Braidoids

Neslihan G\"ug\"umc\"u; Sofia Lambropoulou

REVIEW 4 major objections 7 minor 40 references

Braidoids

T0 review · 4 major / 7 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper proves that planar multi-knotoids are exactly the closures of labeled braidoid diagrams up to L-equivalence.

desk verdict 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. read the letter →

arxiv 1908.06053 v2 pith:GQBRYUK5 submitted 2019-08-16 math.GT math.QA

Neslihan Gügümcü , Sofia Lambropoulou This is my paper · ORCID

classification math.GTmath.QA MSC 57M2757M25

keywords braidoidsknotoidsmulti-knotoidsMarkovtheoremAlexanderL-movesplanarbraidoidingalgorithm

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 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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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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

Dear colleague,

The 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.

What 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.

The 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.

The 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.

Who 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.

Verdict: worth refereeing, with heavy revision.

Best.

Referee Report

4 major / 7 minor

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.

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 (4)

[§5.2.1] 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.
[§5.2.3 (Lemma 3)] 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.
[§6 (proof of Theorem 3, inverse compositions)] 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.
[§6 (Lemmas 7 and 8)] 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.

minor comments (7)

[References [33]] In reference [33], 'Hring-Oldenburg' should be 'Häring-Oldenburg' (or 'Haring-Oldenburg').
[§5.1.5, proof of Lemma 2] 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.'
[§6, Definition 9] 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.
[§5.2.1 / §5.2.3] 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.
[§5.2, Step 1(3)] 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.'
[§6, Definition 11] 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.
[§7.2, proof of Proposition 2] In the surjectivity argument, 'the original braidoid (resp. virtual braidoid) diagram' should read 'the original braid (resp. virtual braid) diagram.'

Circularity Check

0 steps flagged · score 2.0 of 10

Self-citations are minor and non-load-bearing; the §5.2.1 assertion is a proof gap, not circularity.

full rationale

The claimed derivation chain is not circular. The closure operation (Definition 5), L-moves (Definition 8), and fake swing moves (Definition 9) are introduced independently of the theorem they support, and Theorem 1 (Alexander analogue) is proved by an explicit braidoiding algorithm in Section 5 rather than assumed from the authors' earlier papers. Theorem 3's bijection is established by constructing two maps, cl_L and br, with br defined by the same algorithm; the inverse identities are geometric checks on closures, not definitions of the equivalence. Self-citations to [18,19,30,31,32] supply the source of the theory and the classical-braid template for L-move arguments; these are parameter-free results from prior work and are adapted here, not imported as the braidoid Markov theorem. The reviewer concern in §5.2.1—'It can be verified by checking all possible positionings, labelings and orderings of any two up-arcs...'—is a real rigor gap (an exhaustive case check asserted without enumeration), but it is a missing proof of a geometric fact, not a reduction of the conclusion to the premises. Hence no circular step is identified; score 2 reflects minor non-load-bearing self-citation.

Assumptions & free parameters 0 free parameters · 5 assumptions · 3 invented entities

The paper introduces new diagrammatic entities and relies on standard background from knot theory and braid theory. No numerical parameters are fitted. The main assumed background is the classical braid theory of Alexander and Markov, plus the authors' earlier braidoid framework.

assumptions (5)

standard math Reidemeister moves and planar isotopy generate knotoid equivalence
Used throughout Section 2 and in Lemma 1 to compare closures of isotopic braidoids.
standard math Piecewise-linear approximation preserves isotopy classes of knotoid diagrams
Invoked after Definition 1 to work with piecewise-linear diagrams.
standard math Classical Alexander and Markov theorems for braids
Mentioned in the introduction and used as the template; the braidoid theorems are explicitly analogues.
standard math L-move equivalence for classical braids is a one-move version of the Markov theorem
The paper adapts L-moves from [30,31,32] and relies on the classical result for the non-endpoint parts of the proof.
domain assumption General position assumptions for knotoid diagrams (no vertical/horizontal arcs, no alignments)
Section 5.2 assumes small isotopies ensure these conditions; this is a standard genericity assumption.

invented entities (3)

Braidoid diagram
purpose: Generalize braid diagrams to allow free strands with endpoints anywhere in the diagram
Defined in Section 3.1 as a mathematical object; there is no empirical handle outside the theory.
Labeled braidoid diagram
purpose: Support the closure operation by specifying whether joining arcs pass over or under
Introduced in Section 4 as a definitional tool for the closure and Markov theorem.
L-moves and fake swing moves for braidoids
purpose: Generate the L-equivalence relation used in the Markov theorem analogue
Adapted from classical braid L-moves in Section 6; they are new moves on braidoid diagrams.

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Pith. "Pith review of Braidoids." pith.science (2026). https://pith.science/paper/GQBRYUK5

@misc{pith2026190806053,
  author       = {Pith},
  title        = {Pith review of: Braidoids},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GQBRYUK5}},
  note         = {Machine review of arXiv:1908.06053}
}

abstract

Braidoids generalize the classical braids and form a counterpart theory to the theory of planar knotoids, just as the theory of braids does for the theory of knots. In this paper, we introduce basic notions of braidoids, a closure operation for braidoids, we prove an analogue of the Alexander theorem, that is, an algorithm that turns a knotoid into a braidoid, and we formulate and prove a geometric analogue of the Markov theorem for braidoids using the $L$-moves.

Figures

Figures reproduced from arXiv: 1908.06053 by the authors.

**Figure 1.** Knotoid diagrams Definition 1. A piecewise-linear knotoid diagram is a union of finitely many edges: [p1, p2], ..., [pn−1, pn] such that each edge intersects one or two other edges at the vertices, pi , for i = 2, ..., n − 1. The vertices p1 and pn correspond to the endpoints of the diagram. Two edges can also intersect transversely at double points endowed with over/under-data, called crossings of the diagram [PIT… view at source ↗

**Figure 2.** ∆-moves [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 4.** Fake forbidden moves We shall call Ω0, Ω1, Ω2, Ω3 moves together with the swing moves the Ω-moves. Two knotoid diagrams are isotopic to each other if there is a finite sequence of Ω-moves that transforms one into the other. The isotopy generated is clearly an equivalence relation and the isotopy classes of knotoid diagrams are called knotoids in R 2 . The set of all knotoids in R 2 is denoted by K(R 2 ). 2.2. Extend… view at source ↗

Figures from the paper (33 more)

**Figure 5.** Figure 5: Some examples of braidoid diagrams The ends of the strands of B other than the endpoints are called braidoid ends. We assume that braidoid ends lie equidistantly on the top and the bottom lines and none of them is vertically aligned with any of the endpoints. It is cle…

**Figure 6.** Figure 6: A planar ∆-move on a braidoid diagram 3.2.2. Moves of endpoints. Like for knotoid diagrams, we forbid to pull/push an endpoint of a braidoid diagram over or under a strand, as shown in [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 7.** Figure 7: Forbidden braidoid moves We allow the following moves on segments of braidoid strands containing endpoints. (1) Vertical Moves: As shown in [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]

**Figure 8.** Figure 8: A vertical move on h [PITH_FULL_IMAGE:figures/full_fig_p007_8.png]

**Figure 9.** Figure 9: The swing moves for braidoids Definition 3 (braidoid isotopy). It is clear that assuming braidoid ends fixed at the top and bottom lines, the braidoid Ω2 and Ω3- moves together with braidoid Ω0- moves and the swing and vertical moves for the endpoints generate an equiv…

**Figure 10.** Figure 10: and 13, respectively [PITH_FULL_IMAGE:figures/full_fig_p008_10.png]

**Figure 11.** Figure 11: The restricted swing moves for braidoids 1 2 1 2 u u u u ~ ~ ~ [PITH_FULL_IMAGE:figures/full_fig_p009_11.png]

**Figure 12.** Figure 12: The restricted swing moves for braidoids Lemma 1. The closure operation induces a well-defined mapping from the set of labeled braidoids to the set of multi-knotoids in R 2 . Proof. Let b1 and b2 be two labeled braidoid diagrams representing the same labeled braidoid.…

**Figure 13.** Figure 13: An example of non-equivalent labeled closures 5. An algorithm for obtaining labeled braidoids from planar knotoids In this section we present the braidoiding moves on knotoid diagrams that induce algorithms for turning any planar (multi)-knotoid diagram into a labeled…

**Figure 14.** Figure 14: Two up-arcs containing crossings and a free up-arc 5.1.2. Subdivision. We start by marking the local maxima and minima of K with points, which we name as subdividing points. In the process we may need to subdivide further some of the up-arcs of K so that each one con…

**Figure 15.** Figure 15: A braidoiding move [PITH_FULL_IMAGE:figures/full_fig_p011_15.png]

**Figure 16.** Figure 16: The sliding triangle of the up-arc QP with P the cut point A cut-point of an up-arc is defined to be the point where the up-arc is cut to start a braidoiding move. We pick the top-most point P ∈ QP as the cut-point of QP for our algorithm. 5.1.5. A condition on slidin…

**Figure 17.** Figure 17: A further subdivision of the up-arc QP 5.2. Braidoiding algorithm. We now present our algorithm for obtaining a labeled braidoid diagram from a (multi)-knotoid diagram K. The algorithm runs as follows. Step 1: Preparation for eliminating the up-arcs (1) The diagram i…

**Figure 18.** Figure 18: An illustration for the algorithm 5.2.1. Obstructions for the braidoiding algorithm and resolutions. Now, we discuss some bugs of the braidoiding algorithm. In some cases, as exemplified in [PITH_FULL_IMAGE:figures/full_fig_p014_18.png]

**Figure 19.** Figure 19: Obstructions for applying braidoiding moves 5.2.2. Resolutions of the obstructions. Due to the conditions creating obstructions, the resolutions for them can be: • swapping the ordering of the up-arcs; see [PITH_FULL_IMAGE:figures/full_fig_p015_19.png]

**Figure 20.** Figure 20: Swapping the order of up-arcs repairs the obstruction [PITH_FULL_IMAGE:figures/full_fig_p015_20.png]

**Figure 21.** Figure 21: Changing the label of the free up-arc repairs the obstruction o 1 o 2 a new subdividing point o1 o3 o2 o3 braidoiding [PITH_FULL_IMAGE:figures/full_fig_p016_21.png]

**Figure 22.** Figure 22: Adding a subdividing point to the upper up-arc repairs the obstruction 5.2.3. The classical triangle condition. We can make the braidoiding algorithm a simultaneous algorithm that is processed independently of the ordering of the uparcs, by imposing the following co…

**Figure 23.** Figure 23: The classical triangle condition Lemma 3. Let K be a knotoid diagram. There exists a subdivision of K satisfying both the classical triangle condition and the endpoint triangle condition [PITH_FULL_IMAGE:figures/full_fig_p016_23.png]

**Figure 24.** Figure 24: L-moves and the closures of the resulting strands [PITH_FULL_IMAGE:figures/full_fig_p018_24.png]

**Figure 25.** Figure 25: for an example of a fake swing move and a fake forbidden move on a labeled braidoid diagram. o u o u a fake swing move o u a fake forbidden move [PITH_FULL_IMAGE:figures/full_fig_p019_25.png]

**Figure 26.** Figure 26: Adding a subdividing point on the up-arc yields Lequivalence Lemma 5. Labeling a free up-arc either with o or u does not change the resulting labeled braidoid diagram up to the L-equivalence. Proof. The proof is illustrated in [PITH_FULL_IMAGE:figures/full_fig_p021_…

**Figure 27.** Figure 27: Re-labeling the free up-arc and the L-equivalence [PITH_FULL_IMAGE:figures/full_fig_p022_27.png]

**Figure 28.** Figure 28: Re-labeling the free up-arcs Corollary 1. If we have an appropriate choice of relabelling the up-arcs resulting from a further subdivision on K then by Lemmas 4 and 5, the resulting labeled braidoid diagrams are L-equivalent. Lemma 6. Two labeled braidoid diagrams tha…

**Figure 29.** Figure 29: An Ω1-move under braidoiding In this setting, it is crucial to examine specifically the swing moves displacing the endpoints of a knotoid/multi-knotoid diagram. A swing move may displace an endpoint that lies on a down-arc and in a way that the endpoint does not chang…

**Figure 30.** Figure 30: Note that the points P and P ∗ shown in the figure that are chosen for applying the Lo-move on the resulting braidoid diagram and the braidoiding move on the resulting knotoid diagram, respectively, are vertically aligned. A swing move may also cause the endpoint to c…

**Figure 30.** Figure 30: A swing move changing a down-arc to an up-arc swing move braidoiding move braidoiding move u u fake forbidden move u u [PITH_FULL_IMAGE:figures/full_fig_p025_30.png]

**Figure 31.** Figure 31: A swing move displacing an endpoint with respect to a cut-point Lemma 8. A fake forbidden move on a labeled braidoid diagram is composed of a finite number of L-moves and fake or restricted swing moves. Proof. The proof can be verified by [PITH_FULL_IMAGE:figures/ful…

**Figure 32.** Figure 32: A fake forbidden move is decomposed into L-moves and swing moves [PITH_FULL_IMAGE:figures/full_fig_p026_32.png]

**Figure 33.** Figure 33: Abstract examples for the underpass closure 7.2. The induced mappings. We shall now establish that the underpass closure (resp. the virtual closure) defined on braidoid diagrams induces a well-defined map on the set of braidoids (that is, isotopy classes of braidoid d…

**Figure 35.** Figure 35: In this case, the two resulting braid diagrams differ by an [PITH_FULL_IMAGE:figures/full_fig_p029_35.png]

**Figure 34.** Figure 34: A swing move transforms into an L-move on the underpass closure [PITH_FULL_IMAGE:figures/full_fig_p029_34.png]

**Figure 35.** Figure 35: A vertical move transforms into an L-move on the underpass closure By the arguments above the proof of Proposition 2 is completed. 8. Discussion The theory of braidoids is a new diagrammatic setting extending the classical braid theory and giving rise to many new pro…

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