Pith. sign in

REVIEW 3 major objections 5 minor 1 cited by

The monoid structure of singular twisted virtual braids

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper establishes explicit monoid presentations for the three kernels of epimorphisms from the singular twisted virtual braid monoid STVB_n onto the symmetric group, and proves that the monoid embeds in a group.

desk verdict New explicit presentations for submonoids of STVB_n, but the central proofs rely on an unproved transfer of Reidemeister-Schreier to monoids and the group embedding has a one-sided inverse slip. read the letter →

arxiv 2502.08965 v1 pith:NJJ5TEJE submitted 2025-02-13 math.GR math.GT

Prabhakar Madeti , Komal Negi This is my paper · ORCID
classification math.GRmath.GT MSC 57M5057K1220M32
keywords singulartwistedvirtualbraidmonoidpurepresentationReidemeister-Schreiermethodkernelofepimorphismsymmetricgroupembeddingintorepresentationextension
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper is about the algebraic structure behind singular twisted virtual braids: braid diagrams on $n$ strands whose crossings can be classical, virtual, singular, or equipped with bars, considered up to braid-type moves, and forming a monoid—a set with an associative operation and an identity. The authors define three onto maps from the singular twisted virtual braid monoid $\mathrm{STVB}_n$ to the symmetric group $S_n$, and they claim that the kernel of each map—the submonoid of braids that look trivial as permutations—is one of three named submonoids: the singular twisted virtual pure braid monoid $\mathrm{STVP}_n$ and two relatives $\mathrm{STVH}_n$ and $M_n$. For each kernel they give an explicit presentation, listing generators such as $\lambda_{kl}$, $y_{kl}$, $x_{kl}$, $z_{kl}$, and $\gamma_j$, together with defining relations (43)–(70). The paper claims these presentations are complete, giving an explicit description of the normal submonoid structure of $\mathrm{STVB}_n$; it further claims that $\mathrm{STVB}_n$ embeds into a group $\mathrm{STVG}_n$ and that representations of the twisted virtual braid group extend to representations of the singular monoid.

What carries the argument

The central machinery is the Reidemeister–Schreier method transferred from groups to monoids: with a Schreier transversal $\Lambda_n$ (a chosen set of representatives of the $n!$ cosets of the kernel in $\mathrm{STVB}_n$), each kernel generator is written as a rewritten element $s_{\lambda,a}=\lambda a(\lambda a)^{-1}$, and each defining relation of $\mathrm{STVB}_n$ is rewritten as a conjugate $\lambda r_\mu \lambda^{-1}$. The named generators $\lambda_{kl}$ and $y_{kl}$ (built from $\sigma_i$, $\tau_i$, and $\rho_i$) are exactly the nontrivial rewritten elements, and Lemma 3.1—the transitive conjugation action of the symmetric group on the $\gamma_j$—converts the rewritten relations into the uniform index patterns of (44)–(54), (56)–(66), and (67)–(70). The embedding claim is carried by a monoid-embedding criterion (stated as Theorem 4.1 of the paper) that adjoins formal inverses to a chosen set of generators; the representation extension is carried by the substitution $\Phi_{a,b,c}(\tau_i)=a\phi(\sigma_i)+b\phi(\sigma_i^{-1})+ce$, with the relations involving $\tau_i$ checked directly.

What would settle it

Run a convergent rewriting-system completion on the proposed presentations for $\mathrm{STVP}_3$, $\mathrm{STVH}_3$, and $M_3$ using the base presentation of $\mathrm{STVB}_3$ from Theorem 2.4; if the completion finds two words that are equivalent under the original monoid relations but cannot be connected using only the listed kernel relations, the kernel presentation is missing a relation. Since $\Lambda_3$ has only six representatives, this check is finite and explicit.

Watch

Extended reading notes

Core claim

The central claim is that the three normal submonoids of $\mathrm{STVB}_n$ that map onto the symmetric group are exactly the kernels of three epimorphisms $\varphi_1,\varphi_2,\varphi_3$ from $\mathrm{STVB}_n$ to $S_n$, and that each kernel has a concrete monoid presentation. $\mathrm{STVP}_n$ is presented by $\lambda_{kl}^{\pm 1}$, $y_{kl}$, $\gamma_j$ with relations (43)–(54); $\mathrm{STVH}_n$ by $x_{kl}^{\pm 1}$, $z_{kl}$, $\gamma_j$ with relations (55)–(66); and $M_n$ by $\lambda_{kl}^{\pm 1}$, $z_{ij}$, $\gamma_j$ with relations (64)–(70) together with (43)–(47). The proof route is the Reidemeister–Schreier rewriting method applied to a monoid, using the Schreier set $\Lambda_n$ already used for the twisted virtual braid group, and the paper asserts that all remaining relations are obtained from the displayed ones by conjugation. The paper also claims an embedding of the monoid $\mathrm{STVB}_n$ into a group $\mathrm{STVG}_n$, and an extension theorem: any representation $\phi\colon \mathrm{TVB}_n\to G$ extends to a representation $\Phi\colon \mathrm{STVB}_n\to K[G]$ (the group algebra of $G$ over $K$) by sending the singular generator $\tau_i$ to $a\phi(\sigma_i)+b\phi(\sigma_i^{-1})+ce$.

Load-bearing premise

The load-bearing premise is that the Reidemeister–Schreier rewriting procedure, classically stated for subgroups of groups, transfers without modification to these monoids with the chosen Schreier set $\Lambda_n$, so that every kernel relation is a conjugate of one of the listed base relations; if a relation is missed, the presentations could be incomplete.

Editorial extensions

If this is right

  • Because $\mathrm{STVP}_n$, $\mathrm{STVH}_n$, and $M_n$ are kernels of homomorphisms onto $S_n$, each is a normal submonoid of index $n!$ in $\mathrm{STVB}_n$, so the elements of $\mathrm{STVB}_n$ split into $n!$ cosets indexed by permutations.
  • The explicit presentations make it possible to rewrite and compare pure singular twisted virtual braids symbolically, and to compute algebraic invariants such as abelianizations or homomorphism sets for small $n$.
  • The embedding $\mathrm{STVB}_n\hookrightarrow \mathrm{STVG}_n$ turns the non-invertible singular generators $\tau_i$ into invertible elements, so group-theoretic tools apply to the singular monoid.
  • Any representation $\phi\colon \mathrm{TVB}_n\to G$ extends to $\Phi\colon \mathrm{STVB}_n\to K[G]$ by the three-term formula for $\tau_i$, giving a general way to build linear representations of the singular monoid from known twisted virtual braid representations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the Reidemeister–Schreier transfer is valid, the same scheme should yield presentations for the analogous kernels in any singular or virtual braid monoid whose base presentation has the same shape; only the Schreier set and the base relations would change.
  • The three kernels may well be non-isomorphic as monoids; comparing their abelianizations, centers, or growth series would test whether the different epimorphisms capture genuinely different submonoid structures.
  • The representation extension with parameters $a,b,c$ is a natural source of link invariants: taking a closure or a trace of $\Phi$ could produce invariants of singular twisted virtual links, which the paper does not pursue.
  • The group $\mathrm{STVG}_n$ is a plausible setting for a closure-move theorem for singular twisted virtual braids, allowing link invariants that take values in the group rather than the monoid.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper studies the singular twisted virtual braid monoid STVB_n and three of its submonoids defined as kernels of epimorphisms onto the symmetric group S_n: the singular twisted virtual pure braid monoid STVP_n, the submonoid STVH_n, and the submonoid M_n. For each of these submonoids the authors state an explicit presentation: Theorem 3.2 gives generators λ_{kl}, y_{kl}, γ_j with relations (43)-(54); Theorem 3.3 gives generators x_{kl}, z_{kl}, γ_j with relations (55)-(66); and Theorem 3.4 gives generators λ_{kl}, z_{ij}, γ_j with relations (64)-(70). The proofs are based on the Reidemeister-Schreier method as presented in [2], using the same Schreier set Λ_n that was used for the twisted virtual braid group. The paper also claims in Theorem 4.2 that STVB_n embeds into a group STVG_n, constructed by adjoining formal inverses to the generators τ_i, and in Section 4.2 it extends representations of the twisted virtual braid group TVB_n to representations of STVB_n into a group algebra K[G_n].

Significance. If the stated presentations are correct, the paper provides a useful structural description of the singular twisted virtual pure braid monoid and related submonoids, extending known results for braid groups, virtual braid groups, and singular braid monoids. The proposed embedding of STVB_n into a group and the extension of TVB_n representations to STVB_n are natural and potentially valuable results. However, the central derivations rely on an unverified transfer of the Reidemeister-Schreier method from groups to monoids, and the proofs of Theorems 3.3 and 3.4 are reduced to a single sentence. Because the presentation theorems are the main contribution, the current version is not yet a complete proof of the stated claims. The paper does provide explicit candidate generators and relations, which is a concrete starting point, but the missing justifications are load-bearing.

major comments (3)
  1. [Section 4.1, Theorem 4.2 and definition of STVG_n] The proof invokes Theorems 2.7 and 2.9 of [2] (Magnus, Karrass, and Solitar) as if they apply verbatim to the monoid STVB_n. Those theorems are group-theoretic Reidemeister-Schreier results; no justification is given that the same rewriting procedure and the same Schreier set Λ_n work for a monoid and its submonoid. In particular, one must show that Λ_n is a complete set of representatives of the relevant congruence classes in the monoid STVB_n modulo STVP_n, and that the rewritten relations generate the monoid congruence, not merely a group congruence. The assertion that all remaining relations are obtained by conjugation does not address this gap. Since the same method is used in Theorems 3.3 and 3.4, whose proofs are each one sentence ('It is similar...'), the gap propagates to all three central presentation theorems.
  2. [Section 4.1, Theorem 4.2] The application of Keyman's Theorem 4.1 is incomplete. Condition (b) of the theorem requires relations of the form a_i u_i = u_i a_i = 1 for the invertible generators, but the paper's set R2 contains only σ_i σ_i^{-1}=e (along with ρ_i^2=e and γ_i^2=e); the relation σ_i^{-1} σ_i=e is missing. Likewise, the definition of STVG_n includes only the one-sided relation ¯τ_i τ_i = e, whereas Keyman's theorem requires both b_j ¯b_j = 1 and ¯b_j b_j = 1. Without these two-sided inverse relations, STVG_n is not proved to be a group, and the claimed embedding of STVB_n into a group is not established. Additionally, the epimorphism π defined after Theorem 4.2 is not specified on the new generators ¯τ_i, so the kernel ker(π) is not well defined as stated.
  3. [Section 3.1, relation (43)] The presentation of STVP_n treats λ_{kl}^{-1} as a generator but only includes the one-sided relation λ_{ij}λ_{ij}^{-1}=e. In a monoid presentation this does not imply that λ_{ij}^{-1} is a two-sided inverse; the relation λ_{ij}^{-1}λ_{ij}=e is not listed or derived. The same issue occurs in the underlying presentation of STVB_n in Theorem 2.4, relation (14), which lists only σ_iσ_i^{-1}=e. Since the Reidemeister-Schreier rewriting used in the proof manipulates expressions such as λ_{i,i+1}^{-1}, the invertibility of these elements is assumed without being established. This is a technical point that affects the rigor of the generator and relation computations.
minor comments (5)
  1. [Section 3] The phrase 'normal submonoids of index n!' is used without defining 'normal' or 'index' for monoids. The kernel of a monoid homomorphism is a submonoid, but the usual group-theoretic notions of normality and index do not transfer automatically; the short exact sequences displayed are not standard in the monoid category. This framing should be clarified or replaced by a precise statement about congruences or kernels.
  2. [Section 4.2, item (iii)] In the displayed verification of τ_iγ_j = γ_jτ_i, the expression 'aϕ(σ_i^{-1}ρ_j)' should be 'aϕ(σ_i^{-1}γ_j)' (or equivalently the factor should be ϕ(γ_j)), and the subsequent line contains a corresponding typo. The intended equality is clear, but the displayed formula is inconsistent.
  3. [Section 4.2, item (iv)] In the verification of ρ_iτ_iρ_i = γ_{i+1}γ_iτ_iγ_iγ_{i+1}, the middle term 'bϕ(ρ_iσ_{i+1}^{-1}ρ_i)' should read 'bϕ(ρ_iσ_i^{-1}ρ_i)'. The subsequent line uses the correct expression, so this is a typographical error, but it should be fixed.
  4. [Theorem 2.4] Relation (17) is written as σ_iτ_i = τ_iσ_i for i = 1,...,n, but the generators τ_i are only defined for i = 1,...,n-1; the range should be i = 1,...,n-1.
  5. [Section 4.1] There is a typo: 'moniod' should be 'monoid'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the kernel presentations are derived from prior presentations via Reidemeister-Schreier, not from the target results; the main weaknesses are omitted proofs and an incomplete group-embedding check, not circular reduction.

full rationale

The claimed derivations (presentations of STVP_n, STVH_n, M_n and the embedding of STVB_n) start from the known presentation of STVB_n (Theorem 2.4, cited from [7]) and apply the Reidemeister-Schreier method with a Schreier set imported from [6]. The target presentations are not assumed as inputs: the new relations involving y_kl, z_kl and the tau-generators are obtained by rewriting and conjugation from the defining relations of STVB_n, and the short exact sequences define the kernels rather than predicting them. The heavy reliance on [6,7] is self-citation, and it is load-bearing in the sense that the whole paper builds on those presentations, but this is a normal dependency on earlier theorems and not a reduction of the conclusions to the conclusions. No equation in the paper equates a theorem with its own input; no fitted parameter is renamed as a prediction; no uniqueness theorem from the authors is used to forbid alternatives. The genuinely problematic passages are completeness gaps rather than circularity: Theorems 3.3 and 3.4 are each closed by 'It is similar to the proof of Theorem 3.2' (Sec. 3.2, 3.3), Theorem 4.3 says the proof is 'analogous', and Theorem 4.2's application of Keyman's theorem lists only sigma_i sigma_i^{-1}=e, rho_i^2=e, gamma_i^2=e and later defines STVG_n with only bar-tau_i tau_i=e, omitting the opposite inverses; these affect correctness/completeness, not circularity. The RS transfer to monoids is asserted ('As a Schreier set ... we take the same set Lambda_n, which is used in finding the presentation of TVP_n') without proving the monoidal analogue, but an unproved transfer is an omitted argument, not a circular definition. Overall there is no exhibited Eq. X = Eq. Y by construction, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central presentations rest on the assumption that the Reidemeister-Schreier method transfers to monoids with the chosen transversal, and that the short proofs by analogy in Theorems 3.3 and 3.4 capture all relations. No new entities or free parameters are introduced. The embedding and representation results rest on standard theorems from the cited literature.

assumptions (4)
  • domain assumption The Reidemeister-Schreier method, as presented in Magnus-Karrass-Solitar for groups, applies to monoids with a Schreier transversal to a normal submonoid and yields presentations of submonoids.
    Invoked in the proof of Theorem 3.2 via Theorems 2.7 and 2.9 of [2], with no proof that the monoid version is valid as used.
  • domain assumption The set Lambda_n used as a Schreier set in the group TVB_n case is a valid Schreier transversal for the monoid STVB_n modulo STVP_n and the other kernels.
    Taken from [6] with the statement 'As a Schreier set of coset representation of STVP_n in STVB_n we take the same set Lambda_n', but no monoid-specific verification is supplied.
  • ad hoc to paper All defining relations of the kernel are obtained from the defining relations of STVB_n by rewriting the defining relations r_mu and conjugating by representatives; no additional relations arise from monoid-specific behavior.
    The proofs of Theorems 3.2-3.4 conclude that 'the remaining relations r_{1,lambda} can be derived from this relation by conjugation', without enumerating all conjugate relators.
  • standard math Keyman's embedding theorem [9] applies to the presentation of STVB_n with A = {sigma_i^{pm1}, rho_i, gamma_i} and B = {tau_i}.
    Theorem 4.2 checks conditions (a)-(d) of Theorem 4.1; this is a standard application if the check is correct.

how reviews work

0 comments
Cite this review

Pith. "Pith review of The monoid structure of singular twisted virtual braids." pith.science (2026). https://pith.science/paper/NJJ5TEJE

@misc{pith2026250208965,
  author       = {Pith},
  title        = {Pith review of: The monoid structure of singular twisted virtual braids},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NJJ5TEJE}},
  note         = {Machine review of arXiv:2502.08965}
}
abstract

In this paper, we examine specific submonoids within the singular twisted virtual braid monoid $STVB_n$. Notably, we establish that the singular twisted virtual pure braid monoid $STVP_n$ serves as the kernel of an epimorphism from $STVB_n$ onto the symmetric group $S_n$. We identify the generators and defining relations for $STVP_n$. Additionally, we construct other epimorphisms from $STVB_n$ onto $S_n$, whose kernels are analogous to $STVP_n$, and determine their respective generators and defining relations. Furthermore, we demonstrate the embedding of the monoid $STVB_n$ into a group. Also, we provide the extension of the representation of the twisted virtual braid group to the representation of the singular twisted virtual braid monoid.

Figures

Figures reproduced from arXiv: 2502.08965 by the authors.

Figure 1
Figure 1. A singular twisted virtual braid on 3 strands [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Classical braid moves V2 V3 V4 b b b b b b b b b b b b b b b b b b b b b b b b b b b b 1 i i + 1 n 1 i i + 1 n 1 i i + 1 i + 2 n 1 i + 1 i + 2 n 1 i i + 1 i + 2 n 1 i + 1 i + 2 n 1 i i + 1 i + 2 n 1 i + 1 i + 2 n i i i b b b b [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Virtual braid moves Definition 2.3. A singular twisted virtual braid refers to the equivalence class comprising singular twisted virtual braid diagrams. 2.1 Singular twisted virtual braid monoid In a manner similar to singular braids and virtual singular braids, the collection of singular twisted virtual braids forms a monoid, where the operation is defined by concatenation, [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Twisted braid moves S1 b b b b b b b b 1 i i + 1 i + 2 n 1 i i + 1 i + 2 n S2 b b b b b b b b 1 i i + 1 i + 2 n 1 i i + 1 i + 2 n b b b b 1 i i + 1 n bb b b 1 i i + 1 n S3 b b b b 1 i i + 1 n b b b b 1 i i + 1 n S4 [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]
Figure 5
Figure 5. Figure 5: Singular braid moves akin to the structure of the braid group. The set of singular twisted virtual braids with n strands is also a monoid, represented by ST V Bn . i i + 1 i i + 1 i i + 1 i σi σ −1 i ρi γi b b b b b b b b b b b b bbb b b b b b b b 1 n 1 1 n n 1 n i i +…
Figure 6
Figure 6. Figure 6: Generators for the monoid of singular twisted virtual braids [PITH_FULL_IMAGE:figures/full_fig_p004_6.png]
Figure 7
Figure 7. Figure 7: Generators of the monoid of singular twisted virtual pure braids [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 8
Figure 8. Figure 8: Generators of the monoid of ST V Hn . xijγk = γkxij (58) xij = γiγjxjiγjγi (59) xijzkl = zklxij (60) xikxkjzik = zkjxikxkj (61) zikxkjxik = xkjxikzkj (62) xijzij = zijxij (63) zijzkl = zklzij (64) zijγk = γkzij (65) zij = γiγjzjiγjγi (66) Where each distinct letter rep…

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

  1. Matrix representations of the twisted virtual braid group and its extensions

    math.RT 2025-06 reject novelty 6.0 of 10

    Every complex local representation of TVB2 into GL3(C) belongs to one of eight explicit families, and similar families are listed for TVBn into GL_{n+1}(C) and for STVB2 into M3(C).

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages · cited by 1 Pith paper

  1. [2]

    Magnus, A

    W. Magnus, A. Karrass and D. Solitar, Combinatorial group theory, Interscience Publ., New York, (1996)

    work page 1996

  2. [1]

    M. O. Bourgoin, Twisted link theory, Algebr. Geom. Topol. , vol. 8 , no. 3, 1249– 1279(2008)

    work page 2008

  3. [3]

    K. Negi, M. Prabhakar, and S. Kamada, Twisted virtual braids and twisted links, Osaka Journal of Mathematics , vol. 61, Issue 4, (2023)

    work page 2023

  4. [4]

    V. G. Bardakov and T. A. Kozlovskaya, On 3-strand singular pure braid group, J. Knot Theory Ramif., vol. 29, no. 10, 2042001(20 pages)(2020)

    work page 2020

  5. [5]

    V. G. Bardakov and T. A. Kozlovskaya, Singular braids, singular links and subgroups of camomile type, ArXiv:2212.08267

    work page Pith review arXiv

  6. [6]

    V. G. Bardakov, T.A. Kozlovskaya, K. Negi, M. Prabhakar, Twisted Virtual Braid Group, arXiv:2310.04154v1

    work page Pith review arXiv

  7. [7]

    Singular twisted links and singular twisted virtual braids

    K. Negi and M. Prabhakar, Singular twisted links and singular twisted virtual braids, arXiv:2403.17383

    work page Pith review arXiv

  8. [8]

    Caprau, S

    C. Caprau, S. Zepeda, On the virtual singular braid monoid, J. Knot Theory Rami- fications, vol. 30, no.14, Paper No. 2141002, 26 pp(2021)

    work page 2021

Show all 12 references
  1. [9]

    Keyman, A class of monoids embeddable in a group, Turkish J

    E. Keyman, A class of monoids embeddable in a group, Turkish J. Math. , vol. 25, 299-305(2001)

    2001

  2. [10]

    R. Fenn, E. Keyman, C. Rourke, The singular braid monoid embeds in a group, J. Knot Theory Ramif. , vol. 7, no. 7, 881–892(1998)

    1998

  3. [11]

    V. G. Bardakov, N. Chbili, T. Kozlovskaya, Extension of braid group representations to the monoid of singular braids, arXiv:2403.00516

    arXiv

  4. [12]

    V. V. Vershinin, On the singular braid monoid, Algebra i Analiz , vol. 21, no. 5, 19–36(2009)

    2009

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.