Bonded Knots: Topology and Invariants

• Bonded knots are topological objects combining a knot or link with embedded bond arcs, used to model protein chains with intramolecular bridges.
• The theory employs diagrammatic isotopy, Reidemeister-type moves, and skein modules to classify objects and compute invariants such as the Yamada polynomial.
• Algebraic formulations, including bonded braids and Burau representations, extend classical braid theory to capture the complex topology of biochemical structures.

Bonded knots are topological objects consisting of a knot or link together with embedded bond arcs whose endpoints lie on the underlying curve. They were introduced as models for closed protein chains with intramolecular bridges, especially disulfide bonds, and they occupy an intermediate position between classical knot theory and the theory of spatial graphs. In the uncolored setting, a bonded knot may be viewed as a $3$-valent spatial graph with a perfect matching; in colored variants, the bonds also carry labels that encode bond type or other biochemical information. Recent work has organized the subject around isotopy theory, skein modules, braid-theoretic formalisms, and a small-complexity census of uncolored bonded knots and links (Gabrovšek et al., 19 Mar 2026, Gabrovšek et al., 26 Feb 2025, Diamantis et al., 20 Jul 2025).

1. Definition and basic categories

A bonded link is defined as a pair $(L,\mathbf b)$ or, in the colored oriented setting, a triple $(L,\mathbf b,\chi)$, where $L$ is an embedded knot or link in $S^3$, $\mathbf b=\{b_1,\dots,b_n\}$ is a collection of pairwise disjoint embedded closed intervals whose endpoints lie on $L$, and $\chi$ assigns colors to bonds when color is present. A bonded knot is the one-component case. The bonds are not abstract combinatorial ties: they are embedded arcs in the complement of the backbone, with endpoints attached transversely to the link. This makes bonded knots special cases of edge-colored trivalent spatial graphs and distinguishes them from tied-link models, where ties are not embedded and can be moved freely through a diagram (Gabrovšek et al., 19 Mar 2026, Gabrovsek, 2019, Diamantis et al., 20 Jul 2025).

In the uncolored case used for the small-complexity classification, no chemical labels are attached to the bonds. Each bond contributes a pair of $3$-valent vertices, and the resulting graph-theoretic object can be regarded as a $3$-valent graph with a perfect matching. In the colored theory, colors can encode different bond types, and a later refinement allows contact-distance information to be incorporated so that the formalism can distinguish protein circuit-topology data that would otherwise collapse in non-rigid skein theory (Gabrovsek, 2019).

A further geometric organization, introduced in the 2025 synthesis of the subject, separates bonded links into three categories. In long bonded links, bonds may be knotted, linked, and may interact with link strands in complicated ways. In standard bonded links, each bond is isotoped to be unknotted and has no self-crossings. In tight bonded links, bonds are standard and additionally do not cross any link arcs. The same source states that every bonded link diagram can be isotoped into standard form, and every standard bonded link can be isotoped to tight form, so these categories function partly as normal forms for diagrammatic work (Diamantis et al., 20 Jul 2025).

2. Isotopy theory, diagrammatics, and complexity

The ambient equivalence relation is isotopy in $(L,\mathbf b)$0, expressed diagrammatically through generalized Reidemeister moves. In the basic $(L,\mathbf b)$1-valent-graph picture one has the usual Reidemeister moves $(L,\mathbf b)$2, $(L,\mathbf b)$3, and $(L,\mathbf b)$4, together with moves $(L,\mathbf b)$5 and $(L,\mathbf b)$6 governing the interaction of crossings with bond endpoints. Following Kauffman’s distinction, one obtains rigid knotted graphs when move $(L,\mathbf b)$7 is disallowed and topological knotted graphs when it is allowed. The classification of uncolored bonded knots with small singularity number works in the topological setting, while several algebraic constructions develop both topological and rigid versions (Gabrovšek et al., 19 Mar 2026, Cavicchioli et al., 6 Jul 2025).

A parallel formulation for bonded link diagrams uses planar isotopy, classical Reidemeister moves on link arcs away from nodes, Reidemeister-type moves on bonded arcs, mixed Reidemeister-type moves involving link arcs and bonds, vertex slide moves, and either topological vertex twists (TVT) or rigid vertex twists (RVT) depending on the category. For diagrams with isolated bonds, this can be repackaged into bond-preserving move systems such as $(L,\mathbf b)$8 in the topological case, with rigid analogues in the rigid case (Cavicchioli et al., 6 Jul 2025, Diamantis et al., 20 Jul 2025).

For tabulation, the principal complexity measure is the singularity number

$(L,\mathbf b)$9

where $(L,\mathbf b,\chi)$0 is the minimal crossing number over all diagrams of $(L,\mathbf b,\chi)$1 and $(L,\mathbf b,\chi)$2 is the number of bonds. The first systematic census classifies all uncolored bonded knots and links with $(L,\mathbf b,\chi)$3, directly paralleling classical knot tables organized by crossing number (Gabrovšek et al., 19 Mar 2026).

Computer treatment uses Planar Diagram (PD) notation adapted to graphs with vertices. Arcs are numbered consecutively; each $(L,\mathbf b,\chi)$4-valent vertex is recorded by listing the incident arcs in counterclockwise order; and for a crossing, the list begins with the undercrossing arc. A representative example is

$(L,\mathbf b,\chi)$5

This encoding is used for enumeration, simplification, and comparison of candidate diagrams (Gabrovšek et al., 19 Mar 2026).

3. Polynomial and skein-theoretic invariants

The most effective invariant in the small-complexity classification is the Yamada polynomial, an invariant of spatial graphs. In the singularity-$(L,\mathbf b,\chi)$6 census it distinguishes all but two cases; the two exceptional ambiguities are both pairs of order-$(L,\mathbf b,\chi)$7 connected sums built from the same factors. Specifically, $(L,\mathbf b,\chi)$8 and $(L,\mathbf b,\chi)$9 are both order-$L$0 connected sums $L$1, and $L$2 and $L$3 are both order-$L$4 connected sums of two copies of $L$5. The paper presents this as a subtlety of order-$L$6 connected sums rather than a broad failure of the invariant (Gabrovšek et al., 19 Mar 2026).

A second major invariant framework is the bonded Kauffman bracket skein module. In the rigid case, the module $L$7 is defined from the free $L$8-module on rigid-vertex framed bonded links modulo the classical bracket skein relation and the disjoint-union relation, with $L$9, $S^3$0, and $S^3$1 invertible in the coefficient domain. The basic generators are the one-bond $S^3$2-curve $S^3$3 and the handcuff link $S^3$4, and the principal theorem states that the module is freely generated by

$S^3$5

Consequently, the module is infinitely generated and torsion-free. In the topological case, the relation

$S^3$6

eliminates $S^3$7, and the module is freely generated by $S^3$8, again infinitely generated and torsion-free. The associated reduced rigid bonded bracket polynomial is

$S^3$9

lying in Laurent polynomials in $\mathbf b=\{b_1,\dots,b_n\}$0 with polynomial dependence on $\mathbf b=\{b_1,\dots,b_n\}$1 and $\mathbf b=\{b_1,\dots,b_n\}$2 (Gabrovšek et al., 26 Feb 2025).

The colored theory admits a HOMFLYPT skein module of colored bonded knots. In the rigid setting, this module is freely generated by products of colored elementary $\mathbf b=\{b_1,\dots,b_n\}$3-curves and handcuff links. In the non-rigid setting, the elementary bond types collapse to products of $\mathbf b=\{b_1,\dots,b_n\}$4 alone; the paper states explicitly that the non-rigid module “does not provide information about the knottedness of the bonds.” That limitation motivates refined coloring schemes encoding contact distance or circuit-topology information, which can distinguish protein examples such as CN29 and ADWX-1 (Gabrovsek, 2019).

A further invariant package is available in the topological and rigid categories developed for standard and tight bonded links. In that setting, one has the unplugging invariant $\mathbf b=\{b_1,\dots,b_n\}$5, obtained by vertex unpluggings and valued in sets of classical links, and the tangle insertion invariant $\mathbf b=\{b_1,\dots,b_n\}$6, obtained by replacing each bond by a band and inserting a classical $\mathbf b=\{b_1,\dots,b_n\}$7-tangle. For tight rigid bonded links there is also a bonded bracket polynomial $\mathbf b=\{b_1,\dots,b_n\}$8, a Kauffman-bracket-type regular-isotopy invariant. The paper gives, for example,

$\mathbf b=\{b_1,\dots,b_n\}$9

for the unknot with $L$0 bonds (Diamantis et al., 20 Jul 2025).

4. Braid-theoretic and algebraic formulations

Bonded knot theory admits an algebraic counterpart in the form of bonded braids. One approach defines the topological bonded braid monoid $L$1 on $L$2 strands with generators $L$3 and bond generators $L$4, subject to the standard braid relations together with bond relations and mixed relations such as

$L$5

The rigid bonded braid monoid $L$6 adds kink generators $L$7. Inclusions $L$8 recover the classical braid group as the bond-free subtheory (Cavicchioli et al., 6 Jul 2025).

The fundamental structural results are bonded analogues of the Alexander and Markov theorems. Every topological bonded knot is the closure of an element of $L$9, and every rigid bonded knot is the closure of an element of $\chi$0. For topological bonded braids, two closures are equivalent if and only if the corresponding braids are related by a finite sequence of conjugations,

$\chi$1

cyclic permutations of bonds,

$\chi$2

and stabilizations,

$\chi$3

The rigid version is analogous, with cyclic permutations of the $\chi$4 as well (Cavicchioli et al., 6 Jul 2025).

A parallel presentation defines a bonded braid monoid $\chi$5 generated by braid generators and bond generators $\chi$6, with reduced presentations using only adjacent bonds $\chi$7. In that framework, the monoid is isomorphic to the singular braid monoid $\chi$8. The same work formulates bonded $\chi$9-equivalence, a Markov theorem expressed through conjugation, stabilization, and bond commuting, and an enhanced theory with attracting and repelling bonds represented algebraically by $3$0 and $3$1. It also extends the theory to bonded knotoids and bonded braidoids, thereby treating open-chain objects with bonds and defining overpass, underpass, and bonded semi-closure operations (Diamantis et al., 20 Jul 2025).

The braid-theoretic approach also supports matrix representations. The bonded Burau representation

$3$2

extends the classical Burau representation by sending $3$3 to explicit matrices $3$4; a reduced bonded Burau representation $3$5 is obtained by factoring out a trivial $3$6-dimensional summand. The low-dimensional behavior is partially resolved: the reduced representation is faithful for $3$7 and for $3$8, faithfulness is unknown for $3$9, and non-faithfulness holds for $3$0 (Cavicchioli et al., 6 Jul 2025).

5. Classification at singularity number at most seven

The first systematic census concerns uncolored bonded knots and links with singularity number $3$1. The computation was implemented in Python using KnotPy and proceeds by a graph-generation and reduction pipeline. First, all non-isomorphic planar graphs with up to seven vertices were generated using plantri, subject to connectivity at least $3$2, minimum degree $3$3, between $3$4 and $3$5 edges for $3$6 vertices, and only vertices of degree $3$7 or $3$8. Vertices of degree less than $3$9 were parallelized until they fit the allowed local structure; degree-$(L,\mathbf b)$00 vertices were interpreted as crossings and degree-$(L,\mathbf b)$01 vertices as bond endpoints. Graphs violating the parity condition that the number of $(L,\mathbf b)$02-valent vertices be even were discarded. This produced 927 candidate graphs (Gabrovšek et al., 19 Mar 2026).

Each degree-$(L,\mathbf b)$03 vertex was then resolved as one of the two crossing types, producing 20,019 candidate diagrams. After simplification via brute-force Reidemeister moves and reduction to canonical form using KnotPy’s simplify and canonical routines, 1,437 canonical representatives remained. Grouping by Yamada polynomial yielded 47 singleton diagrams, while the remaining 1,390 diagrams fell into 137 groups requiring additional equivalence testing. Running reduce_equivalent_diagrams with increasing crossing-increasing depth produced 92 more unique diagrams at depth $(L,\mathbf b)$04, 13 more at depth $(L,\mathbf b)$05, and 2 more at depth $(L,\mathbf b)$06; depth $(L,\mathbf b)$07 with flypes produced no further reductions. The remaining ambiguous cases were checked manually. After removing disjoint unions, connected sums, and classical knots and links, the authors obtained 51 bonded knot diagrams; after mirror identification and exclusion of one non-bonded graph, the final tables contained 30 types (Gabrovšek et al., 19 Mar 2026).

These $(L,\mathbf b)$08 types divide into three graph-theoretic families:

Family	Notation	Count
Bonded knots	$(L,\mathbf b)$09	20
Bonded handcuff links	$(L,\mathbf b)$10	6
Bonded links	$(L,\mathbf b)$11	4

Among the $(L,\mathbf b)$12 types, 7 are order-$(L,\mathbf b)$13 connected sums, marked by superscript $(L,\mathbf b)$14; 10 are achiral and 20 are chiral. The paper adopts the convention that a bonded knot is prime if it does not decompose as an order-$(L,\mathbf b)$15 or order-$(L,\mathbf b)$16 connected sum into nontrivial pieces, while order-$(L,\mathbf b)$17 connected sums are still treated as prime because factorization is not unique in that case. This differs from Moriuchi’s convention, in which order-$(L,\mathbf b)$18 sums were regarded as non-prime (Gabrovšek et al., 19 Mar 2026).

Representative entries include $(L,\mathbf b)$19, whose Yamada polynomial is

$(L,\mathbf b)$20

$(L,\mathbf b)$21, with

$(L,\mathbf b)$22

and $(L,\mathbf b)$23, whose Yamada polynomial is $(L,\mathbf b)$24. The tables also record chirality, PD code, and explicit diagrams. One generated prime graph was excluded because its non-loop edges cannot be colored so that no two bonds meet, so it is not a bonded knot in the sense adopted for the classification (Gabrovšek et al., 19 Mar 2026).

6. Protein topology, applications, and extensions

The biological motivation is explicit: bonded knots are intended to model protein-like chains with intramolecular bridges. The backbone is represented by a closed curve in $(L,\mathbf b)$25, while the bridges are represented by additional embedded arcs. The classification paper emphasizes that disulfide bridges are common enough to matter topologically, noting that roughly $(L,\mathbf b)$26 of proteins contain at least one such covalent bond. Within this framework, bonded knots model cystine knots, one-bond $(L,\mathbf b)$27-curves, more complicated bonded knots with several bonds, and lasso proteins in which a backbone segment pierces a surface associated to the bond-backbone arrangement. The same paper positions bonded knots as an extension of circuit topology because they encode not only bond arrangement but also the actual spatial knotting of the backbone (Gabrovšek et al., 19 Mar 2026).

Skein-theoretic work reinforces this protein orientation. The bonded Kauffman bracket theory treats proteins with intramolecular bonds, especially disulfide-bonded proteins and other “entangled proteins,” and includes a worked example based on the TRTX-Tp1a toxin. The colored HOMFLYPT theory is likewise motivated by hydrogen bonds, salt bridges, and disulfide bonds, and proposes refined coloring as a way to encode circuit-topology information. These constructions show that bonded-knot invariants are not merely abstract graph invariants: they are designed to separate topologies arising from biologically stabilized folded chains (Gabrovšek et al., 26 Feb 2025, Gabrovsek, 2019).

Recent extensions broaden the scope beyond closed backbones. The theory of bonded knotoids and bonded braidoids models open chains with inter- and intra-chain bonds, defines closure operations adapted to endpoints, and suggests new invariants for biological macromolecules. A plausible implication is that the closed-chain bonded-knot framework and the open-chain bonded-knotoid framework together provide a common topological language for macromolecular structures in which bond placement is part of the topology rather than a secondary annotation (Diamantis et al., 20 Jul 2025).

Bonded knots therefore constitute a distinct research domain: more structured than classical knot theory, more constrained than arbitrary spatial graph theory, and directly calibrated to chemically bonded biomolecular chains. Their current theory combines spatial-graph isotopy, polynomial and skein invariants, braid monoids and Markov-type equivalence, and a completed low-complexity tabulation in the uncolored case (Gabrovšek et al., 19 Mar 2026, Cavicchioli et al., 6 Jul 2025).