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arxiv: 2408.01891 · v2 · submitted 2024-08-04 · 🧮 math.GT

On the determinant of checkerboard colorable virtual knots

Tomoaki Hatano , Yuta Nozaki This is my paper

Pith reviewed 2026-05-23 22:24 UTC · model grok-4.3

classification 🧮 math.GT
keywords virtual knotsdeterminantascending polynomialcheckerboard colorableConway polynomialArf invariantMurasugi theorem
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The pith

The determinant modulo 8 of checkerboard colorable virtual knots is classified by the coefficient of z squared in the ascending polynomial.

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

The paper extends Murasugi's classical theorem, which links the knot determinant modulo 8 to the Arf invariant, to the setting of checkerboard colorable virtual knots. Boden and Karimi defined a determinant in this virtual setting. The authors prove that this determinant modulo 8 is instead determined by the z squared coefficient in the ascending polynomial, an extension of the Conway polynomial. A reader cares because the result supplies a concrete polynomial test that distinguishes these virtual knots in a manner parallel to the classical case.

Core claim

We prove that the determinant of checkerboard colorable virtual knots modulo 8 is classified by the coefficient of z^2 in the ascending polynomial, an extension of the Conway polynomial for classical knots.

What carries the argument

The ascending polynomial, defined as an extension of the Conway polynomial that applies to checkerboard colorable virtual knots and carries the mod-8 classification.

If this is right

  • The mod-8 determinant of these virtual knots can be read off from a single coefficient in the ascending polynomial.
  • The result supplies a direct generalization of Murasugi's theorem that applies beyond classical knots.
  • Any two checkerboard colorable virtual knots with different z squared coefficients must have determinants that differ modulo 8.

Where Pith is reading between the lines

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

  • The ascending polynomial may serve as a practical computational tool for testing virtual knot distinctions that were previously only accessible via the determinant.
  • Further relations between the ascending polynomial and other virtual knot invariants could be checked by evaluating both on the same families of examples.

Load-bearing premise

The ascending polynomial behaves enough like the Conway polynomial on checkerboard colorable virtual knots for the classical mod-8 relation to transfer.

What would settle it

Pick one checkerboard colorable virtual knot, compute its determinant modulo 8 and the coefficient of z squared in its ascending polynomial, and check whether the two quantities satisfy the claimed classification.

Figures

Figures reproduced from arXiv: 2408.01891 by Tomoaki Hatano, Yuta Nozaki.

Figure 1
Figure 1. Figure 1: A virtual knot diagram, the corresponding Gauss diagram, and (projection of) knot in a thickened torus. surface. For its precise definition and related topics, we refer the reader to [14], [2], [4], and [6]. Boden, Chrisman, and Karimi [1] introduced a determinant of a pair (L, F), where L is a link in Σ × [0, 1] with [L] = 0 in H1(Σ × [0, 1]; Z/2Z) and F is a spanning surface for L. Their determinant can … view at source ↗
Figure 2
Figure 2. Figure 2: Colorings of short arcs around positive/negative crossings and virtual crossing. Example 2.2. The virtual knot in [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: On the other hand, one can see that it is not almost classical. [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 3
Figure 3. Figure 3: A mod 2 Alexander numbering of a diagram of 4.90 and labels of crossings and long arcs. is the n × m matrix such that bij = 1 if aj is both the over-crossing arc and one of the under-crossing arcs at ci , otherwise bij =    2 if aj is the over-crossing arc at ci , −1 if aj is one of an under-crossing arc at ci , 0 otherwise. Example 2.7. For the labels of the diagram of K = 4.90 on the right of [PITH… view at source ↗
Figure 4
Figure 4. Figure 4: Oriented virtual links L+, L−, L0 that are iden￾tical except within a 3-ball [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Spanning surfaces F+, F−, F0 that are identical except within a 3-ball. Remark 2.14. When F0 is disconnected, we obtain a spanning surface by attaching a tube to F0. The mock Seifert matrix for the resulting spanning surface satisfies Theorem 2.13 because then det S+ = det S− and det S0 = 0. Proof of Theorem 2.13. Fix a basis of H1(F0; Z) and let S0 be the corre￾sponding mock Seifert surface for F0. Since … view at source ↗
Figure 6
Figure 6. Figure 6: Conway combinations C1, C ′ 1 , C2, and C ′ 2 . Definition 3.3. For a based Gauss diagram G, define the ascending polyno￾mial ∇asc(G)(z) and descending polynomial ∇des(G)(z) ∈ Z[z] respectively by ∇asc(G)(z) = X i≥0 ⟨C2i , G⟩z 2i and ∇des(G)(z) = X i≥0 ⟨C ′ 2i , G⟩z 2i . Remark 3.4. The polynomials ∇asc(G)(z) and ∇des(G)(z) depend on the choice of a basepoint of G in general. For instance, one can observe … view at source ↗
Figure 7
Figure 7. Figure 7: Descending diagram with isolated 2n arrows. Lemma 4.2. Let Gb be a based Gauss diagram of mod p almost classical oriented virtual knot and let a be an arrow. The a divides the circle into two arcs and suppose that the tail of any arrow intersecting with a is attached to the arc containing b. Let Hb be the 2-component based Gauss diagram obtained by smoothing Gb along a. Then, ⟨C1, Hb⟩ ≡ ⟨C ′ 1 , Hb⟩ ≡ 0 mo… view at source ↗
Figure 8
Figure 8. Figure 8: Ascending diagrams with basepoints b and b ′ [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
read the original abstract

For classical knots, Murasugi showed that the determinant modulo $8$ is classified by the Arf invariant. Boden and Karimi introduced a determinant for checkerboard colorable virtual knots. We prove that this determinant modulo $8$ is classified by the coefficient of $z^2$ in the ascending polynomial, an extension of the Conway polynomial for classical knots.

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 / 3 minor

Summary. The manuscript proves that for checkerboard colorable virtual knots the determinant (introduced by Boden and Karimi) modulo 8 is classified by the coefficient of z² in the ascending polynomial, an extension of the Conway polynomial; this extends Murasugi's classical theorem relating the determinant mod 8 to the Arf invariant.

Significance. If the result holds, the work supplies a concrete invariant relation that classifies a family of virtual knots and directly connects classical and virtual knot theory through an extension of the Conway polynomial. The parameter-free character of the claimed mod-8 classification is a strength.

minor comments (3)
  1. The introduction should explicitly recall the definition of the ascending polynomial (including its relation to the Conway polynomial) before stating the main theorem, to aid readers who may not be familiar with virtual-knot extensions.
  2. Notation for the determinant and the coefficient of z² should be introduced with a single consistent symbol set in §2 or §3; current usage appears to switch between det and Δ without a clarifying sentence.
  3. A short table or list comparing the classical Murasugi statement, the Boden-Karimi determinant, and the new virtual statement would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, recognition of the result's significance in connecting classical and virtual knot theory, and recommendation of minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper defines the ascending polynomial independently as an extension of the Conway polynomial and defines the determinant for checkerboard-colorable virtual knots following Boden-Karimi. It then proves a relation between the determinant mod 8 and the z^2 coefficient, extending Murasugi's classical result. No equations reduce one quantity to the other by construction, no fitted parameters are relabeled as predictions, and no load-bearing step relies on a self-citation chain or imported uniqueness theorem. The central claim is a proof relating two separately defined invariants, with the extension justified by explicit construction rather than ansatz smuggling or renaming. This matches the default expectation of a non-circular derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The proof rests on standard algebraic properties of knot polynomials and the prior definitions of the determinant and ascending polynomial; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption The ascending polynomial extends the Conway polynomial while preserving the mod-8 determinant relation on checkerboard colorable virtual knots.
    Invoked to transfer Murasugi's classical result to the virtual setting.

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Reference graph

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