Bonded Knotoids: Theory and Applications

• Bonded knotoids are open knot diagrams enhanced with internal bonds attached at distinct nodes, capturing the topology of open chains without artificial closure.
• The theory distinguishes between topological and rigid types and classifies bonds into long, standard, and tight, extending classical knotoid and braidoid frameworks.
• They provide a robust model for biological macromolecules, such as proteins, by accurately representing both open-chain structure and bonding interactions.

Bonded knotoids are knotoids equipped with bonds: open knot diagrams in an oriented surface $\Sigma$ together with embedded bond arcs whose nodes lie on the knotoid strand and are distinct from the knotoid endpoints. They extend ordinary knotoids—which were introduced to model open curves without artificial closure—by encoding additional internal attachments, and they were developed explicitly as part of a unified theory of bonded knots, bonded braids, and bonded braidoids. In this framework, bonded knotoids model open chains with distinguished endpoints and internal bonds, particularly in settings motivated by biological macromolecules such as proteins (Diamantis et al., 20 Jul 2025).

1. Historical and conceptual setting

Ordinary knotoids were introduced by Turaev as a generalization of knots from closed curves to open curves. In the protein-topology literature, this generalization was motivated by the fact that proteins are open linear chains, whereas classical knot theory is built for closed curves. Closure-based methods can be ambiguous, because different deterministic or stochastic closures can assign different knot types to the same protein, and they can alter the geometry by adding a closing arc. Knotoids avoid this by working directly with projections of the open chain itself, so the topology is inferred without closing the chain and without changing its geometry (Goundaroulis et al., 2017).

Within this broader theory, bonded knotoids add bond information to open-ended diagrams. The survey literature presents them as an application-driven extension of ordinary knotoids for open protein chains with chemical bonds, while emphasizing that they build directly on the ordinary knotoid formalism and the corresponding braidoid theory (Gügümcü et al., 2018). The later algebraic-topological treatment develops bonded knotoids explicitly, together with their closures and their braid-like counterparts, as models for open chains with inter and intra-chain bonds (Diamantis et al., 20 Jul 2025).

A recurrent conceptual distinction is therefore between three levels of description. Ordinary knotoids model open-chain entanglement without closure. Bonded knotoids retain that open-chain structure while recording internal bonds. Bonded braidoids provide an algebraic counterpart to bonded knotoids, in the same way that braidoids serve as a counterpart to ordinary knotoids (Gügümcü et al., 2018).

2. Formal definition and diagrammatic structure

A knotoid diagram in an oriented surface $\Sigma$ is a generic immersion of the unit interval $[0,1]$ into $\Sigma$, with transversal double points carrying over/under information. The images of $0$ and $1$ are the two endpoints, called the leg and the head in the bonded-knotoid formulation; ordinary knotoid literature also uses the terminology tail and head. The endpoints are distinct from each other and from crossings, and the diagram is oriented from one endpoint to the other (Diamantis et al., 20 Jul 2025).

A bonded knotoid diagram in $\Sigma$ is then a knotoid equipped with bonds, defined as in the case of bonded knots, with the additional requirement that the nodes of the bonds differ from the endpoints of the knotoid. A bond is an embedded arc whose endpoints are nodes attached to the knotoid strand. The paper also states that the bonds in a bonded knotoid diagram could also be long bonds (Diamantis et al., 20 Jul 2025).

This definition immediately places bonded knotoids in a family of related open-ended objects. The same source introduces a bonded multi-knotoid diagram, defined as a union of a bonded knotoid diagram and finitely many knot diagrams equipped with bonds, and a bonded linkoid diagram, defined as an immersion of a disjoint union of finitely many intervals, each with bonds (Diamantis et al., 20 Jul 2025). This suggests that bonded knotoids are the one-interval member of a broader endpoint-bearing diagrammatic hierarchy.

The open-ended character of the underlying knotoid remains fundamental. In ordinary knotoid theory, the endpoints cannot be moved through strands by forbidden endpoint moves; if such forbidden moves were admitted, the endpoint structure would lose its topological force. Bonded knotoids inherit this endpoint sensitivity and refine it by requiring that bonds also respect analogous endpoint restrictions (Gügümcü et al., 2018).

3. Equivalence relations, move calculus, and categorical variants

Bonded knotoids occur in two equivalence types: topological bonded knotoids and rigid bonded knotoids. These are induced by the corresponding isotopies for bonded links, namely topological vertex equivalence and rigid vertex equivalence, with all moves performed away from the knotoid endpoints (Diamantis et al., 20 Jul 2025).

The allowed relations include planar isotopies of link arcs, bonds, or nodes; classical Reidemeister moves away from nodes; Reidemeister-type moves involving bonds; mixed Reidemeister moves involving both bonds and link arcs; vertex slide moves; topological vertex twists in the topological category; and rigid vertex twists in the rigid category, replacing topological vertex twists (Diamantis et al., 20 Jul 2025). The distinction between topological and rigid categories therefore persists unchanged when passing from bonded links to bonded knotoids.

The theory also extends the three bond categories from bonded links to bonded knotoids: long bonds, standard bonds, and tight bonds. The paper states explicitly that all three categories of bonded links and both isotopy types extend naturally to bonded knotoids. In this classification, long bonds need not be local, standard bonds are unknotted and in standard form, and tight bonds do not cross any diagram arcs (Diamantis et al., 20 Jul 2025).

The forbidden-move structure of knotoids remains operative. Ordinary knotoid theory forbids pulling a strand adjacent to an endpoint over or under another strand. Bonded knotoid theory adds that the same is true when, instead of a classical strand, one considers a bond. Thus a bond cannot be slid through an endpoint-adjacent forbidden configuration (Diamantis et al., 20 Jul 2025). This is an important point of principle: bonded knotoids are not obtained by freely attaching auxiliary arcs to a knotoid diagram. Their bond structure is constrained by the same endpoint-sensitive topology that makes ordinary knotoids nontrivial.

A common misconception is to treat bonded knotoids as if they were merely ordinary knotoids together with decorative extra arcs. The move calculus shows otherwise. The classification into topological versus rigid vertex types, together with long/standard/tight bond categories and endpoint-sensitive forbidden configurations, makes the bond data part of the isotopy type rather than an auxiliary annotation (Diamantis et al., 20 Jul 2025).

4. Closure operations and semi-closure

Bonded knotoids admit closure operations analogous to Turaev’s underpass and overpass closures for ordinary knotoids, but the closure arc must now pass relative to both strands and bonds. The underpass closure and overpass closure of a bonded knotoid diagram are obtained by connecting the endpoints with an arc that goes under or over each arc and each bond it meets, respectively. The result is a bonded knot (Diamantis et al., 20 Jul 2025).

This operation has two structural features that matter for classification. First, different closures of a bonded knotoid may result in non-isotopic bonded knots. Second, assuming a specific closure type, there is a well-defined surjective map from bonded knotoids to bonded knots (Diamantis et al., 20 Jul 2025). Closure therefore supplies a map to the closed theory, but not a complete invariant of the open theory.

The paper also defines a bonded semi-closure. If one allows the nodes of a bond to coincide with the endpoints of a knotoid, one obtains the bonded semi-closure of a bonded knotoid. More precisely, connecting the endpoints with a bond that goes under or over each arc and each bond it meets yields the underpass semi-closure or overpass semi-closure (Diamantis et al., 20 Jul 2025). The paper interprets this physically as a force bringing the endpoints together, but it does not yield a bonded knot.

These constructions clarify the relation between bonded knotoids and ordinary closure-based topology. In ordinary protein applications of knotoids, avoiding artificial closure is central because closure can change the inferred topology of an open chain (Goundaroulis et al., 2017). Bonded closure does not remove that caution; rather, it formalizes a controlled passage from an open bonded object to a closed bonded object while preserving the distinction between overpass and underpass conventions. A plausible implication is that closure is best viewed here as a functorial device for comparison with bonded-knot invariants, not as a replacement for the intrinsic open-ended theory.

5. Bonded braidoids and the Alexander-type closure theorem

The algebraic counterpart of bonded knotoids is the bonded braidoid. A bonded braidoid diagram is a braidoid diagram equipped with bonds. Its strands consist of classical braid strands and free strands; the bonds are simple horizontal arcs with nodes on strands; and the nodes differ from the braidoid endpoints and braidoid ends (Diamantis et al., 20 Jul 2025).

The isotopy theory for bonded braidoids uses bonded braid moves, bonded vertical moves, and bonded swing moves. For labeled bonded braidoid diagrams, each pair of corresponding ends is labeled $o$ or $u$, and closure is performed by joining corresponding ends with a vertical segment that passes over or under according to the label. This is the braidoid-style closure used to recover bonded knotoids (Diamantis et al., 20 Jul 2025).

The main structural theorem is the bonded-knotoid analogue of the Alexander theorem. The paper states:

Any oriented topological (enhanced) bonded (multi)-knotoid diagram is isotopic to the closure of a labeled (enhanced) bonded braidoid diagram.

It also states that any bonded multi-knotoid diagram is isotopic to the uniform closure of a bonded braidoid diagram, where “uniform” means that all labels are $u$ (Diamantis et al., 20 Jul 2025). Thus every bonded knotoid admits a braided representative.

The same paper develops a full generator-and-relation theory for bonded braids, defining the bonded braid monoid and formulating analogues of the Alexander and Markov theorems for bonded braids, including an $\Sigma$0-equivalence for bonded braids. By contrast, it does not develop a full standalone presentation for bonded braidoids in the same way; rather, bonded braidoids are introduced as the knotoid counterpart and are used to formulate the closure theorem (Diamantis et al., 20 Jul 2025). This asymmetry is explicit in the source and should not be collapsed: the theory of bonded braidoids is structural and closure-oriented, whereas the detailed monoid and group presentations are given for bonded braids.

6. Invariants, biological motivation, and neighboring frameworks

The bonded-knotoid section of the 2025 theory is primarily structural rather than invariant-theoretic. It does not present a dedicated bonded-knotoid polynomial invariant in that section. It does, however, emphasize that different closures can yield different bonded knots and that bonded knotoids model open chains with bonds in a way that may distinguish configurations not captured by ordinary knotoids. Earlier in the same paper, invariants are constructed for bonded links: unplugging invariants in the topological category, tangle insertion invariants in the rigid category, and the bonded bracket polynomial in the tight rigid category. The framework suggests that such invariants can be adapted, but that adaptation is not presented as part of the bonded-knotoid section itself (Diamantis et al., 20 Jul 2025).

The biological motivation is explicit. Bonded knotoids are described as especially relevant for modeling open chains such as proteins, because they allow one to represent both the open ends and internal bonds in a single diagram. The cited bond types include disulfide bridges, hydrogen bonds, salt bridges, and other intra-/inter-chain constraints. The models are said to capture the topology of open chains with inter and intra-chain bonds and to suggest new invariants for classifying biological macromolecules (Diamantis et al., 20 Jul 2025).

This motivation should be read against the ordinary knotoid approach to proteins. Projection-based knotoid methods study global and local entanglement of open protein chains without closure, identify dominant knotoids from many projection directions, and produce knotoid fingerprints for subchain analysis. In that context, knotoids were argued to provide a natural topological language for open protein chains, because proteins are not closed loops and closure can distort the geometry or alter the assigned topology (Goundaroulis et al., 2017). Bonded knotoids refine that paradigm by adding internal bonding structure rather than replacing it.

Several neighboring theories delimit the scope of bonded knotoids. The survey literature places them within the wider knotoid–braidoid framework for open-ended diagrams and their braided counterparts (Gügümcü et al., 2018). The work on forbidden move-distance studies ordinary knotoids, not bonded knotoids, but its analysis of small perturbations of projection direction and its protein-motivated sampling results are clearly adjacent (Barbensi et al., 2019). More general frameworks, such as generalized knotoids with multiple poles and knotoidal graphs, provide a larger ambient category that can encode shared endpoint structure, though they are not presented as the bonded-knotoid theory itself (Adams et al., 2022). Likewise, recent graph-polynomial and state-sum developments for knotoids and linkoids extend foundational machinery for endpoint-bearing diagrams, but do not directly define bonded knotoids (Chmutov et al., 2024).

A final misconception is therefore worth dispelling. Bonded knotoids are neither synonymous with ordinary protein knotoids nor merely a special case of generalized endpoint-bearing diagrams. They are a specific theory of knotoids with bonds, closure operations, isotopy categories, and braid-like counterparts. At the same time, the current literature shows that their invariant theory is less developed in explicit knotoid form than their structural and diagrammatic foundations. This suggests a mathematically natural next stage for the subject, but that suggestion remains an inference rather than a stated theorem (Diamantis et al., 20 Jul 2025).