When is arc crossing change an unknotting operation?

Jingze Song; Qian Liao; Zhiyun Cheng

REVIEW 3 major objections 5 minor 8 references

When is arc crossing change an unknotting operation?

T0 review · 3 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read This paper proves exactly when arc crossing change can unknot a link: one version always works, the other fails only on a parity obstruction that leaves a Hopf link.

desk verdict Worth a serious referee: new parity classification for link unknotting, but Theorem 1.4's induction is missing an alternation-preservation step and one walk argument needs rigor. read the letter →

arxiv 2509.08341 v1 pith:44EMINT5 submitted 2025-09-10 math.GT

Zhiyun Cheng , Qian Liao , Jingze Song This is my paper

classification math.GT MSC 57K1057M15

keywords unknottingoperationarccrossingchangelinkdiagramlinkingnumberHopfalternatingknotadmissibletraildirectedgraph

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 asks exactly when a non-commutative local move on link diagrams—arc crossing change, which switches the two endpoints of an arc rather than a single crossing—can transform any diagram of a link into a diagram of an unlink. The answer is a complete two-case rule: one variant of the move unknots every link diagram, while the other unknots unless the total linking number is odd and every component is free of self-crossings; in that single exceptional case the best possible result is n−2 unknots plus a Hopf link. The paper also proves that in alternating knot diagrams any two crossing points can be switched independently by such moves. If correct, this settles the question for the diagrammatic operation and gives a rare example of a non-commutative unknotting operation whose failure is described by a single parity invariant.

What carries the argument

The arguments are carried by three interlocking objects. First, the total linking number t(L) mod 2, together with the parity identity |U(K_i)| ≡ sum_j lk(K_i,K_j) (mod 2), which turns the local effect of an arc switch into a global parity count. Second, a type C component: a self-crossing-free component with exactly one undercrossing, the only local configuration where arc crossing change II differs from arc crossing change I; it is the obstruction to unknotting. Third, for Theorem 1.4, the directed graph A_G built from a knot shadow, whose vertices are all 2^n diagrams with that shadow and whose edges are single arc crossing changes, plus admissible x-y trails—paths that start and end on u

What would settle it

For Theorem 1.4, an explicit alternating knot diagram with two crossings x and y for which the directed graph has no x′→y or x″→y trail avoiding x would refute the claim. For Theorem 1.3, a connected link diagram with even total linking number that cannot be transformed to the unlink by any finite sequence of arc crossing changes II would refute the characterization.

Watch

Extended reading notes

Core claim

On the paper's terms, the discovery is a dichotomy. Arc crossing change I acts on the crossing that bounds an arc and is a genuine unknotting operation for every connected link diagram. Arc crossing change II, which leaves that boundary crossing unchanged (equivalently switches it twice), is governed exactly by the parity of the total linking number t(L). If t(L) is even, it unknots; if t(L) is odd, it unknots precisely when at least one component has a self-crossing; and if t(L) is odd with all components free of self-crossings, the operation reduces any diagram to n−2 unknots and a Hopf link and can go no further. For alternating knot diagrams, the paper proves a stronger local statement:

Load-bearing premise

Theorem 1.4's induction needs that from any alternating knot diagram with at least five crossings one can delete a crossing not among x, x′, x″ so that the remaining diagram is still an alternating knot diagram; the paper assumes this without proving it.

Editorial extensions

If this is right

If Theorem 1.3 is right, whether arc crossing change II unknots a given link diagram is decided by two visible features: total linking number parity and the existence of a self-crossing on some component.
The exceptional case is rigid: with odd total linking number and all components crossing-free, the operation leaves a Hopf link, so the obstruction is nontrivial and cannot be removed by further arc crossing changes of the same type.
Theorem 1.4 gives a practical recipe: in an alternating knot diagram, any chosen set of crossings can be switched by a sequence of arc crossing changes, so the operation behaves like a complete set of switching generators on alternating shadows.
The non-equivalence result in Section 4 means the arc version is strictly weaker than the semi-arc version when one tries to realize individual semi-arc switches, since some diagrams have indegree 0 in A_G.

Reading between the lines

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

The parity dichotomy likely reflects a Z/2-valued diagram invariant: the operation II preserves t(L) mod 2 when no component has self-crossings, so unknotting is blocked exactly when the invariant differs from 0; one could test whether a similar invariant exists for other nontrivial link types.
The directed graph A_G suggests an algorithm to decide admissibility of a pair of crossings in any knot diagram by search; the paper's examples indicate the alternating condition is sufficient but not necessary, so a natural next step is to characterize all diagrams whose A_G has the needed trails.
Extending Theorem 1.4 to non-alternating diagrams might depend only on the shadow's planarity rather than alternation; the admissible-trail method gives a concrete way to test this on the 8_20 diagram.
For links, one could formulate a version of the result on surfaces; the parity lemmas are local and should carry over, though type C components may interact with handles in that setting.

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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

Bottom line: the paper answers a natural question—when does Cericola's arc crossing change unknot a link?—with a genuinely new parity classification and a clean negative answer to the Kinuno–Cericola equivalence question. The main theorems are worth a serious referee, but Theorem 1.4 has a real gap and one part of Theorem 1.3 is too informal.

The new content is real. The type I/type II distinction for links is useful, and Theorem 1.3's statement is clean: even total linking number always unknots; odd total linking number either unknots when some component has a self-crossing or reduces the diagram to unknots plus a Hopf link. The proof uses a neat parity lemma (|U(K_i)| equals the linking-number sum mod 2) and a felicitous-labeling induction. Theorem 1.2 is straightforward and clear. The directed graph on knot shadows in Section 4 is a good way to see the non-commutativity and gives the figure-8 counterexample that settles Cericola's question. These are solid contributions.

The soft spots. Theorem 1.4's induction does not prove that the chosen resolution of vertex z is alternating. The paper only says 'choose the one such that new diagram is a knot diagram'; being a knot diagram does not force alternating. Smoothing a crossing of the trefoil can yield a 2-crossing unknot whose crossings have the same sign, so the resolved diagram is not alternating. Without a proof that some z can be chosen whose knot resolution is still alternating, the induction hypothesis cannot be applied. The gap is repairable—by choosing z more carefully or adding a short lemma on alternating smoothings—but it is load-bearing. The n≤4 base cases are also dismissed in a sentence. The serial-walk argument in Theorem 1.3(2)(b)(ii) is informal; the claimed trail from J to the Hopf-link crossing is plausible but not rigorously justified. That is a secondary issue, not a reason to distrust the theorem.

The citation pattern is fine; the self-citations are contextual and not load-bearing. This is a paper for knot theorists working on unknotting operations and diagram moves. It deserves peer review, and I would send it out expecting a revise-and-resubmit that tightens Section 4.2 and the walk in 3.2.

Referee Report

3 major / 5 minor

Summary. The paper extends Cericola's arc crossing change from knot diagrams to link diagrams. It defines two local versions, arc crossing change I and II, and proves: (Theorem 1.2) type I is an unknotting operation for all link diagrams; (Theorem 1.3) type II is an unknotting operation when the total linking number is even, and also when the total linking number is odd and at least one component has a self-crossing, while in the remaining odd case the diagram can be transformed into the disjoint union of n-2 unknots and a Hopf link. In Section 4 the authors construct a directed graph from a knot shadow, use it to show that Cericola's arc crossing change is not equivalent to Kinuno's, and prove (Theorem 1.4) that any two crossings of an alternating knot diagram are arc crossing change admissible.

Significance. If the proofs are completed, Theorems 1.2 and 1.3 would give a clean characterization of when a non-commutative local operation is an unknotting operation on link diagrams, and Theorem 1.4 would be a genuine structural result about alternating diagrams. The paper is constructive and builds on Cericola's theorem rather than introducing fitted parameters or circular definitions. Its main weakness is that two load-bearing arguments are not fully rigorous: the induction in Theorem 1.4 rests on an unproved alternation-preservation statement, and the proof of Theorem 1.3(2)(b)(ii) uses an informal serial-walk argument. These gaps appear repairable, so the result is plausible but not yet established as written.

major comments (3)

[Section 4.2, proof of Theorem 1.4] The inductive step chooses a vertex z outside {x,x',x'',y}, resolves z, and asserts that the new diagram is a knot diagram with n-1 crossings to which the induction hypothesis applies. This requires that the resolved diagram is alternating. The paper does not prove this, and it is not automatic: a smoothing of a crossing of an alternating diagram need not preserve the alternating over/under pattern. The paper also does not justify that such a z with a one-component smoothing exists for every n≥5. Consequently the induction hypothesis, which is stated only for alternating diagrams, cannot be applied as written. The base case n≤4 is dismissed with 'verified similarly' after a few examples, which is not a complete proof. This gap is load-bearing for Theorem 1.4.
[Section 3.2, Theorem 1.3 case (2)(b)(ii)] The step that unlinks each maximal connected component L_{c,j} with odd total linking number is described by a serial walk along arcs a1,a2,...,a_{v+2} and illustrated by Figure 9, but it is not rigorously proved. The text asserts that one can keep walking from component to component until reaching K_u, that the required undercrossings exist at each step, and that after the preceding arc crossing changes the next segment is again an arc. It also asserts that 'finally, only the two crossing points c and c' are changed'; however, the later remark that J ≼ L_{c,j} may fail shows that the global effect on other components has not been tracked. Since this is the mechanism that removes the residual Hopf link in the mixed self-crossing case, the argument must be made fully formal.
[Section 4.2, Lemma 4.4] The proof of Lemma 4.4 contains an informal iterative step: after encountering a block of overcrossings v_i,...,v_{i+j}, the text applies arc crossing changes and says to repeat until the trail consists of undercrossings followed by overcrossings. It is not proved that the stated arc crossing changes are well-defined at each stage, nor that the performed operations preserve the admissibility of the trail or the property that the eventual trail connects the desired vertices. Because Lemma 4.4 is the mechanism that converts the directed trail into an actual sequence of arc crossing changes, this lack of detail contributes to the proof gap in Theorem 1.4.

minor comments (5)

[Section 4.1, Proposition 4.2] The proofs of Proposition 4.2 are left to the reader, but part (2) is used to argue that the figure-8 diagram 0000 has indegree 0. Please include at least a sketch of these facts, especially the indegree formula.
[Section 4.2, base cases] The base case n≤4 of the claim in Theorem 1.4 is illustrated only by the examples in Figure 14 and then 'verified similarly'. A finite case check for all alternating shadows with at most four crossings should be listed or replaced by a uniform argument.
[Remark 4.5 / Figure 17] The remark about 8_19, 8_20, and 8_21 refers to 'the shadow area' in Figure 17, but the diagrams are not actually reproduced in the text. As written, the claim about 8_20 is unverifiable.
[Section 2] The notation L_{<j} and L_{>j} is used from the beginning of the paper, but the order on components is only implicit. Please define L_{<j} and L_{>j} explicitly after introducing the labeling.
[Throughout] There are several typographical and formatting artifacts, such as the broken spacing in the title and abstract and the hard-to-read labels in Figures 9 and 10. These should be corrected in the final version.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the main theorems rest on Cericola's external theorem and self-contained combinatorial arguments, with self-citations only contextual.

full rationale

The paper's central derivation chain is not circular. Theorem 1.1 (Cericola's ascending-diagram theorem) is an external result, and the current authors use it as a black box rather than re-deriving it from their own assumptions. Theorems 1.2 and 1.3 are proved by induction on link components, parity arguments for linking numbers, and explicit case analyses; these arguments do not reduce to any fitted parameter or to the conclusion itself. Theorem 1.4 is proved by an internal induction on crossing number using Lemma 4.4 and a directed-trail claim; its proof does not depend on any self-citation. The paper's self-citations (references [2]–[5]) appear only in the introduction as background on region crossing change and in Section 4.1 as a parenthetical 'similar construction' remark; they are not load-bearing for the main claims. The induction step in Theorem 1.4 does contain a potential proof gap: the text asserts that after resolving a chosen vertex z the resulting diagram is a knot diagram to which the induction hypothesis applies, but it does not explicitly justify that the resolved diagram remains alternating. However, a missing justification is a correctness concern, not circularity: the theorem is not assumed in its own proof, and the gap does not consist in defining the conclusion into the premises or citing the authors' own work as the only support. Hence no circular step can be exhibited, and the appropriate finding is 'no significant circularity' with score 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

Pure mathematical topology paper: no fitted parameters, no invented entities. The central claims rest on an imported theorem from [1] and on implicit combinatorial facts about alternating diagrams and vertex resolution that are not fully proved.

assumptions (3)

domain assumption Cericola's Theorem 1.1: every knot diagram can be transformed into an ascending diagram by a sequence of arc crossing changes.
Imported from reference [1] and used in the proof of Theorem 1.2 (Section 3.1) to transform each knot component into an ascending diagram.
ad hoc to paper For an alternating knot diagram with n >= 5, there exists a vertex z outside a specified set whose resolution yields a knot diagram with n-1 crossings, and this new diagram is alternating.
Invoked in the induction proof of Theorem 1.4 (Section 4.2) without proof. The paper states one chooses the resolution so the new diagram is a knot diagram, but neither existence nor alternation preservation is justified.
domain assumption The shadow of a knot diagram admits a two-in two-out directed graph from the knot orientation, and for alternating diagrams this orientation has the 'rationality' property needed for the proof of Theorem 1.4.
Used at the start of the proof of Theorem 1.4 to construct the directed graph G; the precise content of 'rationality' is left implicit.

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Cite this review

Pith. "Pith review of When is arc crossing change an unknotting operation?." pith.science (2026). https://pith.science/paper/44EMINT5

@misc{pith2026250908341,
  author       = {Pith},
  title        = {Pith review of: When is arc crossing change an unknotting operation?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/44EMINT5}},
  note         = {Machine review of arXiv:2509.08341}
}

read the original abstract

This paper extends the study of arc crossing change, a local operation on knot diagrams recently introduced by Cericola, from knot diagrams to link diagrams. We consider two types of arc crossing change on link diagrams and discuss when they are unknotting operations. Furthermore, we show that any two crossing points in an alternating knot diagram are arc crossing change admissible.

Figures

Figures reproduced from arXiv: 2509.08341 by the authors.

**Figure 2.** arc crossing change I arc crossing change II [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** Arc crossing change I on link diagrams Theorem 1.3. Let L = Sn i=1 Ki be a link diagram with n components, and denote the total linking number of L as t(L) = P 1≤i<j≤n lk(Ki , Kj), where lk(Ki , Kj) denotes the linking number of components Ki and Kj. The following statements hold for arc crossing change II: (1) If the total linking number t(L) is even, then arc crossing change II is an unknotting operation for L. (2… view at source ↗

Figures from the paper (14 more)

**Figure 4.** Figure 4: Arc crossing change II on link diagrams Compare the arc crossing change introduced by Kinuno and that introduced by Cericola, one observes that each arc crossing change of Cericola can be realized by that of Kinuno. In [1], Cericola asked whether the converse also hold…

**Figure 5.** Figure 5: A diagrammatic sketch of O(Kj ) and U(Kj ) Following the definition above, we introduce a notation that will be used throughout the remainder of this paper. For two distinct components Ki and Kj of a link diagram L, we write Ki - Kj if Ki ∩Kj ⊆ O(Kj ) (equivalently, K…

**Figure 6.** Figure 6: below shows how to transform a link diagram of 31∪41 into an ascending diagram via arc crossing changes on the bold arcs. arc crossing change 31 ∪ 41 F [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]

**Figure 7.** Figure 7: Applying arc crossing change I to the bold arc Consider the component K1, according to the discussion above, via a sequence of arc crossing changes I, all crossings of U(K1) can be converted into overcrossings. Thus now we have L≥2 - K1. By Theorem 1.1, now K1 can be t…

**Figure 8.** Figure 8: Two examples of type C component Remark 3.2. If there exists a knot diagram K which has exactly one self-crossing and U(K) = ∅, then applying arc crossing change I on K switches this crossing but applying arc crossing change II preserves it. However, this is not essent…

**Figure 9.** Figure 9: Unlink the Hopf link Now let us apply arc crossing changes II to the arcs a1, a2, . . . , av+2 serially. Note that a1 is an arc of L, but a2 is not. However, after applying arc crossing change II to a1, a2 becomes an arc in the new link diagram. Finally, only the two c…

**Figure 10.** Figure 10: The directed graph of the shadow of figure-8 knot Here we list some elementary facts about AG. The proofs are left to the reader. Proposition 4.2. Let G be the shadow of a knot diagram with n crossing points and AG the associated directed graph, then the following sta…

**Figure 11.** Figure 11: When an x − y trail meets a crossing In order to prove Theorem 1.4, we need the following lemma. Lemma 4.4. If two crossing points x, y of a knot diagram K can be connected by an admissible x − y trail, then x, y are arc crossing change admissible. Proof. By ignoring …

**Figure 12.** Figure 12: Switching vi and vi+j+1 Repeat this progress until v1, . . . , vl are undercrossings and vl+1, . . . , vk are overcrossings. Now there exists an admissible trail from y to vl+1 and all the crossings in between are undercrossings. According to the discussion above, y …

**Figure 13.** Figure 13: Obtain a directed graph G from an alternating diagram K Choose two vertices of G, denoted by x and y. Let us use x ′ and x ′′ to denote the two vertices incident to x which are directed toward x. Note that x ′ 6= x ′′ since K is a knot diagram. We claim that there ex…

**Figure 14.** Figure 14: Directed graphs with 2, 3, 4 vertices Now we assume that n ≥ 5 and the claim holds true for all alternating knot diagrams with at most n − 1 crossings and any chosen pair of vertices. Consider an alternating knot diagram K with n crossing points, since n ≥ 5, we can c…

**Figure 15.** Figure 15: Resolution of the vertex z Now the new 4-valent plane graph is a two-in two-out directed graph with n − 1 vertices. By assumption, there exists a directed trail with first vertex x ′ or x ′′ and last vertex y which does not pass through x. Whether the vertex z is invo…

**Figure 16.** Figure 16: A directed path from x ′ to y The proof is finished. Remark 4.5. Following the proof of Theorem 1.4, a natural question arises: if every pair of crossings of a knot diagram K is arc crossing change admissible, must K be an alternating knot diagram? The answer to this …

**Figure 17.** Figure 17: Crossings in the shadow area are not arc crossing admissible We end this paper with the following question. Question 4.6. Is the converse of Lemma 4.4 valid? Or more specific, find a necessary and sufficient condition for two crossings in a knot diagram to be arc cros…

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Works this paper leans on

8 extracted references · 8 canonical work pages

[1]

Knot Theory Rami- ﬁcations 33 (2024), no

Christopher Cericola, Arc crossing change is an unknotting operation , J. Knot Theory Rami- ﬁcations 33 (2024), no. 1, Paper No. 2350091, 14

work page 2024
[2]

3, 2289–2308

Jiawei Cheng, Zhiyun Cheng, Jinwen Xu, and Jieyao Zheng, Region crossing change on sur- faces, Mathematische Zeitschrift 300 (2022), no. 3, 2289–2308

work page 2022
[3]

7, 1487–1495

Zhiyun Cheng and Hongzhu Gao, On region crossing change and incidence matrix , Science China Mathematics 55 (2012), no. 7, 1487–1495

work page 2012
[4]

Zhiyun Cheng and Jingze Song, Region crossing change on nonorientable surfaces , arXiv:2506.05885 (2025)

work page Pith review arXiv 2025
[5]

Zhiyun Cheng, When is region crossing change an unknotting operation? , Math. Proc. Cam- bridge Philos. Soc 155 (2013), no. 2, 257–269

work page 2013
[6]

Russell, Equivalence of edge bicolored graphs on surfaces , Electron

Oliver Dasbach and Heather M. Russell, Equivalence of edge bicolored graphs on surfaces , Electron. J. Combin. 25 (2018), no. 1, Paper 1.59, 15 pp

work page 2018
[7]

Knot Theory Ramiﬁcations 31 (2022), no

Rin Kinuno, Structures of homomorphisms induced by arc selection game a nd arc freeze se- lection game , J. Knot Theory Ramiﬁcations 31 (2022), no. 14, Paper No. 2250100, 19

work page 2022
[8]

Ayaka Shimizu, Region crossing change is an unknotting operation , J. Math. Soc. Japan 66 (2014), no. 3, 693–708. School of Mathematical Sciences, Laboratory of Mathematic s and Complex Systems, MOE, Beijing Normal University, Beijing, 100875, China Email address : czy@bnu.edu.cn School of Mathematical Sciences, Beijing Normal University, Beijing 100875, ...

work page 2014

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [1]

Knot Theory Rami- ﬁcations 33 (2024), no

Christopher Cericola, Arc crossing change is an unknotting operation , J. Knot Theory Rami- ﬁcations 33 (2024), no. 1, Paper No. 2350091, 14

work page 2024

[2] [2]

3, 2289–2308

Jiawei Cheng, Zhiyun Cheng, Jinwen Xu, and Jieyao Zheng, Region crossing change on sur- faces, Mathematische Zeitschrift 300 (2022), no. 3, 2289–2308

work page 2022

[3] [3]

7, 1487–1495

Zhiyun Cheng and Hongzhu Gao, On region crossing change and incidence matrix , Science China Mathematics 55 (2012), no. 7, 1487–1495

work page 2012

[4] [4]

Zhiyun Cheng and Jingze Song, Region crossing change on nonorientable surfaces , arXiv:2506.05885 (2025)

work page Pith review arXiv 2025

[5] [5]

Zhiyun Cheng, When is region crossing change an unknotting operation? , Math. Proc. Cam- bridge Philos. Soc 155 (2013), no. 2, 257–269

work page 2013

[6] [6]

Russell, Equivalence of edge bicolored graphs on surfaces , Electron

Oliver Dasbach and Heather M. Russell, Equivalence of edge bicolored graphs on surfaces , Electron. J. Combin. 25 (2018), no. 1, Paper 1.59, 15 pp

work page 2018

[7] [7]

Knot Theory Ramiﬁcations 31 (2022), no

Rin Kinuno, Structures of homomorphisms induced by arc selection game a nd arc freeze se- lection game , J. Knot Theory Ramiﬁcations 31 (2022), no. 14, Paper No. 2250100, 19

work page 2022

[8] [8]

Ayaka Shimizu, Region crossing change is an unknotting operation , J. Math. Soc. Japan 66 (2014), no. 3, 693–708. School of Mathematical Sciences, Laboratory of Mathematic s and Complex Systems, MOE, Beijing Normal University, Beijing, 100875, China Email address : czy@bnu.edu.cn School of Mathematical Sciences, Beijing Normal University, Beijing 100875, ...

work page 2014