{"id":"9205501c-193a-4321-a535-71301d659429","arxiv_id":"2502.08965","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The singular twisted virtual pure braid monoid and two related submonoids are characterized as kernels of epimorphisms to the symmetric group, with explicit presentations, plus a monoid-to-group embedding and representation extension result.","lead":"This paper gives algebraic presentations for three submonoids of the singular twisted virtual braid monoid, and proves this monoid embeds in a group. It also extends representations of twisted virtual braid groups to this larger monoid.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central presentations rest on an unproved transfer of Reidemeister-Schreier to monoids; the proof asserts rather than verifies the rewritten relation set.","rationale":"I agree with the reader's weakest assumption: the central presentations for STVP_n, STVH_n, and M_n depend on an unverified transfer of the Reidemeister-Schreier method from groups to monoids, with the proof of Theorems 3.3 and 3.4 reduced to \"similar\" arguments. This is the most load-bearing concern because the strongest claim is precisely the correctness of those presentations. I add a second, more concrete red flag in Theorem 4.2: the application of Keyman's embedding theorem states only one-sided inverse relations, so the object STVG_n may not be a group as presented. The paper does give explicit, checkable presentation data and a plausible framework, and I see no evidence of a false result; the issue is proof completeness, not a known counterexample. A complete RS rewriting computation for n=3 would directly settle whether the missing relations exist, so the reader's CONDITIONAL verdict is the right calibration.","tokens_in":13070,"tokens_out":11867,"duration_ms":125858,"concrete_test":"For n=3, compute by hand or with a monoid rewriting engine the complete set of Reidemeister-Schreier relations {f(lambda r lambda^{-1}) : lambda in Lambda_3, r a defining relation of Theorem 2.4}, and check that each is derivable from the displayed relations (43)-(54). If any rewritten relation fails, Theorem 3.2 is incomplete; if they all pass, the burden shifts to proving that the same transversal and rewriting transfer to all n.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3's three main theorems (STVP_n, STVH_n, M_n) are obtained by applying the group-theoretic Reidemeister-Schreier method of [2, Thm 2.7/2.9] to the monoid STVB_n, using \"the same set Lambda_n\" as for the group TVB_n, and by asserting that all remaining relations are obtained by conjugation. No proof is supplied that for a monoid the chosen Lambda_n is a complete Schreier transversal for STVP_n in STVB_n, nor that the rewriting map f preserves the monoid congruence when the defining relations contain non-invertible generators tau_i. This is not a cosmetic gap: submonoids of monoids are not covered by the Magnus-Karrass-Solitar theorems, and a missing rewritten relation would make the presentations incomplete while leaving the epimorphism and kernel statements true. Theorems 3.3 and 3.4 are each proved by a single \"similar\" sentence, so the gap propagates to all three central claims. A separate but concrete symptom appears in Theorem 4.2: the application of Keyman's theorem lists only one-sided inverse relations (sigma_i sigma_i^{-1}=e and bar{tau_i} tau_i=e) and omits the opposite sides, so the group STVG_n as written is not guaranteed to be a group.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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].","tokens_in":13327,"tokens_out":6052,"duration_ms":56142,"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":[{"comment":"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.","section":"Section 4.1, Theorem 4.2 and definition of STVG_n"},{"comment":"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.","section":"Section 4.1, Theorem 4.2"},{"comment":"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.","section":"Section 3.1, relation (43)"}],"minor_comments":[{"comment":"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.","section":"Section 3"},{"comment":"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.","section":"Section 4.2, item (iii)"},{"comment":"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.","section":"Section 4.2, item (iv)"},{"comment":"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.","section":"Theorem 2.4"},{"comment":"There is a typo: 'moniod' should be 'monoid'.","section":"Section 4.1"}],"recommendation":"major_revision","confidential_remarks":"The paper's central claims are plausible but rest on an unproved extension of the Reidemeister-Schreier method to monoids and on incomplete verification of the group axioms in Theorem 4.2. These are fixable in principle, but the current proofs do not yet establish the main theorems. The heavy reliance on [6] and [8] for the rewriting computations, combined with the one-sentence proofs of Theorems 3.3 and 3.4, suggests that a substantial revision is needed rather than minor polishing."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Let me give you my take on Madeti and Negi's paper. The concrete new output is a set of explicit presentations for STVP_n, STVH_n, and M_n, the three kernel submonoids of STVB_n, plus an embedding of STVB_n into a group and a representation extension. Those statements are checkable and genuinely new; the authors borrow the machinery from Bardakov et al.'s twisted virtual braid group paper and from Caprau–Zepeda's virtual singular braid monoid paper, but the specific results don't appear there. If the presentations are correct, this is a useful contribution.\n\nThe soft spot is the proof of the central Theorem 3.2 and its siblings. The authors invoke the Reidemeister-Schreier method from Magnus–Karrass–Solitar and apply it directly to the monoid STVB_n, using the same Schreier transversal Lambda_n as in the group case. That is not legitimate without argument. The R-S method is for groups; to use it for a monoid you need to prove that a monoid version of the Schreier rewriting works, or show the submonoid is actually a subgroup. The paper does neither. The proof asserts that all remaining relations are obtained by conjugation, and Theorems 3.3 and 3.4 are each dismissed with 'similar'. That is a load-bearing gap: if one rewritten relation is missing, the presentations are incomplete.\n\nThere's also a concrete slip in Theorem 4.2. Keyman's theorem requires, for the set A of invertible generators, both a_i u_i = 1 and u_i a_i = 1. The paper only lists sigma_i sigma_i^{-1} = e (and rho_i^2 = e, gamma_i^2 = e), omitting sigma_i^{-1} sigma_i = e. As written, the group STVG_n is not guaranteed to be a group. This is easy to fix, but it needs fixing.\n\nThe representation extension section has some typos in the displayed verifications (e.g., sigma^{-1}_i rho_j where gamma_j is meant), but the method is standard and the idea is sound.\n\nOverall, the paper is worth a serious referee. The claims are explicit and can be checked with enough computation. But my honest assessment is that the current version is not ready: the R-S transfer needs a real proof, and the group embedding needs the missing inverse relations. A knowledgeable referee could sort this out, but the authors should be asked to fill the gaps.","headline":"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.","tokens_in":13851,"tokens_out":5219,"would_cite":false,"duration_ms":47440,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["57M50","57K12","20M32"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["singular twisted virtual braid monoid","singular twisted virtual pure braid monoid","monoid presentation","Reidemeister-Schreier method","kernel of epimorphism","symmetric group","monoid embedding into group","representation extension"],"falsifier":"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.","tokens_in":12827,"feed_emoji":"🪢","tokens_out":18656,"duration_ms":153664,"temperature":0.7,"pith_summary":"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.","feed_headline":"Singular twisted virtual braid kernels get explicit presentations","feed_subtitle":"The three kernels of the map to S_n get explicit generators and relations, opening the way to group methods","key_machinery":"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.","core_discovery":"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 $a\\phi(\\sigma_i)+b\\phi(\\sigma_i^{-1})+ce$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Provides the presentation of the base monoid STVB_n whose relations are rewritten throughout Section 3.","marker":"[7]"},{"why":"Supplies the Schreier set Lambda_n and the conjugacy computations for the twisted virtual braid group that the monoid proof transfers.","marker":"[6]"},{"why":"Gives the generator identities s_{lambda,sigma_i}=lambda_{i,i+1}^{-1} and s_{lambda,tau_i}=y_{i,i+1} used to identify the kernel generators.","marker":"[8]"},{"why":"States the Reidemeister-Schreier rewriting theorems that the paper invokes for the monoid presentations.","marker":"[2]"},{"why":"States the monoid-embedding criterion used in Theorem 4.2 to embed STVB_n into a group.","marker":"[9]"},{"why":"Establishes the representation extension Phi_{a,b,c} for the singular braid monoid, which the paper adapts to STVB_n.","marker":"[11]"}],"fun_headline_variants":["Explicit presentations for three STVB kernels","Three kernels of STVB get explicit monoid presentations","STVB kernel monoids: generators and defining relations","Monoid kernels of STVB: explicit presentations"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Explicit presentations for three STVB kernels","Three kernels of STVB get explicit monoid presentations","STVB kernel monoids: generators and defining relations","Monoid kernels of STVB: explicit presentations"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000595,"raw_usage":{"total_tokens":2825,"prompt_tokens":1024,"completion_tokens":1801,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":640,"completion_tokens_details":{"reasoning_tokens":1741}},"tokens_in":640,"tokens_out":1801,"duration_ms":13000,"temperature":1.0,"reasoning_tokens":1741,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T23:02:38.503164+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Singular twisted links and singular twisted virtual braids","cited_arxiv_id":"2403.17383","evidence_quote":"Provides the presentation of the base monoid STVB_n whose relations are rewritten throughout Section 3."},{"cited_title":"Twisted Virtual Braid Group","cited_arxiv_id":"2310.04154","evidence_quote":"Supplies the Schreier set Lambda_n and the conjugacy computations for the twisted virtual braid group that the monoid proof transfers."},{"cited_title":"Caprau, S","cited_arxiv_id":null,"evidence_quote":"Gives the generator identities s_{lambda,sigma_i}=lambda_{i,i+1}^{-1} and s_{lambda,tau_i}=y_{i,i+1} used to identify the kernel generators."},{"cited_title":"Magnus, A","cited_arxiv_id":null,"evidence_quote":"States the Reidemeister-Schreier rewriting theorems that the paper invokes for the monoid presentations."},{"cited_title":"Keyman, A class of monoids embeddable in a group, Turkish J","cited_arxiv_id":null,"evidence_quote":"States the monoid-embedding criterion used in Theorem 4.2 to embed STVB_n into a group."},{"cited_title":"Extensions of braid group representations to the monoid of singular braids","cited_arxiv_id":"2403.00516","evidence_quote":"Establishes the representation extension Phi_{a,b,c} for the singular braid monoid, which the paper adapts to STVB_n."}],"review_version":1}