{"id":"acdb8f9c-d5d1-434e-8728-b283ab7fb4ae","arxiv_id":"2502.08955","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper claims virtual links with all pairwise virtual linking numbers even can be unknotted by arc shifts, and that all other links are classified by the parity pattern of these numbers.","lead":"This paper studies arc shift moves on virtual link diagrams, rewiring an arc between two crossings. It claims that links with even pairwise virtual linking numbers can be untangled, giving a finite classification by parity.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The converse of Theorem 3.21 depends on an unproved normalization algorithm: Proposition 3.16 does not provide an explicit arc shift sequence for odd-parity 2-component links, and no canonical representative or reduction is constructed for n components, so the parity classification is not…","rationale":"The forward invariant (Lemma 3.6) and the forward direction of Theorem 3.12 are credible: parity changes by even integers. The failure point is the converse and its extension. The proof of Theorem 3.12 sketches an alignment of self-chords and mixed chords, but the steps are not given as rules; the non-homogeneous case in Proposition 3.16 is asserted in a sentence, and Theorem 3.21 simply cites 'the reducing algorithm' without explaining how it handles diagrams whose mixed chords are not all removable. This is the same load-bearing gap identified by the reader's weakest_assumption. I therefore do not change the reject verdict, but the core concern is an incompleteness of proof rather than a demonstrated contradiction; the proposed search/derivation would distinguish a repairable gap from a false theorem.","tokens_in":12979,"tokens_out":18088,"duration_ms":196904,"concrete_test":"Work out the missing reduction in Proposition 3.16 for a two-component Gauss diagram with no self-chords and exactly three mixed chords with parities (1,0) (e.g., one K2 to K1 chord and two K1 to K2 chords of opposite signs), and then for the same chord multiset with the endpoints interleaved on the two circles. If no explicit arc shift plus RII sequence reduces the interleaved diagram to the single-chord representative L(1,0), Proposition 3.16 is false; if a sequence exists, publish it as the proof. Repeating the check on a three-component diagram with one chord per ordered pair tests the n-component claim.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 3.21 (ii) to (i) is justified only by 'the reducing algorithm, depicted in Theorem 3.12'. That algorithm is described only for n-homogeneous proper links, where every mixed chord is eventually canceled by an RII pair. For the non-homogeneous case, Proposition 3.16 is the sole bridge, but its proof says 'we will ultimately obtain' and then asserts the three cases without giving the move sequence; it also treats only two components. No n-component normal form is even defined, so the count 2^(n choose 2) in Remark 3.23 has no demonstrated upper bound. The paper's note that arc shifts cannot invert chords is not itself a contradiction, because RII can cancel same-direction opposite-sign mixed chords; the real defect is that the alignment and sign-pairing procedure is never shown to terminate or to be independent of the initial diagram. Until that procedure is supplied, the central 'if' direction of Theorem 3.21, Proposition 3.16, and the finite-classification claim rest on an unsupported assertion.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":13193,"tokens_out":33597,"duration_ms":328027,"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":[{"comment":"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.","section":"Theorem 3.12 (⇐) proof"},{"comment":"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.","section":"Proposition 3.16"},{"comment":"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.","section":"Theorem 3.21 (ii)⇒(i)"},{"comment":"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.","section":"Remark 3.23"},{"comment":"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.","section":"Theorem 3.28 proof"}],"minor_comments":[{"comment":"The reference 'Corollary 3.16' (twice) should be 'Proposition 3.16'.","section":"Proposition 3.20 proof"},{"comment":"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.","section":"Theorem 3.21 statement"},{"comment":"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.","section":"Proposition 3.24"},{"comment":"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.","section":"Corollary 3.14 proof"},{"comment":"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.","section":"Theorem 3.29 proof"},{"comment":"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.","section":"Example 3.31"},{"comment":"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.","section":"Definitions 3.2–3.3"},{"comment":"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.","section":"Abstract"}],"recommendation":"reject","confidential_remarks":"The manuscript's central claim — the arc shift classification of n-component virtual links — is not proved: the converse of Theorem 3.21 rests on an informally described reduction that is the real substance of the paper, and Proposition 3.16 merely asserts its conclusion. The class count in Remark 3.23 contradicts the paper's own two-component result, which suggests the manuscript has not been checked as a whole. The parity invariance lemma and the worked examples suggest the underlying idea may be correct and repairable, but a repair would require substantial new proof content. The topic fits the scope of a knot theory journal; the paper is not suitable in its present form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThe 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.\n\nStill, 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.","headline":"A parity invariant and a lower bound for arc shift number are solid, but the classification theorem rests on an unproved normalization algorithm.","tokens_in":13687,"tokens_out":2566,"would_cite":false,"duration_ms":25407,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["57K10","57K12"],"pacs":[],"model":"deepseek-v4-flash","headline":"Two virtual links are arc shift equivalent exactly when the parities of their virtual linking numbers match.","keywords":["arc shift number","virtual links","odd writhe","arc shift equivalence","virtual linking numbers","Gauss diagrams","unknotting operation","n-homogeneous proper links"],"falsifier":"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.","tokens_in":12772,"feed_emoji":"🔗","tokens_out":17694,"duration_ms":167665,"temperature":0.7,"pith_summary":"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.","feed_headline":"Arc shift equivalence is just parity of virtual linking numbers","feed_subtitle":"All even: arc shifts unknot the link; parity pattern picks the class among 2^(n choose 2).","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"It defines the arc shift move on Gauss diagrams, proves it unknots virtual knots, and supplies the odd writhe lower bound that the paper extends to links.","marker":"[7]"},{"why":"It constructs infinitely many virtual knots with arc shift number n, the sequence result this paper parallels for link diagrams.","marker":"[6]"},{"why":"It gives the Xi-move classification and the normal forms L(a1,a2,b1,b2;k,l) used in Proposition 3.24 to connect arc shift classes with Xi classes.","marker":"[10]"},{"why":"It supplies the virtual linking number invariants for virtual links whose parities appear in Theorems 3.12 and 3.21.","marker":"[9]"},{"why":"It introduces virtual link diagrams, Gauss diagrams, and generalized Reidemeister moves on which the whole setup rests.","marker":"[8]"},{"why":"It establishes that diagrams related by generalized Reidemeister moves represent the same virtual link, justifying the Gauss diagram passage.","marker":"[4]"}],"fun_headline_variants":["Arc shift class determined by virtual linking number parities","All-even parities turn arc shift into an unknotting operation","Arc shift equivalence splits into 2^(n choose 2) parity classes","Odd writhe bounds arc shift number from below","Exact arc shift number via virtual torus link family"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Arc shift class determined by virtual linking number parities","All-even parities turn arc shift into an unknotting operation","Arc shift equivalence splits into 2^(n choose 2) parity classes","Odd writhe bounds arc shift number from below","Exact arc shift number via virtual torus link family"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000401,"raw_usage":{"total_tokens":2059,"prompt_tokens":875,"completion_tokens":1184,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":491,"completion_tokens_details":{"reasoning_tokens":1099}},"tokens_in":491,"tokens_out":1184,"duration_ms":11702,"temperature":1.0,"reasoning_tokens":1099,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T23:08:17.857964+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It defines the arc shift move on Gauss diagrams, proves it unknots virtual knots, and supplies the odd writhe lower bound that the paper extends to links."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It constructs infinitely many virtual knots with arc shift number n, the sequence result this paper parallels for link diagrams."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It gives the Xi-move classification and the normal forms L(a1,a2,b1,b2;k,l) used in Proposition 3.24 to connect arc shift classes with Xi classes."},{"cited_title":"Invariants of virtual links and twisted links using affine indices","cited_arxiv_id":"2312.05489","evidence_quote":"It supplies the virtual linking number invariants for virtual links whose parities appear in Theorems 3.12 and 3.21."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It introduces virtual link diagrams, Gauss diagrams, and generalized Reidemeister moves on which the whole setup rests."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It establishes that diagrams related by generalized Reidemeister moves represent the same virtual link, justifying the Gauss diagram passage."}],"review_version":1}