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arxiv: 1907.08709 · v1 · pith:WHR7LDASnew · submitted 2019-07-19 · 🧮 math.GT · math.GN

Virtual Parity Alexander Polynomial

Heather A. Dye , Aaron Kaestner This is my paper

Pith reviewed 2026-05-24 18:38 UTC · model grok-4.3

classification 🧮 math.GT math.GN
keywords virtual knotsAlexander polynomialparityknot invariantsunknottingcrossing changesvirtual knot theory
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The pith

The parity virtual Alexander polynomial shows many virtual knots resist unknotting by odd-crossing changes alone.

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

The paper defines a parity version of the virtual Alexander polynomial for virtual knots and studies its basic properties through explicit computations. The central result is that this invariant obstructs unknotting operations that change only odd crossings, proving that many virtual knots cannot be reduced to the unknot under that restriction. A sympathetic reader cares because the obstruction supplies a concrete test that distinguishes virtual knots according to which crossings can be altered. The work follows prior constructions of virtual Alexander polynomials but adds parity to capture this new limitation on unknotting.

Core claim

The parity virtual Alexander polynomial is defined by incorporating parity data into the virtual Alexander polynomial; its values on examples demonstrate that many virtual knots cannot be unknotted by crossing changes performed exclusively on odd crossings.

What carries the argument

The parity virtual Alexander polynomial, which augments the standard virtual Alexander polynomial with parity information to produce an obstruction for restricted unknotting.

If this is right

  • Non-trivial values of the polynomial on a given virtual knot imply it cannot be unknotted by odd-crossing changes.
  • The invariant can be evaluated on families of virtual knots to classify which ones admit the restricted unknotting operation.
  • Properties such as invariance under virtual knot moves are preserved while the parity component adds the new obstruction.
  • Computed examples establish that the obstruction applies to multiple distinct virtual knots.

Where Pith is reading between the lines

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

  • The same parity mechanism might be applied to other polynomial invariants to produce further obstructions.
  • This approach could be tested on virtual links to see whether the obstruction generalizes beyond single-component knots.
  • If the polynomial factors in a recognizable way, it might relate the parity obstruction to classical Alexander polynomial factors.

Load-bearing premise

The definition of the parity virtual Alexander polynomial is well-defined and its values on examples correctly detect the claimed unknotting obstruction.

What would settle it

An explicit computation of the polynomial on a virtual knot that is known to unknot under odd-crossing changes alone, yielding a non-trivial value, would falsify the obstruction claim.

Figures

Figures reproduced from arXiv: 1907.08709 by Aaron Kaestner, Heather A. Dye.

Figure 1
Figure 1. Figure 1: Reidemeister moves 1 arXiv:1907.08709v1 [math.GT] 19 Jul 2019 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Virtual Reidemeister moves a b c [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Labeled knot diagram crossings (indicated by a circled crossing). Two virtual knot diagrams, K1 and K2, are equivalent if one can be transformed into the other by a sequence of Reidemeister moves and virtual Reidemeister moves [14]. A virtual knot is an equivalence class of virtual link diagrams determined by the Reidemeister moves and the virtual Reidemeister moves. For convenience, we collectively refer … view at source ↗
Figure 4
Figure 4. Figure 4: Labeled crossings even. The parity of a crossing abstractly gives information about the planarity of the dia￾gram. For more information about even and odd crossings and their interactions with the Reidemeister moves, see Chrisman and Dye [4], or Kaestner and Kauffman [10]. 3. The virtual parity group The virtual parity group of a virtual knot diagram K, P GK, is a free group modulo relations determined by … view at source ↗
Figure 5
Figure 5. Figure 5: Labeled Reidemeister II x y z f a b c d e ↔ c d e a b x y z [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Labeled Reidemeister III Proof. We show invariance under diagrammatic that include odd crossings. Invariance under moves involving only even crossings or even and virtual crossings are shown in BDGGHN [1]. The case of a Reidemeister III move involving even and odd crossings is analogous to the case of a virtual Reidemeister IV move that includes an even crossing. We begin by examining a Reidemeister II mov… view at source ↗
Figure 7
Figure 7. Figure 7: Virtual knot examples Φ∆(K3.1) = 1 q + q st − θ 2 q + θ 2 stq (15) Φ∆(K4.7) = 1 q (16) − q Φ∆(K4.9) = −1 + 1 s 2t 2 + 1 sq − 1 s 2tq − q st2 + q t (17) Φ∆(K6.32008) = 1 − 1 st + 1 sq − t q − sq + q t − q θ + stq θ (18) We see that the virtual parity Alexander polynomial does not vanish on the knot 6.32008, whereas the parity Alexander polynomial (Kaestner and Kauffman [10]) does vanish. 5. Properties We st… view at source ↗
Figure 8
Figure 8. Figure 8: Crossing under symmetry Example 5.2.1. The diagram of K3.1 contains two odd crossings and two virtual crossings. The sign of either of the two crossings can be switched without changing the value of the polynomial. The diagram of K4.7 contains four odd crossings and two virtual crossings; This diagram cannot be turned into a classical diagram by changing the sign of the odd crossings. The diagram of K4.9 c… view at source ↗
read the original abstract

In this paper, we define the parity virtual Alexander polynomial following the work of BDGGHN [1] and Kaestner and Kauffman [10]. The properties of this invariant are explored and some examples are computed. In particular, the invariant demonstrates that many virtual knots can not be unknotted by crossing change on only odd crossings.

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

1 major / 0 minor

Summary. The manuscript defines the parity virtual Alexander polynomial following BDGGHN and Kaestner-Kauffman, explores its properties, computes examples, and claims that the invariant shows many virtual knots cannot be unknotted by crossing changes on only odd crossings.

Significance. If the obstruction property holds, the invariant supplies a computable tool for detecting parity-constrained unknotting obstructions in virtual knot theory, extending existing Alexander-type invariants with parity filtration.

major comments (1)
  1. The central claim (that the polynomial is invariant under virtual Reidemeister moves yet differs from the unknot value precisely when no odd-crossing unknotting sequence exists) is asserted via examples in the abstract, but the manuscript supplies neither the explicit module definition nor the parity-filtered Fox calculus nor a lemma establishing the obstruction property; without these, non-trivial example values do not entail the claimed obstruction.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the single major comment below.

read point-by-point responses

  1. Referee: The central claim (that the polynomial is invariant under virtual Reidemeister moves yet differs from the unknot value precisely when no odd-crossing unknotting sequence exists) is asserted via examples in the abstract, but the manuscript supplies neither the explicit module definition nor the parity-filtered Fox calculus nor a lemma establishing the obstruction property; without these, non-trivial example values do not entail the claimed obstruction.

    Authors: We agree that the obstruction property must be established rigorously rather than illustrated only by examples. The manuscript defines the parity virtual Alexander polynomial by direct adaptation of the module and Fox calculus constructions from the cited references BDGGHN and Kaestner-Kauffman, but we acknowledge that an explicit spelling-out of the parity-filtered version and a dedicated lemma showing invariance under odd-crossing changes (hence equality to the unknot polynomial whenever such a sequence exists) would make the central claim self-contained. We will incorporate both the expanded definition and the lemma in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

Minor self-citation for definition; central claim via explicit example computations

full rationale

The paper adopts its definition of the parity virtual Alexander polynomial directly from cited external works [1] (BDGGHN) and [10] (Kaestner-Kauffman, overlapping one author). The claimed obstruction property for odd-crossing unknotting is asserted via computed examples on specific virtual knots rather than any derivation that reduces by construction to the definition itself, a fitted parameter, or a self-citation chain. No self-definitional equations, renamed known results, or load-bearing uniqueness theorems appear. This constitutes normal citation practice with independent computational content.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Based on abstract only; the paper relies on background from cited references for the definition of virtual Alexander polynomials and parity.

axioms (1)
  • domain assumption Properties of virtual knot diagrams and the virtual Alexander polynomial as defined in BDGGHN [1] and Kaestner-Kauffman [10]
    The new polynomial is defined following those works.
invented entities (1)
  • parity virtual Alexander polynomial no independent evidence
    purpose: New invariant for virtual knots incorporating parity
    Introduced in this paper as the central object

pith-pipeline@v0.9.0 · 5563 in / 1156 out tokens · 18890 ms · 2026-05-24T18:38:11.666881+00:00 · methodology

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Reference graph

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