Bonded Braidoids: Algebraic and Topological Extensions

• Bonded braidoids are planar diagrams combining classical braid strands, free endpoints, and horizontal bonds to model open-chain interactions.
• They extend ordinary braidoids by incorporating additional generators, geometric constraints, and closure techniques analogous to Alexander–Markov correspondences.
• The framework supports robust invariant constructions and computational methods for classifying open polymer chains and biological macromolecules.

Searching arXiv for the cited papers to ground the article in the relevant literature. First, I’ll look up the 2025 paper on bonded knots and braids, since it appears to introduce bonded braidoids. Now I’ll verify the earlier survey on knotoids and braidoids to situate bonded braidoids historically and terminologically. Bonded braidoids are the algebraic counterparts of bonded knotoids: open knot diagrams with bonds, modeled in a braid-like planar form with both classical strands and free strands terminating at interior endpoints. In the framework developed in "Topology and Algebra of Bonded Knots and Braids" (Diamantis et al., 20 Jul 2025), they extend ordinary braidoids by allowing embedded bonds represented by horizontal arcs attached transversely to strand nodes, thereby encoding open chains with inter- and intra-chain interactions. The construction places bonded braidoids at the intersection of braid theory, knotoid theory, and the topology of bonded open chains, while preserving Alexander- and Markov-type correspondences with bonded knotoids (Diamantis et al., 20 Jul 2025).

1. Historical and conceptual placement

The immediate antecedent of bonded braidoids is the theory of knotoids and braidoids. The 2018 survey "A survey on knotoids, braidoids and their applications" describes knotoids as open-ended knot diagrams in surfaces and braidoids as geometric objects analogous to classical braids, forming a counterpart theory to planar knotoids (Gügümcü et al., 2018). However, that survey does not introduce or develop any notion of bonded braidoids; its braidoid theory is explicitly confined to ordinary, un-bonded braidoids (Gügümcü et al., 2018).

Bonded braidoids therefore arise as a later extension of the braid–knotoid correspondence. In the 2025 framework, bonded knots are classical knots endowed with embedded bonding arcs modeling physical or chemical bonds, and bonded braidoids are introduced only after the development of bonded knots, bonded braids, enhanced bonded knots and braids, and bonded knotoids (Diamantis et al., 20 Jul 2025). This places bonded braidoids not as an isolated generalization of braids, but as the open-ended, endpoint-sensitive analogue of bonded braids.

A common misconception is that bonded braidoids are already implicit in earlier braidoid theory. The literature cited here does not support that view: ordinary braidoids and bonded braidoids are distinct notions, with the latter requiring additional geometric data, additional generators, and a distinct closure theory (Gügümcü et al., 2018).

2. Geometric definition and isotopy categories

Let $\Sigma$ be an oriented surface, with $\Sigma = \mathbb{R}^2$ or $S^2$, and fix nonnegative integers $n$, $p$, and $q$, where $n$ is the number of classical braid strands and $p,q$ are the numbers of top and bottom free strands. A bonded braidoid diagram on $(n,p,q)$ strands is a planar diagram in the rectangle $[0,1]\times[0,1]\subset \mathbb{R}^2$ consisting of four types of data (Diamantis et al., 20 Jul 2025):

• $\Sigma = \mathbb{R}^2$0 classical strands, monotonically directed downward from distinct top-boundary points to distinct bottom-boundary points;
• $\Sigma = \mathbb{R}^2$1 upper free strands, each running from a specified top-boundary point to an interior endpoint;
• $\Sigma = \mathbb{R}^2$2 lower free strands, each running from a specified bottom-boundary point to an interior endpoint;
• $\Sigma = \mathbb{R}^2$3 embedded bonds, namely disjoint horizontal arcs attached transversely at two nodes on the strands, with nodes not allowed to coincide with endpoints or boundary points.

All strand crossings are generic transversal double points equipped with standard over/under information. If $\Sigma = \mathbb{R}^2$4, the construction reduces to the ordinary $\Sigma = \mathbb{R}^2$5 braidoid diagrams of Gügümcü–Lambropoulou (Diamantis et al., 20 Jul 2025).

The theory distinguishes several geometric subcategories. A standard bonded braidoid is one in which each bond is presented in an $\Sigma = \mathbb{R}^2$6-neighborhood with no self-crossings. A tight bonded braidoid forbids any crossings between bonds and strands. The isotopy theory is also bifurcated. In the topological vertex isotopy category, nodes may move freely along strands, so TVT is allowed. In the rigid vertex isotopy category, nodes are carried along rigid 3-balls, so TVT is disallowed and only RVT is permitted (Diamantis et al., 20 Jul 2025).

These distinctions are structurally significant. They determine which local moves are admissible, which invariants are well defined, and whether bond–strand crossings can appear at all. A plausible implication is that the theory is intentionally stratified so that different physical or geometric bonding constraints can be represented within a single formalism.

3. Algebraic structures: monoids, groups, generators, and relations

Let $\Sigma = \mathbb{R}^2$7 denote isotopy classes of bonded braidoid diagrams on $\Sigma = \mathbb{R}^2$8 strands under topological vertex isotopy. Concatenation, obtained by gluing bottom to top, gives $\Sigma = \mathbb{R}^2$9 a monoid structure; in the tight category, the same operation defines a submonoid. In the enhanced setting, where bonds are distinguished as attracting or repelling, one obtains a group $S^2$0 by adjoining inverses for each elementary bond (Diamantis et al., 20 Jul 2025).

For $S^2$1, one may take the following generators (Diamantis et al., 20 Jul 2025):

• $S^2$2, for $S^2$3, the usual braid generators acting on the $S^2$4 classical strands;
• $S^2$5, for $S^2$6, elementary bonds attached to strands $S^2$7;
• $S^2$8, for $S^2$9, upper-endpoint creation on some classical strand;
• $n$0, for $n$1, lower-endpoint creation.

In the enhanced group $n$2, one also has $n$3, interpreted as repelling bonds, with $n$4 (Diamantis et al., 20 Jul 2025).

The defining relations are the following, for all admissible indices and with $n$5 denoting the monoid identity (Diamantis et al., 20 Jul 2025): $n$6

$n$7

$n$8

$n$9

$p$0

$p$1

$p$2

$p$3

$p$4

together with the usual braidoid endpoint relations of Gügümcü–Lambropoulou. In $p$5, one further has $p$6 (Diamantis et al., 20 Jul 2025).

A key structural remark is that the submonoid generated by $p$7 is isomorphic to the bonded braid monoid $p$8, and hence to the singular braid monoid. The $p$9 and $q$0 generators are what enrich the structure from bonded braids to bonded braidoids (Diamantis et al., 20 Jul 2025). This suggests that bonded braidoids interpolate between two established algebraic regimes: braid-like bond algebra on one side and endpoint-bearing braidoid algebra on the other.

4. Closure, braiding procedures, and Alexander–Markov theory

If $q$1 is a bonded knotoid diagram in the plane, possibly multi-component and equipped with specified standard bonds, it can be converted into a bonded braidoid by a braiding procedure. The diagram is first brought into braidoid general position, meaning that it has no horizontal segments and no vertical alignments of nodes or endpoints. One then subdivides all up-arcs, with respect to the vertical direction, and performs the usual $q$2-braiding moves of Lambropoulou–Rourke while keeping bonds horizontal; TVT may be applied to reorient at nodes as needed. The outcome is a labeled bonded braidoid $q$3 whose oriented closure recovers the bonded knotoid $q$4 (Diamantis et al., 20 Jul 2025).

The resulting closure map is denoted

$q$5

Its closure convention joins each free-strand end at the top to its partner at the bottom by a small vertical segment passing entirely under or over according to the original knotoid closure convention (Diamantis et al., 20 Jul 2025).

Within this framework, the Alexander-type theorem states that every oriented bonded (multi)-knotoid in the topological vertex category arises as the closure of some standard bonded braidoid. The proof adapts the classical up-arc braiding algorithm of Lambropoulou–Rourke: one slides and twists at nodes to eliminate up-arcs through TVT moves, braids the underlying link, and brings bonds to horizontal position (Diamantis et al., 20 Jul 2025).

The Markov-type theorem characterizes when two labeled bonded braidoids have isotopic closures as bonded knotoids. This occurs if and only if they are related by braidoid isotopy, $q$6-moves on any strand away from nodes or endpoints, bond commuting $q$7, endpoint commuting $q$8 and $q$9, together with the usual braidoid endpoint relabelings. Equivalently, the $n$0-moves may be replaced by classical Markov stabilizations $n$1 and conjugations (Diamantis et al., 20 Jul 2025).

These theorems place bonded braidoids in the same formal role that ordinary braids occupy for knots and ordinary braidoids occupy for knotoids: they provide a diagrammatic-algebraic normal form for an open-ended bonded topology.

5. Invariants

Three invariant constructions are singled out for bonded braidoids, each tied to a particular isotopy regime or geometric category (Diamantis et al., 20 Jul 2025).

The first is the unplugging invariant in the topological vertex category. A bonded braidoid is viewed as a trivalent graph, and each node is unplugged by removing one incident edge. This produces a finite set of classical braidoids $n$2. Any chosen braid invariant, such as HOMFLY-PT, applied to all members of $n$3 yields an invariant of the original bonded braidoid (Diamantis et al., 20 Jul 2025). The method reduces bonded data to a controlled family of classical objects.

The second is the tangle-insertion invariant, defined for the rigid vertex standard or tight category. Each bond is replaced by a band, and a fixed collection of 2-tangles is inserted. The resulting classical link or braid then carries invariants such as the Jones polynomial or bracket polynomial, which become rigid-vertex invariants of the bonded braidoid (Diamantis et al., 20 Jul 2025). Here the bond is not deleted but resolved through a prescribed local replacement.

The third is the bonded bracket polynomial, defined in the rigid-tight category. It extends the Kauffman bracket by including bonds through skein rules of the form

$n$4

with formal coefficients $n$5, followed by the usual evaluation on the resolved classical diagram. This polynomial is invariant under $n$6, $n$7, and braided-vertex slide moves, hence a regular isotopy invariant of tight bonded braidoids (Diamantis et al., 20 Jul 2025).

Taken together, these constructions show that bonded braidoid invariants may be obtained either by reduction to classical braidoids, by local tangle substitution, or by direct skein extension. A plausible implication is that the theory is designed to accommodate both purely algebraic and diagrammatic-combinatorial approaches to classification.

6. Examples, embeddings, and modeled applications

The theory includes explicit examples illustrating both notation and distinguishing power. One two-strand example with one upper endpoint, one lower endpoint, and one bond is

$n$8

Under resolution of the bond by the bonded-bracket skein, one obtains three classical braidoids with brackets

$n$9

so that

$p,q$0

Since these three are distinct, the resulting bonded bracket distinguishes $p,q$1 from a different bond arrangement (Diamantis et al., 20 Jul 2025).

A three-strand example with two adjacent bonds is

$p,q$2

It is stated not to be bond-commuting equivalent to

$p,q$3

as can be seen either from their bonded-bracket expansions or from plugging-unplugging, since the sets of classical closures differ (Diamantis et al., 20 Jul 2025). These examples demonstrate that the position and order of bonds can encode information not removable by the admissible commuting relations.

There is also a natural injection of the bonded-braid monoid $p,q$4 into the bonded-braidoid monoid $p,q$5, obtained by adjoining a trivial upper and lower endpoint with no crossings or bonds. Under this embedding, braid invariants extend to braidoid invariants (Diamantis et al., 20 Jul 2025). This makes bonded braidoids a genuine extension rather than merely a parallel theory.

The intended application domain is the classification of open polymer chains modeled by knotoids, augmented by intra-chain and inter-chain interactions represented by bonds. More broadly, the framework is presented as capturing the topology of open chains with inter and intra-chain bonds and as suggesting new invariants for classifying biological macromolecules (Diamantis et al., 20 Jul 2025). This suggests a topology-first formal language for open bonded structures that cannot be fully represented either by closed bonded knots or by ordinary, un-bonded braidoids alone.