Classification of virtual links by arc shift move

Aastha Sahore; Amrender Singh Gill; Komal Negi; Madeti Prabhakar

REVIEW 5 major objections 8 minor 11 references

Classification of virtual links by arc shift move

T0 review · 5 major / 8 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Two virtual links are arc shift equivalent exactly when the parities of their virtual linking numbers match.

desk verdict A parity invariant and a lower bound for arc shift number are solid, but the classification theorem rests on an unproved normalization algorithm. read the letter →

arxiv 2502.08955 v1 pith:JQDOCZ2O submitted 2025-02-13 math.GT

Aastha Sahore , Komal Negi , Amrender Singh Gill , Madeti Prabhakar This is my paper · ORCID

classification math.GT MSC 57K1057K12

keywords arcshiftnumbervirtuallinksoddwritheequivalencelinkingnumbersGaussdiagramsunknottingoperationn-homogeneousproper

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 proves that the arc shift move, known to unknot virtual knots, does not unknot all virtual links: a link can be trivialized by arc shifts exactly when every virtual linking number between its components is even (the n-homogeneous proper case). More generally, two n-component virtual links are arc shift equivalent precisely when the parities of all their virtual linking numbers agree, so there are $2^{\binom{n}{2}}$ equivalence classes. The paper also defines the arc shift number for n-homogeneous proper links, proves it is bounded below by half the total odd writhe, and gives sequences of diagrams for which the upper bound of the arc shift number is exactly n. The interest is that an operation too weak to unknot all links still yields a finite, computable classification, in contrast to forbidden-move and Xi-move settings where classes are infinite.

What carries the argument

The machinery is the arc shift move on Gauss diagrams plus the parity of virtual linking numbers. In a Gauss diagram, each crossing is a directed chord from the over-passing component to the under-passing component; an arc shift slides one endpoint past an adjacent endpoint and flips both chord signs, and the paper's key observation is that every variant of the move preserves the parity of each $L^i_j$ while changing the value by an even integer. A link is n-homogeneous proper when all these parities are even, which is exactly the condition under which the reduction algorithm can align self-chords, remove them by RI, then align mixed chords and cancel them in opposite-sign RII pairs. The canonical representatives $L(p,q)$ for $p,q \in \{0,1\}$ encode the surviving parity patterns, and the odd writhe $J(L)$ supplies the lower bound for the arc shift number.

What would settle it

Build the Gauss diagram of a two-component link with four mixed chords, all parallel, all directed from component 1 to component 2, and all with the same sign; then $L^1_2=4$ and $L^2_1=0$, so the link is 2-homogeneous proper and Theorem 3.12 predicts arc shift triviality. Since arc shifts, by the paper's own observation, never invert a chord's direction, and RII cancellation requires a pair of opposite directions, this diagram could not be reduced to the empty diagram. If such a diagram exists as a virtual link, the 'if' direction of the classification fails; if it can be trivialized, the reduction handles a case the written argument does not cover.

Watch

Extended reading notes

Core claim

The central claim is Theorem 3.21: for ordered n-component virtual links, arc shift equivalence is classified by the parity of the virtual linking numbers $L^i_j$ and $L^j_i$, where $L^i_j$ is the sum of signs of the crossings in which component $i$ overpasses component $j$. The forward direction is invariance of parity under every arc shift variant; the reverse direction is a reduction argument on Gauss diagrams. When all these numbers are even, the link is n-homogeneous proper and arc shift becomes an unknotting operation (Theorem 3.12 and Corollary 3.14), so the two-component pattern L(0,0), L(1,0), L(0,1), L(1,1) generalizes to $2^{\binom{n}{2}}$ classes. Non-homogeneous links survive as representatives $L(p,q)$ with $p,q \in \{0,1\}$, and the mirror image is arc shift equivalent to the original iff all parities agree. The paper's arc shift number results show the odd writhe $J(L)$ controls the cost of unknotting from below, while (2,4n)-virtual torus links give explicit upper bounds.

Load-bearing premise

The proof depends on the reduction step that every Gauss diagram can, using only arc shifts and Reidemeister moves, be brought to a canonical form with all self-chords parallel and all mixed chords arranged in canceling opposite-sign pairs; if some diagram resists this reduction, the classification collapses.

Editorial extensions

If this is right

For n-component virtual links, arc shift equivalence has exactly $2^{\binom{n}{2}}$ classes, one for each parity pattern of the ordered virtual linking numbers.
A virtual link can be unknotted by arc shifts if and only if it is n-homogeneous proper, meaning every virtual linking number is even; any odd entry makes trivialization impossible.
For n-homogeneous proper links, the arc shift number is an invariant and is at least $J(L)/2$, where $J(L)$ is the total odd writhe.
There are explicit sequences of 2-homogeneous proper link diagrams, such as the $(2,4n)$-virtual torus links, whose arc shift number is at most n.
The mirror image of a link is arc shift equivalent to the original exactly when all virtual linking numbers of the link have the same parity.

Reading between the lines

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

If the classification is right, arc shift equivalence becomes a purely algebraic quotient: the parity matrix of the virtual linking numbers is a complete invariant, so no diagram-level search is needed to tell two links apart.
The same parity data gives a practical route to arc shift numbers: the paper's reduction supplies an upper bound from the number of moves used, while $J(L)/2$ supplies a lower bound, leaving equality as a testable condition on specific families.
The proof's reduction step is the place to look for a sharper statement: because arc shifts cannot reverse chord directions, any parity-even diagram whose mixed chords all point one way could be an obstruction to the RII pairing step, suggesting an extra invariant beyond parity.
The finite quotient may generalize to welded or fused settings, where virtual linking numbers are already known to classify, turning an infinite classification into a coarse but computable one.

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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

Take this one for the parity invariant and the arc shift number lower bound, but not for the classification theorem as written. The forward direction is clean: Lemma 3.6 proves parity of virtual linking numbers is invariant under arc shifts, and Corollary 3.7 shows odd parity prevents trivialization. The lower bound A(L) ≥ J(L)/2 in Theorem 3.29 is a nice extension of the knot case. The examples, especially the virtual Hopf link and the (2,4n)-virtual torus links, are instructive and clearly drawn.

The problem is the converse. The central claims—Theorem 3.12 for 2-homogeneous proper links, Proposition 3.16 for odd 2-component links, and Theorem 3.21 for n-component links—all rely on a normalization algorithm that is described only informally. In the proof of 3.12, you are told to align self-chords, then mixed chords, then sort by direction, then pair signs with As, but the procedure is not proved to terminate or to be independent of the initial diagram, and the sign-pairing step is asserted. Proposition 3.16 says "we will ultimately obtain" and then lists the three parity diagrams without giving the move sequence. Theorem 3.21 simply cites "the reducing algorithm, depicted in Theorem 3.12," which only handles the even case. No canonical representative for n-component odd-parity links is even defined, so the finite classification with 2^(n choose 2) classes has no demonstrated upper bound.

The stress-test note is right: the "cannot invert chords" remark is not by itself a contradiction, because As changes signs but leaves directions, and RII cancels opposite-sign parallel chords. The real defect is that the paper never proves every diagram can be brought to the required parallel form. That gap is load-bearing.

Still, the paper is not nonsense. The invariant is genuine, the definitions are natural, and the examples are useful. It deserves a serious referee—the classification claim is plausible and the gap may be fillable—but as submitted the central theorem is not supported. I would send it to peer review with a clear request for a complete proof of the reduction algorithm, and I would not cite the classification in my own work until that appears.

Referee Report

5 major / 8 minor

Summary. This paper studies the arc shift move for n-component virtual links. The main claims are: (1) an n-component virtual link is unknottable by arc shift moves exactly when it is n-homogeneous proper, i.e. when all virtual linking numbers are even (Theorem 3.12 for n=2 and Corollary 3.14 by induction); (2) arc shift equivalence of n-component virtual links is classified by the parities of the virtual linking numbers, giving finitely many classes (Theorem 3.21 and Remark 3.23); and (3) the arc shift number A(L) of an n-homogeneous proper link is bounded below by J(L)/2, where J is the odd writhe (Theorem 3.29), with explicit diagram sequences satisfying A(D) ≤ n (Theorem 3.30 and Example 3.31). The invariance half of the story (Lemma 3.6 and the forward directions of the theorems) is straightforward and correct. The converse of the classification, however, is the entire substance of the paper and it rests on informally described reduction procedures that are not proved.

Significance. If the classification were proved, it would be a clean and striking result: arc shift equivalence would partition n-component virtual links into finitely many classes, in contrast to the infinite classifications under Ξ-moves and forbidden moves cited in the paper. The parity invariance lemma (Lemma 3.6) is correct and clean, and the forward directions of Theorems 3.12 and 3.21 follow directly from it. The worked reductions in Remarks 3.9–3.11 and Examples 3.13 and 3.27 are reproducible and illustrate a plausible reduction mechanism, and the upper-bound constructions for the arc shift number (Theorem 3.30 and Example 3.31) are explicit. The paper's central contribution is not, however, established: the converse direction of the classification is asserted rather than proved, and the class count in Remark 3.23 is internally inconsistent. The paper is not suitable for publication in its present form, but the underlying idea may be repairable.

major comments (5)

[Theorem 3.12 (⇐) proof] The converse direction of Theorem 3.12 is the load-bearing step for the unknotting claim, but the proof is a procedural description, not a proof. The claims 'Repeating this process for all self-chords yields a Gauss diagram where no two chords intersect' and 'Continue this process until the selected chord occupies the same position in both components' assert that certain arc shift sequences exist, terminate, and do not disturb previously sorted chords, but no argument is given for any of these properties; Remarks 3.9–3.11 illustrate the procedure only on the specific diagrams in Figures 13–15. In particular, the final step requires the RII-removability conditions (adjacent endpoints, opposite signs, compatible directions) to hold simultaneously, and it is never shown that the sorting procedure produces a configuration satisfying them.
[Proposition 3.16] Proposition 3.16 is the only passage from the homogeneous-proper case to arbitrary 2-component links, but its proof consists of the assertions 'we will ultimately obtain the Gauss diagram of L(1,1)' and 'the Gauss diagram can be transformed into either L(1,0) or L(0,1)', with no move sequence and no argument that the crossings can be made to have the uniform signs required by Definition 3.15. Since Theorem 3.21 and Remark 3.23 both depend on this proposition, the non-homogeneous case of the classification is unsupported.
[Theorem 3.21 (ii)⇒(i)] The converse of the classification theorem is justified only by the sentence 'By the reducing algorithm, depicted in Theorem 3.12, both L and L′ can be reduced... to same virtual link L′′.' That algorithm is described, incompletely, for n-homogeneous proper links (all virtual linking numbers even); for general parity data no n-component normal form is even defined, and Proposition 3.16 treats only the two-component case. Thus the if-direction of Theorem 3.21 is not established, and the finiteness claim of Remark 3.23 has no demonstrated upper bound.
[Remark 3.23] The count 2^(n choose 2) is internally inconsistent: for n = 2 it gives 2, contradicting the paper's own conclusion (Remark 3.17 and Figure 19) that there are exactly four 2-component classes, a fact forced by Lemma 3.6 since L(0,0), L(1,0), L(0,1), and L(1,1) have mutually different parity pairs. Theorem 3.21(ii) concerns the n(n−1) ordered-pair parities, so the formula consistent with the stated invariant would be 2^(n(n−1)); the remark as written cannot be correct under either reading of the invariant.
[Theorem 3.28 proof] The proof of Theorem 3.28 considers only the sign and parity change of the selected crossing c1, but an arc shift moves a chord endpoint past other chords, which changes the odd/even status of every chord whose endpoint-count interval is affected; these additional contributions to J(K_i) are not accounted for. In the self-crossing case the proof invokes Theorem 2.18, but Theorem 2.18 is a consequence of a per-move bound on J (via the telescoping argument), not a proof of such a bound. Since Theorem 3.29's inequality A(L) ≥ J(L)/2 telescopes the bound |J(L_{i+1}) − J(L_i)| ≤ 2 obtained from Theorem 3.28, the lower bound result inherits this gap.

minor comments (8)

[Proposition 3.20 proof] The reference 'Corollary 3.16' (twice) should be 'Proposition 3.16'.
[Theorem 3.21 statement] The formula 'Li_j = (L′i_j mod 2)' should be written as a congruence, e.g., L^i_j ≡ L′^i_j (mod 2), and the phrase 'related by finite sequence of arc shift moves' should be 'arc shift equivalence' (Definition 3.5), which includes generalized Reidemeister moves.
[Proposition 3.24] This proposition is asserted without proof, and the virtual linking numbers of L(a1,a2,b1,b2;k,l) and M(a1,a2,b1,b2;k,l) in terms of the parameters are not computed, so the four cases cannot be verified; a derivation should be supplied or the statement demoted to a remark.
[Corollary 3.14 proof] The induction step requires one to 'use arc shifts to parallel all the mixed chords between the first two components, ensuring that there are no self-chords of the second component located between the mixed chords'; the existence of such an interleaved sorting is exactly the unproven step of Theorem 3.12 and needs separate justification.
[Theorem 3.29 proof] The symbol n is used both for the number of components and for the value of the arc shift number, and the sentence 'There are exactly n terms in the inequality (4)' counts arc shift moves; a distinct symbol (e.g., m or k) should be used for the arc shift number.
[Example 3.31] There is a spelling error ('Condiser'), and the claim that the upper bound for D_n is exactly n is only sketched; the Gauss diagram in Figure 23 appears to show n−1 mixed chords, and the move sequence is not described.
[Definitions 3.2–3.3] The definition of an odd self-crossing counts 'odd number of real crossings from c to c'; it should be stated explicitly whether mixed crossings are included in this count, since the behavior of odd writhe under arc shifts (Theorem 3.28) is sensitive to this.
[Abstract] The phrase 'the upper bound of the arc shift number is exactly n' is imprecise: Theorem 3.30 and Example 3.31 establish upper bounds A(D_n) ≤ n, not exact values.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the parity classification is not definitionally forced; the main weakness is an omitted normalization proof, which is a rigor gap rather than a circular reduction.

full rationale

The derivation chain is not circular in the sense of the seven enumerated patterns. Lemma 3.6 proves parity invariance of virtual linking numbers under arc shifts from the local sign changes of the move, giving the 'only if' direction of Theorem 3.21 independently. The converse is attempted constructively: Theorem 3.12 supplies an arc-shift/RII reduction for 2-homogeneous proper diagrams, and Proposition 3.16 is intended to produce the normal forms L(0,0), L(1,0), L(0,1), L(1,1) from the parities of the virtual linking numbers. Theorem 3.21 then combines Lemma 3.6 with these normal forms; the equivalence classes are not defined in terms of the theorem being proved, nor is a fitted parameter renamed as a prediction. The citations to the authors' prior work ([6], [7]) are used as tools—the virtual-knot arc-shift unknotting theorem and the arc-shift/odd-writhe inequality—and they concern knots, not the n-component link classification claimed here, so they are independent of the target conclusion. The genuine defect is an omitted proof, located in Proposition 3.16 ('we will ultimately obtain the Gauss diagram of L(1, 1) ... even after applying the arc shift moves and RII move') and again in Theorem 3.21's appeal to 'the reducing algorithm, depicted in Theorem 3.12' for arbitrary parity patterns and arbitrary n. No explicit terminating move sequence or n-component normal form is given, so the converse of Theorem 3.21 is not established. That is a correctness/rigor concern, not a circularity: the claimed reduction is absent, but it is not identical to the input by construction.

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

No numerical fitting is used. The load-bearing content is geometric: the central claim rests on the existence of an arc shift reduction algorithm and on the invariance of odd writhe, both of which are asserted rather than fully derived.

assumptions (4)

standard math Generalized Reidemeister moves on diagrams correspond to the depicted moves on Gauss diagrams.
Used throughout the paper as the translation between diagrams and Gauss diagrams, attributed to [4].
domain assumption Odd writhe as defined in Definition 3.3 is an invariant of n-homogeneous proper links.
Stated without proof; the lower bound in Theorem 3.29 and the behavior in Theorem 3.28 depend on it.
ad hoc to paper Every Gauss diagram of a homogeneous proper link can be aligned by arc shifts into a parallel configuration, and every parallel configuration with even mixed chords can be paired and removed by RII.
This is the core algorithm of the converse of Theorem 3.12 and of Theorem 3.21; it is described informally and not proven.
domain assumption Arc shift changes signs of chords but preserves the count of heads and tails on each component.
Stated in the proof of Theorem 3.12 to justify pairing; it is not reconciled with the cancellation of one-directional mixed chords.

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Pith. "Pith review of Classification of virtual links by arc shift move." pith.science (2026). https://pith.science/paper/JQDOCZ2O

@misc{pith2026250208955,
  author       = {Pith},
  title        = {Pith review of: Classification of virtual links by arc shift move},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JQDOCZ2O}},
  note         = {Machine review of arXiv:2502.08955}
}

abstract

In this paper, we establish that the arc shift operation on a $n$-component virtual link diagram acts as an unknotting operation when the virtual link is $n$-homogeneous proper, aiding in the classification of $ n $-component virtual links up to arc shift equivalence. We explore the connection between the arc shift number and the odd writhe of virtual links which are homogeneous proper. Additionally, we identify sequences of virtual link diagrams $ L_n $ for which the upper bound of the arc shift number is exactly $ n $.

Figures

Figures reproduced from arXiv: 2502.08955 by the authors.

**Figure 2.** Detour move. c c ′ [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 4.** Gauss diagram under generalized Reidemeister moves. [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗

Figures from the paper (19 more)

**Figure 5.** Figure 5: Gauss diagram for virtual link. 2.2.1 Forbidden Moves Forbidden moves are not unknotting operation for virtual links and classical links. Allowing both of the forbidden moves Fo and Fu gives rise to fused isotopy [8]. Specifically, two virtual links L1 and L2 are calle…

**Figure 6.** Figure 6: Forbidden moves for virtual link. Theorem 2.6. [5] A classical link L with n-components is completely determined by the linking numbers of each pair of components under fused isotopy. Theorem 2.7. [3] Two welded links L and L ′ are F–equivalent if and only if, for all …

**Figure 7.** Figure 7: Ξ-move If P1 and P3 are connected by a self-chord, then the chord is called a shell. Definition 2.10. [10] The two Gauss diagrams G and G′ are Ξ-equivalent if they are related by a finite sequence of Reidemeister moves RI-RIII and Ξ-moves. Two virtual links are Ξ-equiv…

**Figure 8.** Figure 8: Gauss diagrams G(a1, a2, b1, b2; k, l) and H(a1, a2, b1, b2; k, l). 2.2.4 Arc shift move Definition 2.15. [7] Given a virtual knot diagram D, consider an arc (a, b) passing through the pair of crossings (c1, c2), where atleast one of the crossings is classical. Then, t…

**Figure 9.** Figure 9: Arc shift moves on (a, b). Moves corresponding to arc shift moves in the Gauss diagram are shown in [PITH_FULL_IMAGE:figures/full_fig_p006_9.png]

**Figure 10.** Figure 10: Gauss diagrams analogues to arc shift moves. [PITH_FULL_IMAGE:figures/full_fig_p007_10.png]

**Figure 11.** Figure 11: Virtual Hopf link 8 [PITH_FULL_IMAGE:figures/full_fig_p008_11.png]

**Figure 12.** Figure 12: 2-homogeneous proper link. Ah At Ath Aht RI (a) (b) (c) (d) (e) c1 c5 c6 c3 c2 c7 c4 c9 c8 c10 c1 c2 c3 c3 c3 c3 c4 c4 c4 c4 c4 c1 c1 c1 c1 c2 c2 c2 c3 c5 c5 c5 c5 c6 c6 c6 c6 c7 c7 c7 c7 c8 c8 c8 c8 c9 c9 c9 c9 c10 c10 c10 c10 c2 [PITH_FULL_IMAGE:figures/full_fig_p0…

**Figure 13.** Figure 13: Parallel alignment of self-chords of a Gauss diagram. [PITH_FULL_IMAGE:figures/full_fig_p009_13.png]

**Figure 14.** Figure 14: Parallel alignment of mixed-chords of a Gauss diagram. [PITH_FULL_IMAGE:figures/full_fig_p010_14.png]

**Figure 15.** Figure 15: Trivializing the Gauss diagram. In the final step, the arc shift move As is applied to the mixed crossings based on their signs, ensuring that consecutive pairs have opposite signs. Subsequently, by applying the RII move, all the chords can be removed, as the mixed ch…

**Figure 16.** Figure 16: The 2-homogeneous proper link diagram L. shown in [PITH_FULL_IMAGE:figures/full_fig_p012_16.png]

**Figure 17.** Figure 17: Unknoting sequence of 2-homogeneous proper link diagram [PITH_FULL_IMAGE:figures/full_fig_p012_17.png]

**Figure 18.** Figure 18: Gauss diagrams of 2-component virtual links upto arc shift move. [PITH_FULL_IMAGE:figures/full_fig_p013_18.png]

**Figure 19.** Figure 19: 2-component virtual links upto arc shift move. [PITH_FULL_IMAGE:figures/full_fig_p014_19.png]

**Figure 20.** Figure 20: A class of virtual link L(2n − 1). Definition 3.19. Let D be a virtual knot diagram and D∗ is said to be mirror image of D if D∗ is obtained from D by changing the crossing type at every classic crossing point. Proposition 3.20. Let L = K1 ∪ K2 be a 2-component virtua…

**Figure 21.** Figure 21: Arc shift on virtual link Theorem 3.28. If L = K1 ∪ · · · ∪ Kn and L ′ = K′ 1 ∪ · · · ∪ K′ n are n-homogeneous proper link diagrams that differ by an arc shift move, then J(K′ i , L′ ) = J(Ki , L) or J(K′ i , L′ ) = J(Ki , L) ± 1 or J(K′ i , L′ ) = J(Ki , L) ± 2, for …

**Figure 22.** Figure 22: (2, 4n)-virtual torus link diagram and the corresponding Gauss diagram. number is 1, as shown in [PITH_FULL_IMAGE:figures/full_fig_p017_22.png]

**Figure 23.** Figure 23: The n-homogeneous proper link diagram Dn. Example 3.32. Consider a 2-homogeneous proper link is shown in [PITH_FULL_IMAGE:figures/full_fig_p018_23.png]

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Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

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Pith tools

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

[1] [1]

D. Ali, Z. Yang, M. I. Sheikh, The H(n)-move is an unknotting operation for virtual and welded links, J. Knot theory ramifications 32(09), 2350061(2023)

work page 2023

[2] [2]

Audoux, P

B. Audoux, P. Bellingeri, J. B. Meilhan, E. Wagner, Extensions of some classical local moves on knot diagrams , Michigan Math. J.67(03)(2018), 647–672

work page 2018

[3] [3]

Extensions of some classical local moves on knot diagrams

B. Audoux, P. Bellingeri, J. B. Meilhan, E. Wagner, On forbidden moves and the Delta move , arXiv:1510.04237(2015)

work page Pith review arXiv 2015

[4] [4]

J. S. Carter, S. Kamada, M. Saito, Stable equivalence of knots on surfaces and virtual knot cobordisms, J. Knot Theory Ramifications 11(03)(2002), 311–322

work page 2002

[5] [5]

A. Fish, E. Keyman, Classifying links under fused isotopy , J. Knot Theory Ramifica- tions(07)(25), 1650042(2016)

work page 2016

[6] [6]

A. Gill, K. Kaur, M. Prabhakar, Arc shift number for some virtual knots , Trends in Algebraic Topology and related topics, Trends Math., Birkhauser/Springer, 2018

work page 2018

[7] [7]

A. Gill, K. Kaur, M. Prabhakar, Arc shift number and region arc shift number for virtual knots, J. Korean Math. Soc. 56(04) (2019), 1063–1081

work page 2019

[8] [8]

L. H. Kauffman, Virtual knot theory , European J. Combin. 20(07) (1999), 663–691

work page 1999