 1, . . . , n − 1. 2.2 Some properties of the virtual singular braid group Given a group G we define the lower central series of G recursively as Γ1(G) = G and for i ≥ 2 Γi(G) = [Γi−1(G), G]. Let P be any group-theoretic property. We say that a group is residually P if for any (non-trivial) element g ∈ G, there exists a group H with the property P and a surjective homomorphism ϕ… view at source ↗
Figure 5
Figure 5. Figure 5: Three point relations, for |i − j| = 1. = [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: Quotients of virtual singular braid groups are quotient homomorphisms obtained by adding relations according to the enumeration (1) viσi+1σi = σi+1σivi+1, for i = 1, . . . , n − 2; (2) vi+1σiσi+1 = σiσi+1vi , for i = 1, . . . , n − 2; (3) viτi+1τi = τi+1τivi+1, for i = 1, . . . , n − 2; (4) vi+1τiτi+1 = τiτi+1vi , for i = 1, . . . , n − 2; (5) σ 2 i = 1, for i = 1, . . . , n − 1; (6) σivi = viσi , for i = … view at source ↗
read the original abstract

The virtual singular braid group arises as a natural common generalization of classical singular braid groups and virtual braid groups. In this paper, we study several algebraic properties of the virtual singular braid group $VSG_n$. We introduce numerical invariants for virtual singular braids arising from exponent sums of words in $VSG_n$, and describe explicitly the kernels of the associated homomorphisms onto abelian groups. We then determine all group homomorphisms, up to conjugation, from $VSG_n$ to the symmetric group $S_n$, and obtain corresponding semi-direct product decompositions. In the particular case $n=2$, we provide explicit presentations and algebraic descriptions of the kernels. Moreover, we show that certain relations are forbidden in $VSG_n$, and we introduce and study natural quotients of the virtual singular braid group, including welded and unrestricted versions, for which analogous structural results are obtained.

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

0 major / 2 minor

Summary. The paper studies the virtual singular braid group VSG_n, a common generalization of singular braid groups and virtual braid groups. It introduces numerical invariants from exponent sums of words and describes the kernels of the associated homomorphisms to abelian groups. The central results are a complete classification (up to conjugation) of all group homomorphisms VSG_n → S_n together with the resulting semidirect-product decompositions of VSG_n; explicit presentations and kernel descriptions are given for the case n=2. The paper also identifies certain forbidden relations in VSG_n and introduces and analyzes natural quotients (welded and unrestricted versions) for which analogous structural results hold.

Significance. If the classification of homomorphisms holds, the work supplies a concrete structural description of how VSG_n maps onto S_n and the resulting splittings, which is useful for understanding representations of these generalized braid groups. The explicit n=2 case supplies an independent verification point, and the treatment of quotients extends the results in a natural direction.

minor comments (2)
  1. [Abstract] The abstract states that 'certain relations are forbidden' but does not name them; a brief parenthetical list of the forbidden relations would improve readability.
  2. Notation for the generators of VSG_n (e.g., the virtual, singular, and classical crossings) is introduced inline; a short table or dedicated paragraph collecting the full presentation would aid readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive evaluation of the manuscript. The referee's summary accurately captures the main results on homomorphisms from VSG_n to S_n, the semidirect product decompositions, the n=2 case, and the treatment of quotients. We are pleased that the work is recommended for acceptance.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines VSG_n by a presentation extending singular and virtual braid groups, then classifies homomorphisms VSG_n → S_n (up to conjugation) by checking which assignments of generators to S_n elements satisfy every defining relation, followed by kernel descriptions and semidirect-product splittings. This is a direct enumeration from the relations with no fitted parameters, no self-citation chains invoked as uniqueness theorems, and no renaming of known results as new derivations. The n=2 case supplies an explicit independent verification. The derivation chain remains self-contained against the presentation without any step reducing to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work builds on established axioms from braid group theory without introducing new free parameters or entities.

axioms (1)
  • domain assumption VSG_n is generated by the standard generators of singular and virtual braid groups with the usual relations plus additional ones for the combination.
    This definition is the starting point for all subsequent results described in the abstract.

pith-pipeline@v0.9.0 · 5664 in / 1164 out tokens · 29005 ms · 2026-05-24T11:45:09.250341+00:00 · methodology

discussion (0)

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

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Universal virtual braid groups

    math.GR 2026-04 unverdicted novelty 8.0

    UV_n(c) contains a finite-index right-angled Artin subgroup and has S_n as its smallest non-abelian finite quotient for n≥5, with LERF and Howson properties holding only for n≤3.

  2. On Universal Virtual and Welded Braid Groups and Their Linear Representations

    math.RT 2026-04 unverdicted novelty 7.0

    Universal virtual and welded braid groups are constructed to unify prior variants, with classifications of their complex homogeneous local representations and proofs of abelianization, perfect commutator, trivial cent...

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages · cited by 2 Pith papers

  1. [1]

    Artin, Theory of braids, Ann

    E. Artin, Theory of braids, Ann. Math. 48 (1947), 101–126

    work page 1947

  2. [2]

    Baez, Link invariants of finite type and perturbation th eory, Lett

    J. Baez, Link invariants of finite type and perturbation th eory, Lett. Math. Phys. 26 (1992), 43–51

    work page 1992

  3. [3]

    Bar-Natan, Vassiliev homotopy string link invariants , J

    D. Bar-Natan, Vassiliev homotopy string link invariants , J. Knot Theory Ramifications 4 (1995) 13–32

    work page 1995

  4. [4]

    V. G. Bardakov, The virtual and universal braids, Fundamenta Mathematicae 184 (2004), 1–18

    work page 2004

  5. [5]

    V. G. Bardakov and P. Bellingeri, Combinatorial propertie s of virtual braids, Topology and its Applications 156, 6 (2009), 1071–1082

    work page 2009

  6. [6]

    V. G. Bardakov, P. Bellingeri and C. Damiani, Unrestricted virtual braids, fused links and other quotients of virtual braid groups, Journal of Knot Theory and Its Ramifications 24, 12 (2015), 1550063

    work page 2015

  7. [7]

    Bellingeri and L

    P. Bellingeri and L. Paris, Virtual braids and permutatio ns, Annales de l’Institut Fourier 70, 3 (2020), 1341–1362

    work page 2020

  8. [8]

    Birman, New points of view in knot theory, Bull

    J. Birman, New points of view in knot theory, Bull. Amer. Math. Soc. 28 (1993), 253–287

    work page 1993

  9. [9]

    Caprau, A

    C. Caprau, A. Pena and S. McGahan, Virtual singular braid s and links, Manuscripta Math. 151(1) (2016) 147–175

    work page 2016

  10. [10]

    Caprau and A

    C. Caprau and A. Yeung, Algebraic structures among virt ual singular braids, (2022) arXiv:2201.09187v1

    work page arXiv 2022

  11. [11]

    Caprau and S

    C. Caprau and S. Zepeda, On the virtual singular braid mo noid, J. Knot Theory Ramifica- tions 30, 14 (2021), 2141002

    work page 2021

  12. [12]

    B. A. Cisneros de la Cruz and G. Gandolfi, Algebraic, combi natorial and topological proper- ties of singular virtual braid monoids, J. Knot Theory Ramifications 28, 10 (2019), 1950069

    work page 2019

  13. [13]

    Damiani, A journey through loop braid groups, Expositiones Mathematicae 35, 3 (2017), 252–285

    C. Damiani, A journey through loop braid groups, Expositiones Mathematicae 35, 3 (2017), 252–285

    work page 2017

  14. [14]

    Dey and K

    S. Dey and K. Gongopadhyay, Commutator subgroups of sin gular braid groups, J. Knot Theory Ramifications 31, 05 (2022), 2250033. 18

    work page 2022

  15. [15]

    R. Fenn, E. Keyman and C. Rourke, The singular braid mono id embeds in a group, J. Knot Theory Ramifications 7(7) (1998) 881–892

    work page 1998

  16. [16]

    Kamada, Braid presentation of virtual knots and welde d knots, Osaka J

    S. Kamada, Braid presentation of virtual knots and welde d knots, Osaka J. Math. 44, 2 (2007), 441–458

    work page 2007

  17. [17]

    L. H. Kauffman, Virtual knot theory, Eur. J. Comb. 20, 7 (1999), 663–690

    work page 1999

  18. [18]

    L. H. Kauffman and S. Lambropoulou, Virtual braids, Fund. Math. 184 (2004), 159–186

    work page 2004

  19. [19]

    Magnus, A

    W. Magnus, A. Karrass and D. Solitar, Combinatorial gro up theory. Presentation of groups in terms of generators and relations. 2nd revised edition, D over Publications, New York, 1976

    work page 1976

  20. [20]

    V. V. Vershinin, On homology of virtual braids and Burau r epresentation, J. Knot Theory Ramifications 10, 5 (2001), 795–812

    work page 2001

  21. [21]

    Zhu, On singular braids, J

    J. Zhu, On singular braids, J. Knot Theory Ramifications 6 (1997) 427–440. 19

    work page 1997