Biquandle Virtual Brackets and Virtual Knotoids

Hamdi Kayaslan; Neslihan G\"ug\"umc\"u

REVIEW 3 major objections 5 minor 17 references

Biquandle Virtual Brackets and Virtual Knotoids

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The biquandle virtual bracket matrix is an invariant of virtual knotoids and properly enhances the biquandle counting invariant, the counting matrix, and the biquandle virtual bracket polynomial.

desk verdict A solid, incremental extension of biquandle virtual brackets to virtual knotoids with a genuinely new matrix invariant, but the key examples rest on unverified 'Python computations' that a referee should ask to see. read the letter →

arxiv 2507.07612 v1 pith:UCE7CTIE submitted 2025-07-10 math.AT

Neslihan Gügümcü , Hamdi Kayaslan This is my paper · ORCID

classification math.AT MSC 57K1257K14

keywords virtualknotoidsbiquandlebracketstate-suminvariantcoloringcountingmatrixenhancementknots

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

The paper is trying to establish that biquandle virtual brackets, originally defined for virtual knots, can be turned into invariants of virtual knotoids—open curves with a tail and head that may carry virtual crossings—and that the matrix form $M^\beta_X(K)$ records enough endpoint data to separate virtual knotoids that the standard counting invariants cannot. The key move is to fix the colors of the two endpoint semi-arcs and assemble the bracket state sums into a matrix indexed by pairs of biquandle elements. If the construction is right, it yields a whole family of computable invariants, since the biquandle, the ground ring, and the coefficient table can be varied. Explicit examples show virtual knotoids with identical counting invariant, counting matrix, and bracket polynomial but different bracket matrices.

What carries the argument

The central object is the biquandle virtual bracket matrix $M^\beta_X(K)$. A biquandle is a set with two binary operations whose axioms mirror the Reidemeister moves, letting semi-arcs of a diagram be labeled consistently; a biquandle virtual bracket is a state-sum weight system with six coefficient maps $A,B,V,C,D,U$ and scalars $\delta,\omega$ that satisfy 23 equations so that the sum over vertical, horizontal, and virtual smoothings is invariant under Reidemeister moves. The matrix stores these state sums by boundary data: the $(i,j)$ entry is the formal sum over colorings with the tail semi-arc colored by $x_i$ and the head semi-arc colored by $x_j$ of $u^{\beta(K_f)}$, and this organization is what reveals distinctions the aggregate invariants miss.

What would settle it

Substitute the coefficient tables from Examples 4.0.1 and 4.0.4 into equations (1)–(23) of Definition 4.0.1 and check every identity in $Z_5$ and $Z_{37}$; the claim stands only if all 23 hold. A single failure would mean the state sum changes under a Reidemeister move, so the matrix would not be an invariant.

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Extended reading notes

Core claim

The central claim is that for a finite biquandle $X$, a commutative ring $R$, and a biquandle virtual bracket $\beta$, the matrix $M^\beta_X(K)$ with entries $\sum_{f\in \operatorname{Hom}_{ij}(B(K),X)} u^{\beta(K_f)}$ is an invariant of virtual knotoids, and it is a proper enhancement of the biquandle counting invariant, the biquandle counting matrix, and the biquandle virtual bracket polynomial. The proof of invariance is adapted from the virtual-knot setting treated in the cited biquandle virtual bracket paper, and the enhancement is demonstrated by computation: the pair 2.1.1 and 3.1.1 share all three weaker invariants but have different matrices, as do 3.1.1 and 3.1.3. A table computed over $Z_{37}$ separates nearly all listed virtual knotoids up to five crossings.

Load-bearing premise

The examples stand on the assertion that the two displayed coefficient tables satisfy all 23 equations of Definition 4.0.1; the paper reports a computer check but does not show the verification, and if that assertion fails the invariant examples collapse.

Editorial extensions

If this is right

The biquandle virtual bracket matrix is an invariant of virtual knotoids for every finite biquandle, ring, and coefficient table satisfying the 23 axioms, so it provides a tunable family of invariants rather than a single fixed one.
It strictly refines the biquandle counting invariant, the biquandle counting matrix, and the biquandle virtual bracket polynomial, as shown by pairs such as 2.1.1 versus 3.1.1.
For number rings the matrix entries are polynomials in one variable $u$, making the invariant concrete and computable by state-sum enumeration.
Applied to the standard table of small virtual knotoids, the matrix distinguishes most entries up to five crossings, leaving only three unseparated couples in the displayed table.

Reading between the lines

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

The endpoint-fixing trick used here is likely to extend to linkoids and to bonded knotoids, where several distinguished arcs exist, giving analogous matrix invariants.
The main computational bottleneck is finding coefficient tables that satisfy the 23 defining equations; automating such searches over finite fields would make the invariant practical for larger biquandles and rings than the small examples used in the paper.
Because the invariant is parameterized by a biquandle and a ring, it may interpolate between purely algebraic coloring data and quantum invariants, suggesting a natural comparison with boundary-colored homology theories for open curves.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: this is a real extension of biquandle virtual brackets to virtual knotoids, with a genuinely new matrix invariant and evidence it beats the existing counting and polynomial invariants on a few low-crossing pairs. The main soft spot is that the coefficient matrices in Examples 4.0.1 and 4.0.4—the load-bearing examples—are simply asserted to satisfy the 23 axioms on the strength of 'Python computations,' with no code and no certificate. That's exactly the kind of gap that should be fixed in revision, and it's fixable, but as written a reader can't independently confirm the central enhancement claim.

What's actually new: the biquandle virtual bracket matrix M^β_X (Definition 4.0.6) and its multi-set/polynomial variants for virtual knotoids. The authors combine the biquandle virtual bracket from [16] with the tail-head coloring matrix idea from [12], and they carry out the definitions carefully. The enhancement relations in Remark 4.0.1 are clear, and the examples genuinely show that the matrix distinguishes some virtual knotoids that the counting invariant and counting matrix don't—most convincingly 2.1.1 vs 3.1.1 in Example 4.0.4, where the polynomial invariant also fails. That part is solid and useful.

The soft spots are not deal-breakers but they are real. First, the verification gap above: the 23 equations are nonlinear in the coefficients, so 'our Python computations show' is not enough for a reader to trust the examples. Second, the invariance proof for knotoids is deferred to [16], which is reasonable for a direct extension but leaves a hole in this paper. Third, trivial stuff: the text has some obvious typos/artifacts (e.g., an isolated 'moo' before Definition 4.0.1) that should be cleaned up. The citations look appropriate; the paper is transparent about building on [12] and [16], with no sign of circular fitting.

Who is this for: researchers working on knotoids and quantum invariants; it's a niche but legitimate audience. It doesn't reorganize the subject, but it gives a new computable tool that resolves a few more pairs in Bartholomew's table. I'd send it to peer review—a good referee will ask for the coefficient verification (code or a certificate) and a fuller invariance argument, and the authors can likely provide both. After those fixes, it's a publishable contribution.

Referee Report

3 major / 5 minor

Summary. The paper introduces invariants of virtual knotoids based on biquandle virtual brackets. It defines the biquandle virtual bracket value multi-set, the biquandle virtual bracket polynomial, and the biquandle virtual bracket matrix, and claims that the matrix is a proper enhancement of the biquandle counting invariant, the biquandle counting matrix, and the biquandle virtual bracket polynomial. The main evidence consists of explicit computations on virtual knotoids from Bartholomew's table, using two specific biquandle virtual brackets over finite fields. The paper also discusses the relationship between these invariants and presents a table of bracket matrices for several virtual knotoids.

Significance. If the invariants are well-defined and the computations are correct, the paper provides a useful new family of invariants for virtual knotoids, with concrete examples demonstrating that the matrix invariant is strictly stronger than previously known enhancements. The use of small biquandles and finite ground rings makes the examples computable, and the table of values for low-crossing virtual knotoids is a valuable resource. However, the central computational claim — that the displayed coefficient matrices satisfy the 23 axioms of a biquandle virtual bracket — is not independently verifiable from the manuscript, and the formal definition of the polynomial invariant does not cover the finite-field examples used. These gaps currently limit the paper's contribution to a plausible but not fully substantiated set of invariants.

major comments (3)

[§4.0.1, Examples 4.0.1 and 4.0.4] The paper asserts that the 3×18 coefficient matrices displayed in Examples 4.0.1 and 4.0.4 satisfy all 23 equations in Definition 4.0.1, with the statement 'Our Python computations show' and no further evidence. Since the definition of β(K_f) and hence of M^β_X(K) depends critically on these matrices being biquandle virtual brackets, a single failed equation would invalidate the invariant and the enhancement examples that form the paper's central claim. Please provide a verifiable certificate of the axiom verification, such as the code used, a written check of each equation, or an appendix with the verification details.
[§4.0.2, Definition 4.0.2 and the paragraph after Figure 13] The invariance of β(K_f) under the generalized Reidemeister moves for virtual knotoids is not proved; the text only says the verification is 'similar' to the virtual knot case and refers the reader to [16]. While I expect this adaptation to be routine, the knotoid setting involves endpoints and distinguished semi-arcs, and the proof should at least be sketched, especially for the virtual Reidemeister moves and the detour move, since the bracket includes virtual smoothings. Please state and prove the invariance theorem for β(K_f) or give a precise argument showing how the proof in [16] carries over.
[Definition 4.0.5 and Examples 4.0.1–4.0.4] The polynomial invariant Φ^β_X(K) and the matrix M^β_X(K) are defined only for a number ring R, but all examples in §4 use R = Z_5 or R = Z_37, which are finite fields and not number rings. The notation u^{β(K_f)} is not well-defined when β(K_f) is an element of a finite field, unless one chooses a specific representative for each residue class. The manuscript does not state such a convention, and the resulting polynomial would depend on the choice of representatives. Please either restrict the examples to genuine number rings or extend the definition with an explicit convention for finite rings and explain why the invariant is well-defined under that convention.

minor comments (5)

[§4, heading] The word 'moo' appears on the line before 'Biquandle virtual brackets were introduced...'; this appears to be a stray insertion and should be removed.
[Definition 3.1.1] The exchange laws are not clearly typeset; the two binary operations are not distinguished in the plain text, making the axioms hard to read. Please ensure the two operation symbols are visually distinct in the final version.
[Example 4.0.1] The operations x ▷ y = 2x + 1 = x ▷ y are described as defining an 'Alexander biquandle', but this is not an Alexander biquandle as defined in Example 3.1.1, since the second operation is not linear in y. Either correct the terminology or remove the characterization.
[Table 2] The table is titled 'Φ^β_X (K)' in the header, but the entries are matrices of the form M^β_X(K); the caption should be changed to reflect that the table lists biquandle virtual bracket matrices.
[§3.2, Proposition 3.2.1 and surrounding text] The proofs of Proposition 3.2.1 and Theorem 3.2.1 are extremely brief, and Proposition 3.2.2 has no proof. While these are standard arguments, a short justification for each would improve self-containedness.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the new invariants are constructed from prior biquandle virtual bracket and counting-matrix machinery, and the enhancement claims are checked on explicit diagrams rather than being forced by definition.

full rationale

I find no circular step. The biquandle virtual bracket matrix M^beta_X(K) (Definition 4.0.6) is defined as a state-sum over X_ij-colorings using the biquandle virtual bracket beta from prior work [16] and the matrix-format idea from [11,12]; the claimed enhancement relations are established first by construction (the cardinality of each entry's multiset gives the counting matrix entry) and, for proper enhancement, by explicit examples such as Example 4.0.4 and Table 2, where specified biquandles, rings, and coefficient matrices are fixed and the invariant values are then computed. No fitted parameter is relabeled as a prediction: the coefficient matrices in Examples 4.0.1 and 4.0.4 are exhibited as solutions of the 23 axioms, and the invariant values are computed from them, not used to define them. The self-citations in the introduction and Section 2 (e.g., [10]) provide background and a topological embedding theorem, but the invariant construction and enhancement claims do not depend on those cited results. Two support gaps deserve note but are not circularity: the verification that the displayed coefficient matrices satisfy Definition 4.0.1 is asserted only as "Our Python computations show" (Examples 4.0.1 and 4.0.4), and the invariance proof for knotoids is deferred to [16] in the paragraph before Definition 4.0.2. These are correctness and verification concerns, not instances where a claimed output equals an input by construction.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central examples rely on two specially chosen sets of bracket coefficient maps and on prior definitions and results. The satisfiability assertion for the coefficient matrices is the main unverified input; the rest are standard domain assumptions.

free parameters (2)

Biquandle virtual bracket coefficients (A,B,V,C,D,U) in Example 4.0.1 = 3x18 matrix over Z5 with delta=2, omega=4
Chosen by Python search to satisfy the 23 bracket axioms; the multi-set and matrix values for 2.1.1 and 4.1.1 depend on this choice.
Biquandle virtual bracket coefficients (A,B,V,C,D,U) in Example 4.0.4 = 3x18 matrix over Z37 with delta=5, omega=9
Chosen by Python search to satisfy the bracket axioms; the table of matrix invariants (Table 2) is computed with this bracket.

assumptions (5)

standard math Biquandle axioms (Definition 3.1.1): the operations satisfy the three conditions (right/left inverses, exchange laws).
Standard algebraic structure used as the coloring set; not proved in the paper.
domain assumption Biquandle virtual bracket axioms (Definition 4.0.1, equations (1)-(23)) taken from [16].
The paper assumes these equations define a valid bracket; they are quoted from the prior work of Nelson et al. and not re-derived.
domain assumption The biquandle virtual bracket state-sum is invariant under generalized Reidemeister moves for virtual knotoids.
The paper states the verification is similar to [16] and does not provide it; this invariance underpins the invariant status of the new constructions.
ad hoc to paper The coefficient matrices in Examples 4.0.1 and 4.0.4 satisfy all equations of Definition 4.0.1.
This is asserted on the basis of Python computations that are not shown; the correctness of the example values depends on it.
domain assumption Bartholomew's table of virtual knotoids and the labeled peer codes [2] correctly represent the virtual knotoids.
The computational examples and Table 2 rely on diagrams generated from this table.

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Cite this review

Pith. "Pith review of Biquandle Virtual Brackets and Virtual Knotoids." pith.science (2026). https://pith.science/paper/UCE7CTIE

@misc{pith2026250707612,
  author       = {Pith},
  title        = {Pith review of: Biquandle Virtual Brackets and Virtual Knotoids},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UCE7CTIE}},
  note         = {Machine review of arXiv:2507.07612}
}

read the original abstract

In this paper, we introduce invariants of virtual knotoids based on biquandles and biquandle virtual brackets. We show that one of these invariants, namely biquandle virtual bracket matrix, is a proper enhancement of the other invariants introduced in this paper.

Figures

Figures reproduced from arXiv: 2507.07612 by the authors.

**Figure 1.** Extended Reidemeister moves. Definition 2.0.2. A knotoid is an equivalence class of knotoid diagrams in a surface with respect to the equivalence relation induced by Reidemeister moves and isotopy of the surface. The set of knotoids on a surface Σ is denoted by K(Σ). A knotoid in S 2 is called a classical knotoid. Classical knotoids were extended to virtual knotoids in [7]. Let us make a brief review of this extensi… view at source ↗

**Figure 2.** A virtual knotoid diagram. Two virtual knotoid diagrams are considered to be equal if one can be turned into another via a 3 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** A virtual arc in a thickened torus whose endpoints are attached to two line segments [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (9 more)

**Figure 4.** Figure 4: Product of two virtual knotoid diagrams in [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

**Figure 5.** Figure 5: Biquandle labelings around classical and virtual crossings. [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: A virtual knotoid diagram and its crossing relations. [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 8.** Figure 8: Fundamental biquandle elements around a crossing and their images under a [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]

**Figure 9.** Figure 9: Distinguishing two virtual knotoids by biquandle colorings. [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]

**Figure 11.** Figure 11: Vertical, horizontal and virtual smoothings of a classical crossing. [PITH_FULL_IMAGE:figures/full_fig_p014_11.png]

**Figure 12.** Figure 12: A labeled virtual knotoid diagram and its states. [PITH_FULL_IMAGE:figures/full_fig_p014_12.png]

**Figure 13.** Figure 13: Skein relations for X-virtual bracket. Definition 4.0.3. We compute the X-virtual bracket value of all possible X-colorings of K and form a multi-set of these values, which is denoted by Φβ,M X (K). More formally, Φ β,M X (K) = {β(Kf ) | f ∈ Hom(B(K), X)}. It is clear…

**Figure 14.** Figure 14: A labeled virtual knotoid diagram and its state contributions. [PITH_FULL_IMAGE:figures/full_fig_p016_14.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [16]

Biquandle virtual brackets

Sam Nelson, Kanako Oshiro, Ayaka Shimizu, and Yoshiro Yaguchi. Biquandle virtual brackets. Journal of Knot Theory and Its Ramifications , 28(11):1940003, 2019

work page 2019

[2] [12]

Biquandle brackets and knotoids

Neslihan G¨ ug¨ umc¨ u, Sam Nelson, and Natsumi Oyamaguchi. Biquandle brackets and knotoids. Journal of Knot Theory and Its Ramifications , 30(09):2150064, 2021

work page 2021

[3] [1]

Knots related by knotoids

Colin Adams, Allison Henrich, Kate Kearney, and Nicholas Scoville. Knots related by knotoids. The American Mathematical Monthly , 2019

work page 2019

[4] [2]

A table of virtual links

Andrew Bartholomew. A table of virtual links. https://www.layer8.co.uk/maths/ virtual-links/index.htm, 2022

work page 2022

[5] [3]

Thistlethwaite Theorems for Knotoids and Linkoids

Sergei Chmutov, Qingying Deng, Joanna A Ellis-Monaghan, Sergei Lando, and Wout Moltmaker. Thistlethwaite theorems for knotoids and linkoids. arXiv preprint arXiv:2412.12357, 2024

work page Pith review arXiv 2024

[6] [4]

Studies of global and local entanglements of individual protein chains using the concept of knotoids

Dimos Goundaroulis, Julien Dorier, Fabrizio Benedetti, and Andrzej Stasiak. Studies of global and local entanglements of individual protein chains using the concept of knotoids. Scientific reports, 7(1):6309, 2017

work page 2017

[7] [5]

Topological models for open-knotted protein chains using the concepts of knotoids and bonded knotoids

Dimos Goundaroulis, Neslihan G¨ ug¨ umc¨ u, Sofia Lambropoulou, Julien Dorier, Andrzej Stasiak, and Louis Kauffman. Topological models for open-knotted protein chains using the concepts of knotoids and bonded knotoids. Polymers, 9(9):444, 2017

work page 2017

[8] [6]

Invariants of bonded knotoids and applications to protein folding

Neslihan G¨ ug¨ umc¨ u, Bostjan Gabrovsek, and Louis H Kauffman. Invariants of bonded knotoids and applications to protein folding. Symmetry, 14(8):1724, 2022

work page 2022