{"id":"05e81322-255a-40dd-bb87-556cb64529f2","arxiv_id":"1908.06053","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Braidoids are defined and shown to satisfy analogues of the Alexander and Markov theorems, so planar knotoids can be braided and braidoid equivalence corresponds to knotoid isotopy.","lead":"This paper introduces braidoids, a braid-like counterpart to knotoids, and proves versions of the classical Alexander and Markov theorems for them. The results give a way to encode open-ended knot diagrams as braid-like objects, which may help in studying protein chain topology.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3 rests on an unproved exhaustive obstruction check in §5.2.1; the two bullet conditions are asserted by inspection, and a missed case would break the braidoiding map.","rationale":"In good faith, the paper is a competent diagrammatic development, and the main constructions (closure, L-moves, braidoiding algorithm) are plausible and largely consistent with the classical braid analogues. However, the proof of Theorem 3's only-if direction genuinely depends on the braidoiding algorithm working for every multi-knotoid, and the exhaustiveness of the obstruction classification in §5.2.1 is asserted rather than demonstrated. This is exactly the weakest assumption identified by the reader. The concern does not amount to an observed contradiction or a known counterexample, so it does not warrant rejection; it reinforces the conditional verdict. A formal enumeration or a proof of the obstruction classification would settle the point.","tokens_in":17999,"tokens_out":9512,"duration_ms":96462,"concrete_test":"Perform an exhaustive case enumeration of all relative configurations of two up-arcs satisfying the general-position conditions of Definition 11, with all choices of labels (o/u), orderings, and positions of the top-most point of the first-ordered up-arc relative to the sliding triangle of the second; for each configuration, simulate the braidoiding move and check whether an uncompletable clasp occurs. The check passes only if the obstructed configurations are exactly those satisfying both bullet conditions in §5.2.1. A single counterexample outside the bullets invalidates the resolution step and hence the only-if direction of Theorem 3.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is Theorem 3, whose 'only if' direction is proven by constructing the braidoiding map br and showing br = cl_L^{-1}. For br to be defined on all multi-knotoids, the braidoiding algorithm of §5 must terminate for every diagram in general position. The only place where this is argued is §5.2.1: after an example of a clasp obstruction, the paper states 'It can be verified by checking all possible positionings, labelings and orderings of any two up-arcs that this type of obstruction may occur only if' the top-most point of the first-ordered up-arc lies in the sliding triangle of the second-ordered up-arc, and the two up-arcs have the same label. This is an assertion by inspection, not a proof or formal enumeration. It is load-bearing because any obstruction outside these two conditions cannot be resolved by the listed fixes (swapping order, changing label, further subdivision), and then br is not well-defined and the equation cl_L∘br=id is not established. The manuscript itself flags the gap by using 'can be verified' rather than providing the case analysis; moreover the check is phrased only for pairs of up-arcs, while the ordered algorithm processes many up-arcs and could in principle encounter interactions not reducible to a pair.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops the theory of braidoids, a diagrammatic counterpart to planar knotoids. A braidoid diagram is a set of descending strands in a rectangle in which one or two 'free strands' terminate at the leg and head endpoints, located anywhere in the diagram. The authors define labeled braidoid diagrams and a closure operation joining corresponding top/bottom ends by arcs running entirely over or under the rest of the diagram, yielding planar (multi-)knotoids. The main results are Theorem 1, an Alexander-type theorem: every (multi-)knotoid diagram in R^2 is isotopic to the closure of some labeled braidoid diagram, via an explicit braidoiding algorithm; and Theorem 3, a Markov-type theorem: closures of two labeled braidoid diagrams are isotopic (multi-)knotoids if and only if the diagrams are L-equivalent, L-equivalence being generated by labeled braidoid isotopy, L-moves, and fake swing moves. The proof of Theorem 3 constructs a braidoiding map br on multi-knotoids and shows it inverts the closure map cl_L on L-equivalence classes. A final section defines underpass and virtual closures and proves they induce well-defined surjective maps from (virtual) braidoids to L-equivalence classes of (virtual) braids.","tokens_in":18302,"tokens_out":28159,"duration_ms":244971,"significance":"If Theorem 3 is correct, it gives a complete geometric bridge between planar (multi-)knotoids and labeled braidoid diagrams up to L-moves, extending the Alexander/Markov paradigm to the open-ended setting. The L-move formulation is well chosen, since braidoids do not obviously carry an algebraic structure. The paper is explicit about provenance: Theorem 1 was already proved in the authors' earlier work [18,19] and is reproved in a more rigid form, while Theorem 3 is stated as coming from [19]; the contribution here is the written proof. The definitions of restricted swing moves, fake swing moves, and L-equivalence are natural and will likely be useful in subsequent work on knotoids and protein topology. The braidoiding algorithm and the inverse-map strategy for Theorem 3 are concrete and checkable, which is a strength. The main weakness is that several load-bearing technical points are asserted by inspection or deferred to the classical papers [31,32]: the obstruction classification in §5.2.1, the proof of Lemma 3, and the verification that br and cl_L are mutual inverses.","major_comments":[{"comment":"The obstruction classification is load-bearing and is asserted rather than proved: the text states that the clasp obstruction 'may occur only if' the two bullet conditions hold, with the justification 'It can be verified by checking all possible positionings, labelings and orderings of any two up-arcs.' No enumeration or formal case analysis is given, and the check is phrased for pairs of up-arcs, whereas the algorithm processes an ordered list of all up-arcs of the diagram; the paper does not argue that interactions involving more than two up-arcs, or between a newly braidoided strand and several previously processed strands, reduce to the pairwise case. This matters because br, which the proof of Theorem 3 needs as the inverse of cl_L, is defined only if the braidoiding algorithm terminates for every (multi-)knotoid diagram, and the resolutions of §5.2.2 cover only the stated obstruction type; the same gap affects the termination claim in the proof of Theorem 1 (§5.2.4). An unlisted obstruction would break the equation cl_L ∘ br = id. I recommend a complete case analysis of the possible obstruction configurations, or an argument showing that a subdivision satisfying the classical triangle condition (Lemma 3) makes the algorithm order-free so that the pairwise classification is a consequence rather than an assumption.","section":"§5.2.1"},{"comment":"Lemma 3 is the main existence result ensuring that a subdivision satisfying both the classical triangle condition and the endpoint triangle condition exists, and the proof of Theorem 1 relies on it. The proof is a sketch: it states that the lemma 'is proved similarly with Lemma 1 in [31]' and then asserts that a subdivision with step ε < ½ min{d1,d2} 'provides a subdivision of K satisfying the classical triangle condition and also the triangle condition for the endpoints.' The adaptation is not carried out: the argument bounds the length of sub-arcs, but the classical triangle condition concerns pairs of non-adjacent sliding triangles, and the endpoint triangle condition concerns the two endpoints, so a uniform bound on sub-arc length does not by itself establish either condition. A complete proof, or a precise reduction to [31, Lemma 1] that accounts for the endpoints, is needed.","section":"§5.2.3 (Lemma 3)"},{"comment":"The proof that br and cl_L are mutual inverses is too terse. For br ∘ cl_L = id, the paper states that the closure B̂ of a labeled braidoid diagram B 'is a knotoid diagram in general position whose only up-arcs are the connection arcs' and that braidoiding these arcs yields a diagram 'isotopic to B.' This requires checking that the connection arcs (which run close to vertical lines and pass entirely over or under the rest of the diagram) satisfy the general-position requirements of Definition 11, that braidoiding multiple connection arcs of different labels can be performed without new obstructions, and that the result is L-equivalent to B rather than merely isotopic as a closure. The sentence establishing cl_L ∘ br = id is likewise asserted. Since these two identities are the core of Theorem 3, the argument should be spelled out in detail.","section":"§6 (proof of Theorem 3, inverse compositions)"},{"comment":"Proposition 1, which is needed for the well-definedness of br, rests on Lemmas 7 and 8, but both are justified largely by figures and by analogy with the classical case. Lemma 7 illustrates only the Ω1-move (Figure 29) and states that the Ω2 and Ω3 cases follow 'similarly' to [32,31]; in the presence of endpoints, free strands, and multiple components, the reduction is not automatic. Lemma 8 is said to be 'verified' by Figure 32. Because these lemmas carry the independence of the braidoiding map from knotoid isotopy, the endpoint and free-strand configurations should be treated explicitly rather than by reference to the classical setting.","section":"§6 (Lemmas 7 and 8)"}],"minor_comments":[{"comment":"In reference [33], 'Hring-Oldenburg' should be 'Häring-Oldenburg' (or 'Haring-Oldenburg').","section":"References [33]"},{"comment":"The phrase 'the small line segment whose boundary is the union of the two intersection points' should be replaced by 'the line segment whose endpoints are the two intersection points.'","section":"§5.1.5, proof of Lemma 2"},{"comment":"The definitions of fake forbidden moves and fake swing moves on labeled braidoid diagrams are formulated through their effect on the closure. Since these moves are primitive generators of L-equivalence, a purely local description (beyond the examples in Figures 25 and 32) would improve clarity.","section":"§6, Definition 9"},{"comment":"The logical relationship between the obstruction-resolution discussion of §5.2.1-5.2.2 and the classical triangle condition of §5.2.3 is not explained: the former resolves obstructions after they occur, while the latter aims to prevent them by subdivision. Because the proof of Theorem 1 uses both, the authors should state explicitly which role each plays and why the classical triangle condition does not make the obstruction analysis redundant.","section":"§5.2.1 / §5.2.3"},{"comment":"The condition 'no subdividing points are vertically aligned with each other unless they share a common edge and neither with the endpoints or with any of the crossings' should be rephrased for clarity, e.g., 'no subdividing point is vertically aligned with another subdividing point, an endpoint, or a crossing, unless the two points share an edge of the subdivision.'","section":"§5.2, Step 1(3)"},{"comment":"The note after Definition 11 says a (multi-)knotoid diagram 'can be always brought to general position by small ∆-moves'; this conflates isotopy moves with the choice of subdivision. The endpoint and classical triangle conditions depend on the subdivision (Lemma 3), not only on the diagram's isotopy class, so the two procedures should be stated separately.","section":"§6, Definition 11"},{"comment":"In the surjectivity argument, 'the original braidoid (resp. virtual braidoid) diagram' should read 'the original braid (resp. virtual braid) diagram.'","section":"§7.2, proof of Proposition 2"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is part of a program developed largely by the same authors: braidoids were introduced in [18,19], Theorem 1 was already proved in [18], and Theorem 3 is attributed to [19]. The genuinely new content of this submission is the written proof of Theorem 3 and the closure/L-equivalence framework. The heavy reliance on the authors' own prior work and on [31,32] (both by the second author) is acceptable in context, but the editor may wish to ensure that the paper is not accepted on the strength of Theorem 3's statement alone, since the proof currently has the gaps described in the major comments. If the authors can supply a complete obstruction analysis and fully detailed proofs of Lemma 3 and the inverse-map identities, the paper would be suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe thing to know: this paper gives a Markov theorem analogue for braidoids, and the main idea is sound. The framework was already in the authors' thesis and earlier paper, but Theorem 3 — closures of labeled braidoid diagrams are isotopic multi-knotoids iff the diagrams are L-equivalent — is new here, and the proof strategy is sensible. They adapt the classical L-move machinery and define fake swing moves to handle endpoint issues. That is real progress for knotoid theory.\n\nWhat I like: the paper is honest. It explicitly points to the parts that are analogous to classical results [31,32] and does not pretend to be fully self-contained. The closure operation is carefully defined, and the figures actually illustrate the moves rather than hiding them. The authors also flag the obstruction problem in §5.2.1, but they do not solve it rigorously.\n\nThe soft spot is exactly there. The braidoiding algorithm must terminate for every (multi-)knotoid in general position. The only argument that obstructions do not occur beyond the listed cases is: 'It can be verified by checking all possible positionings, labelings and orderings of any two up-arcs...' That is an assertion by inspection, not a proof. It is load-bearing because the inverse map br depends on the algorithm being defined everywhere. If there is an unlisted obstruction, br is not well-defined and Theorem 3's only-if direction collapses. The check is also phrased only for pairs of up-arcs, while the algorithm processes many; the reduction to pairs is not shown. I suspect the case analysis is finite and easy, but it has to be written out. This is not a fatal flaw in the conception, but it is a real gap in rigor.\n\nThe rest of the proof of Theorem 3 defers to classical L-move arguments. That is fine as a research paper, but it means the result is not fully verified in this document.\n\nWho should read it: knot theorists working on knotoids and open string diagrams, and people applying knotoids to proteins. It will not reshape the field, but it is a solid extension. I would send it to a serious referee who knows classical Markov proofs — the referee should demand the missing enumeration and enough detail to confirm the braidoiding algorithm really terminates.\n\nVerdict: worth refereeing, with heavy revision.\n\nBest.","headline":"A credible analogue of Markov's theorem for braidoids, with a load-bearing obstruction check in §5.2.1 that is asserted rather than proved; worth refereeing but needs a real case analysis.","tokens_in":18789,"tokens_out":2538,"would_cite":false,"duration_ms":25024,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["57M27","57M25"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that planar multi-knotoids are exactly the closures of labeled braidoid diagrams up to L-equivalence.","keywords":["braidoids","knotoids","multi-knotoids","Markov theorem","Alexander theorem","L-moves","planar knotoids","braidoiding algorithm"],"falsifier":"An exhaustive computer search over all small planar multi-knotoid diagrams with three or four up-arcs whose sliding triangles intersect, checking whether every configuration either satisfies the two bullet conditions or is resolved by reordering, relabeling a free up-arc, or subdividing, would settle the claim: one unresolvable configuration would refute the 'only if' direction of Theorem 3, and a complete enumeration supporting the two-pattern list would confirm it.","tokens_in":17824,"feed_emoji":"🪢","tokens_out":8380,"duration_ms":75277,"temperature":0.7,"pith_summary":"This paper establishes that planar knotoids—open-ended knot diagrams with two free endpoints—have a braid-theoretic counterpart, braidoids, and that the correspondence is exact. The authors define braidoid diagrams as descending strands in a vertical strip, with two free strands ending at a leg and a head, and they introduce a closure operation that turns a labeled braidoid into a planar multi-knotoid. They prove an Alexander-type theorem: every planar multi-knotoid can be changed, by an explicit algorithm, into the closure of a labeled braidoid. Their main result is a Markov-type theorem: two labeled braidoid diagrams close to isotopic multi-knotoids exactly when the diagrams are related by L-moves, braidoid isotopy moves, and fake swing moves. If correct, this gives a complete geometric dictionary between the open-ended world of knotoids and a braid-like combinatorial object, making braidoid representatives a legitimate tool for studying planar knotoids.","feed_headline":"Braidoids and knotoids match exactly under L-moves","feed_subtitle":"Every planar multi-knotoid is the closure of a labeled braidoid, and isotopic closures are exactly L-equivalent diagrams.","key_machinery":"The central object is the labeled braidoid diagram: a finite set of downward-oriented strands in a vertical strip, with exactly two free strands ending at the leg and head, and with corresponding top and bottom ends labeled $o$ or $u$ to say whether the closure arcs pass over or under. The argument runs on the braidoiding algorithm, which cuts every up-arc of a planar multi-knotoid diagram at its topmost point, pulls the two pieces to the top and bottom lines all over or all under the rest of the diagram, and records the label $o$ or $u$; the sliding triangle of each up-arc, the endpoint triangle condition, and the classical triangle condition guarantee that no forbidden move is forced. The equivalence side is carried by the L-moves: cut a strand at an interior point and pull the two new ends to top and bottom, both over ($L_o$) or both under ($L_u$), labeling the new pair accordingly, together with restricted swing moves and the fake swing moves that become ordinary isotopy after closure. These moves are exactly what the closure operation cannot see, which is what makes the equivalence theorem work.","core_discovery":"The central claim of the paper is that the closure operation gives a bijection between L-equivalence classes of labeled braidoid diagrams and isotopy classes of planar multi-knotoids. In the paper's formulation, Theorem 3 states that the closures of two labeled braidoid diagrams are isotopic multi-knotoids in the plane if and only if the labeled braidoid diagrams are related by L-equivalence moves. The forward direction is direct from the definitions: an L-move creates a pair of strands whose closure is isotopic to the original arc. The reverse direction uses the braidoiding map, which turns any multi-knotoid diagram, after putting it in general position and subdividing and labeling its up-arcs, into a labeled braidoid diagram; the paper shows this map is well-defined on L-classes and is inverse to the closure map on L-classes. A corollary is the uniform closure theorem: every planar multi-knotoid is isotopic to the closure of a braidoid diagram with all joining arcs running under.","pith_inferences":["Beyond the paper: the bijection invites a search for braidoid-specific invariants, such as representations, polynomials, or categorifications, that are invariant under L-moves, mirroring how braid closures once generated new knot polynomials.","Beyond the paper: the obstruction classification could be stress-tested by exhaustive computer enumeration of all configurations of up to three or four up-arcs; confirming the two-pattern list would strengthen the theorem, while a single unlisted obstructing configuration would narrow its scope.","Beyond the paper: the elementary-block decomposition of braidoids could be developed into an actual tabulation scheme for open protein chains once L-equivalence classes are known to admit finite normal forms.","Beyond the paper: since the paper leaves the underlying algebraic structure of braidoids open, the action of L-equivalence on elementary blocks is a natural place to look for a group or monoid structure."],"forward_implications":["Every planar multi-knotoid can be represented by a labeled braidoid diagram up to L-equivalence, so invariants of knotoids can be studied through braidoid representatives.","The L-moves give a single move family that detects equality of closures: two labeled braidoid diagrams close to isotopic multi-knotoids exactly when they are L-equivalent.","The underpass closure maps braidoids onto L-equivalence classes of classical braids, and the virtual closure maps virtual braidoids onto virtual L-equivalence classes, connecting planar knotoids to classical and virtual braid theory.","The uniform closure theorem ensures every multi-knotoid has a braidoid representative whose closing arcs all run under the diagram, simplifying the closure operation."],"supporting_citations":[{"why":"Supplies the definition of knotoids, multi-knotoids, and the swing-move equivalence that braidoid equivalence mirrors.","marker":"[37]"},{"why":"Introduces braidoids and their closure and a first braidoiding algorithm; the paper's second proof adapts this material.","marker":"[18]"},{"why":"Develops the braidoiding algorithms and the elementary-block building blocks used here.","marker":"[19]"},{"why":"Original short proofs of the Alexander and Markov theorems that the L-move treatment is modeled on.","marker":"[30]"},{"why":"The study where L-moves for braids are developed and where the classical triangle-condition argument originates.","marker":"[31]"},{"why":"Proves the one-move Markov theorem in 3-manifolds using L-moves, the move set adapted to braidoids.","marker":"[32]"},{"why":"Develops virtual braids and the virtual L-move, supporting the virtual closure part of the paper.","marker":"[27]"},{"why":"The closure of braids in handlebodies, the pattern for closing labeled braidoid ends by joining arcs.","marker":"[33]"},{"why":"Introduces invariants of knotoids and their lifting to open curves in 3-space, motivating why planar knotoids are worth braidoiding.","marker":"[17]"}],"fun_headline_variants":["Braidoids and knotoids: L-moves give exact match","Every planar knotoid is a braidoid closure","L-moves make braidoids and knotoids equivalent","Braidoids: the braid analogue for knotoids","Closure maps braidoids onto all knotoids"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof depends on the claim, verified by inspection rather than by formal enumeration, that the only obstructions to the braidoiding algorithm are the two listed patterns of interacting up-arcs; if some other obstruction exists, the inverse braidoiding map would not be defined for every multi-knotoid.","fun_headline_variants_meta":{"raw":{"variants":["Braidoids and knotoids: L-moves give exact match","Every planar knotoid is a braidoid closure","L-moves make braidoids and knotoids equivalent","Braidoids: the braid analogue for knotoids","Closure maps braidoids onto all knotoids"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000251,"raw_usage":{"total_tokens":1495,"prompt_tokens":824,"completion_tokens":671,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":440,"completion_tokens_details":{"reasoning_tokens":590}},"tokens_in":440,"tokens_out":671,"duration_ms":6880,"temperature":1.0,"reasoning_tokens":590,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:56:14.302408+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"An exhaustive computer search over all small planar multi-knotoid diagrams with three or four up-arcs whose sliding triangles intersect, checking whether every configuration either satisfies the two bullet conditions or is resolved by reordering, relabeling a free up-arc, or subdividing, would settle the claim: one unresolvable configuration would refute the 'only if' direction of Theorem 3, and a complete enumeration supporting the two-pattern list would confirm it.","supporting_citations":[{"cited_title":"Knotoids","cited_arxiv_id":null,"evidence_quote":"Supplies the definition of knotoids, multi-knotoids, and the swing-move equivalence that braidoid equivalence mirrors."},{"cited_title":"Knotoids, Braidoids and Applications.Symmetry, Special Issue:Knot Theory and Its Applications (2017)","cited_arxiv_id":null,"evidence_quote":"Introduces braidoids and their closure and a first braidoiding algorithm; the paper's second proof adapts this material."},{"cited_title":"On Knotoids, Braidoids and Their Applications.PhD thesis, 2017","cited_arxiv_id":null,"evidence_quote":"Develops the braidoiding algorithms and the elementary-block building blocks used here."},{"cited_title":"Short proofs of Alexander’s and Markov’s theorems","cited_arxiv_id":null,"evidence_quote":"Original short proofs of the Alexander and Markov theorems that the L-move treatment is modeled on."},{"cited_title":"A study of braids in 3-manifolds","cited_arxiv_id":null,"evidence_quote":"The study where L-moves for braids are developed and where the classical triangle-condition argument originates."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proves the one-move Markov theorem in 3-manifolds using L-moves, the move set adapted to braidoids."},{"cited_title":"Virtual braids and the L-move, J","cited_arxiv_id":null,"evidence_quote":"Develops virtual braids and the virtual L-move, supporting the virtual closure part of the paper."},{"cited_title":"Knot Theory Ramif","cited_arxiv_id":null,"evidence_quote":"The closure of braids in handlebodies, the pattern for closing labeled braidoid ends by joining arcs."},{"cited_title":"New invariants of knotoids.European J","cited_arxiv_id":null,"evidence_quote":"Introduces invariants of knotoids and their lifting to open curves in 3-space, motivating why planar knotoids are worth braidoiding."}],"review_version":1}