Selective Synthesis of Ruthenium(II) Metalla[2]Catenane via Solvent and Guest-Dependent Self-Assembly

The Topological Transformation of Trefoil Knots to Solomon Links via Diels-Alder Click Reaction

The quest for more efficient, user-friendly, and less wasteful topological transformations remains a significant challenge in the realm of postassembly modifications. In this article, high yields of two molecular trefoil knots (Rh-1, Ir-1) were obtained using ligand 3,6-bis(3-(pyridin-4-yl)phenyl)-1,2,4,5-tetrazine (L1) with reactive tetrazine units and binuclear half-sandwich organometallic units [Cp*2M2(μ-TPPHZ)(OTf)2](OTf)2 (Rh-B, M = RhIII; Ir-B, M = IrIII). 2,5-Norbornadiene was used as an inducer of the Diels-Alder click reaction to modulate rapidly and efficiently the transformation of Trefoil knots to Solomon links. However, the key to achieving this topological structural change is the subtle increase in site steric of the pyridazine fragments (L2), which allows the molecular structures to spread and bend in three-dimensional space, as confirmed by single-crystal X-ray diffraction, ESI-TOF/MS, elementary analysis and detailed solution-state NMR techniques.

Unravelling the Roles of Solvophobic Effects and π⋅⋅⋅π Stacking Interactions in the Formation of [2]Catenanes Featuring Di-(N-Heterocyclic Carbene) Building Blocks

A series of [2]catenanes has been prepared from di-NHC building blocks by utilizing solvophobic effects and/or π⋅⋅⋅π stacking interactions. The dinickel naphthobiscarbene complex syn-[1] and the kinked biphenyl-bridged bipyridyl ligand L2 yield the [2]catenane [2-IL](OTf)4 by self-assembly. Solvophobic effects are pivotal for the formation of the interlocked species. Substitution of the biphenyl-linker in L2 for a pyromellitic diimide group gave ligand L3 , which yielded in combination with syn-[1] the [2]catenane [3-IL](OTf)4 . This assembly exhibits enhanced stability in diluted solution, aided by additional π⋅⋅⋅π stacking interactions. The π⋅⋅⋅π stacking was augmented by the introduction of a pyrene bridge between two NHC donors in ligand L4 . Di-NHC precursor H2 -L4 (PF6 )2 reacts with Ag2 O to give the [Ag2 L4 2 ]2 [2]catenane [4-IL](PF6 )4 , which shows strong π⋅⋅⋅π stacking interactions between the pyrene groups. This assembly was readily converted into the [Au2 L4 2 ]2 gold species [5-IL](PF6 )4 , which exhibits exceptional stability based on the strong π⋅⋅⋅π stacking interactions and the enhanced stability of the Au-CNHC bonds.

Coordination-directed self-assembly of nano-cages: metal ion-change, ligand-extending, shape-control and transdermal drug delivery

The combination of different pyridyl ligands and metal ions has proven to be a very reliable strategy for controlling the coordination mode of the heterometallic coordination nano-cages. Adjusting the length of the ligands could result in the selective synthesis of several heterometallic coordination nano-cages, either [8Rh + 2M]-4L, [8Rh + 2M]-5L or [8Rh + 4M]-6L cages, derived from the very same precursors (LH3tzdc) through half-sandwich rhodium self-assembly. Moreover, a series of [8Rh + 4M]-6L cages was chosen to exemplify the preparation. The rigidity of various pyridyl donor ligands caused the vertical nano-cage to be energetically preferred and was able to change the self-assembly process through ligand flexibility to selectively give the inclined nano-cage and cross nano-cage.

Photo- and Temperature-Induced Reversible Structural Transformation between Dodecanuclear and Pentadecanuclear Gold(I) Sulfido Complexes

Stimuli-responsive structural transformation has attracted much attention for its potential to mimic the behavior of biological transformations and functions. Here, two unprecedented dodecanuclear and pentadecanuclear gold(I) sulfido clusters (denoted trans-Au₁₂ and trans-Au₁₅, respectively) with impressive stimuli-responsive interconversion have been obtained by taking advantage of the judiciously designed tridentate phosphine ligand L_trans as the building block. Both UV light and temperature can be applied to trigger the structural conversions between trans-Au₁₂ and trans-Au₁₅. In addition, NMR, high-resolution electrospray ionization mass spectrometry, and UV-vis absorption spectroscopy have been employed to monitor the transformation process and decipher the mechanism of structural conversion. This work not only provides a paradigm to investigate photo-induced cluster-to-cluster transformation based on polydentate phosphine ligands but also offers a new direction for the construction of the stimuli-responsive materials.

Synthesis, characterization, and heparin-binding study of a self-assembled p-cymene-Ru(II) metallocycle based on a 4-amino-1,8-naphthalimide Tröger's base supramolecular scaffold

We report the very first example of a self-assembled p-cymene-Ru(II) metallocycle based on a green emitting 4-amino-1,8-naphthalimide Tröger's base (TBNap) supramolecular scaffold. A new cleft-shaped TBNap-derived di-4-picolyl donor was synthesized and reacted in a 2 : 2 stoichiometry ratio with a dinuclear Ru(II) acceptor (Ru-A) to generate a [2 + 2] self-assembled metallocycle (TBNap-Ru-MC) in good yield. Both TBNap and TBNap-Ru-MC showed positive solvatochromism in different solvents with varying polarities. In addition, the binding propensity of cationic TBNap-Ru-MC toward the heparin polyanion was determined using fluorescence titration studies. The initial fluorescence emission of TBNap-Ru-MC was quenched upon the gradual addition of the heparin polyanion, and the Stern-Volmer quenching constant (K_SV) was calculated to be 3.97 × 10⁵ M^-1.

Solvato-Controlled Assembly and Structural Transformation of Emissive Poly-NHC-Based Organometallic Cages and Their Applications in Amino Acid Sensing and Fluorescence Imaging

Stimuli-induced structural transformation of supramolecular cages has drawn increasing attention because of their sensitive feature to external variations as model systems to simulate biological processes. However, combining structural transformation and useful functions has remained a difficult task. This study reports the solvato-controlled self-assembly of two unique topologies with different emission characteristics, a water-soluble Ag₈ L₄ cage (A) and an Ag₄ L₂ cage (B), produced from the same sulfonate-pendant tetraphenylethene (TPE) bridged tetrakis-(1,2,4-triazolium) ligand. Both cages show interesting solvent-responsive reversible structural transformation, and the change of fluorescence signals can efficiently track the process. Additionally, water-soluble cage A exhibits unique properties in thermochromism, thiol amino acid sensing, and subcellular imaging in aqueous media.

Mapping the Assembly of Neutral Tetrahedral Cages Tethered by Oximido Linkers and Their Guest Encapsulation Studies

A primary criterion for the design of polyhedral metal-organic cages is the requirement of geometrically matched pairs of metal ions and ligand moieties. However, understanding the pathway it takes to reach the final polyhedral structure can provide more insights into the self-assembly process and improved design strategies. In this regard, we report two neutral tetrahedral cages with the formulas {[Pd₃(NⁱPr)₃PO]₄(L¹)₆} (1-TD) and {[Pd₃(NⁱPr)₃PO]₄(L²)₆} (2-TD) starting from the acetate-bridged cluster {[Pd₃(NⁱPr)₃PO]₂(OAc)₂(OH)}₂·2(CH₃)₂SO (HEXA-Pd) and the respective oxamide precursors (L¹H₂: [C₂(NH₂)₂O₂]) and (L²H₂: (C₂(NHMe)₂O₂]). When subtle variations in the reaction conditions were made, two new tetrameric Pd₁₂ assemblies, {[Pd₃(NⁱPr)₃PO]₄(L¹)₂(OAc)₄(OMe)₄} (1-TM) and {[Pd₃(NⁱPr)₃PO]₄(L²)₂(OAc)₄(OMe)₄} (2-TM), were obtained from the same precursors. Detailed investigations using NMR, mass spectrometry, X-ray crystallography, and computational studies indicate that the macrocyclic complexes 1-TM and 2-TM are the reaction intermediates involved in the formation of the tetrahedral cages 1-TD and 2-TD, respectively. Moreover, the tetrahedral cages 1-TD and 2-TD exhibited intrinsic cavities of volume ∼85 ų. Guest encapsulation studies revealed that the cage 1-TD can encapsulate a wide range of guest molecules such as CH₂Cl₂, CHCl₃, CCl₄, C₆H₆, and C₆H₅F. Interestingly, 1-TD was shown to exhibit a preferential binding of C₆H₅F and C₆H₆ over other halogenated guest molecules, as determined from NMR titrations and computational studies.

Anthracene-Containing Metallacycles and Metallacages: Structures, Properties, and Applications

Due to its highly conjugated panel-like structure and unique photophysical and chemical features, anthracene has been widely used for fabricating attractive and functional supramolecular assemblies, including two-dimensional metallacycles and three-dimensional metallacages. The embedded anthracenes in these assemblies often show synergistic effects on enhancing the desired supramolecular and luminescent properties. This review focuses on the metallasupramolecular architectures with anthracene-containing building blocks, as well as their applications in host-guest chemistry, stimulus response, molecular sensing, light harvesting, and biomedical science.

An Adaptable Water-Soluble Molecular Boat for Selective Separation of Phenanthrene from Isomeric Anthracene

Anthracene crude oil is a common source of phenanthrene for its industrial use. The isolation of phenanthrene from this source is a challenging task due to very similar physical properties to its isomer anthracene. We report here a water-soluble Pd(II) molecular boat (MB1) with unusual structural topology that was obtained by assembling a flexible tetrapyridyl donor (L) with a cis-Pd(II) acceptor. The flexible backbone of the boat enabled it to breathe in the presence of a guest optimizing the fit within the cavity. The boat binds phenanthrene more strongly than anthracene, which enabled separation of phenanthrene with an >98% purity from an equimolar mixture of the two isomers using MB1 as an extracting agent. MB1 represents a unique example of a coordination receptor suitable for selective aqueous extraction of phenanthrene from anthracene with reusability of several cycles.

Silver(I)-Carbene Bond-Directed Rigidification-Induced Emissive Metallacage for Picric Acid Detection

A new triphenylamine-based tetraimidazolium salt L was developed for silver(I)-carbene bond-directed synthesis of tetranuclear silver(I) octacarbene ([Ag₄(L)₂](PF₆)₄) metallacage 1. Interestingly, after assembly formation, metallacage 1 showed a nine-fold emission enhancement in dilute solution while ligand L was weakly fluorescent. This is attributed to the rigidity induced to the system by metal-carbene bond formation where the metal center acts as a rigidification unit. The enhanced emission intensity in dilute solution and the presence of the triphenylamine core made 1 a potential candidate for recognition of picric acid (PA). This recognition can be ascribed to the dual effect of ground-state charge-transfer complex formation and resonance energy transfer between the picrate and metallacage 1. For metallacage 1, a considerable detection limit toward PA was observed. The use of such metal-carbene bond-directed rigidification-induced enhanced emission for PA sensing is noteworthy.

Selective construction and stability studies of a molecular trefoil knot and Solomon link

Two novel compounds, a molecular trefoil knot and a Solomon link, were constructed successfully through the cooperation of multiple π-π stacking interactions. A reversible transformation between the trefoil knot and the corresponding [2 + 2] macrocycle could be achieved by solvent- and guest-induced effects. However, the Solomon link maintains its stability in different concentrations, solvents and guest molecules. Single-crystal X-ray crystallographic data, NMR spectroscopic experiments and ESI-MS support the synthesis and structural assignments. These synthesis methods open the door to the further development of smart materials, which will push the advancement of rational design of biomaterials.

Exciton Coupling in Redox-Active Salen based Self-Assembled Metallacycles

The incorporation of a redox-active nickel salen complex into supramolecular structures was explored via coordination-driven self-assembly with homobimetallic ruthenium complexes (bridged by oxalato or 5,8-dihydroxy-1,4-naphthoquinato ligands). The self-assembly resulted in the formation of a discrete rectangle using the oxalato complex and either a rectangle or a catenane employing the larger naphthoquinonato complex. The formation of the interlocked self-assembly was determined to be solvent and concentration dependent. The electronic structure and stability of the oxidized metallacycles was probed using electrochemical experiments, UV-Vis-NIR absorption, EPR spectroscopy and DFT calculations, confirming ligand radical formation. Exciton coupling of the intense near-infrared (NIR) ligand radical intervalence charge transfer (IVCT) bands provided further confirmation of the geometric and electronic structures in solution.

Reversible pH-Controlled Catenation of a Benzobisimidazole-Based Tetranuclear Rectangle

The development of methodologies to control on demand and reversibly supramolecular transformations from self-assembled metalla-structures requires the rational design of architectures able to answer to an applied stimulus. While solvent or concentration changes, light exposure or addition of a chemical have been largely explored to provide these transformations, the case of pH sensitive materials is less described. Herein, we report the first example of a pH-triggered dissociation of a coordination-driven self-assembled interlocked molecular link. It incorporates a pH sensitive benzobisimidazole-based ligand that can be selectively protonated on its bisimidazole moieties. This generates intermolecular electrostatic repulsions that reduces drastically the stability of the interlocked structure, leading to its dissociation without any sign of protonation of the pyridine moieties involved in the coordination bonds. Importantly, the dissociation process is reversible through addition of a base.

A Self-Assembled Palladium(II) Barrel for Binding of Fullerenes and Photosensitization Ability of the Fullerene-Encapsulated Barrel

Fullerene extracts obtained from fullerene soot lack their real application due to their poor solubility in common solvents and difficulty in purification. Encapsulation of these extracts in a suitable host is an important approach to address these issues. We present a new Pd₆ barrel (1), which is composed of three 1,4-dihydropyrrolo[3,2-b]pyrrole panels, clipped through six cis-Pd^II acceptors. Large open windows and cavity make it an efficient host for a large guest. Favorable interactions between the ligand and fullerene (C₆₀ and C₇₀ ) allows the barrel to encapsulate fullerene efficiently. Thorough investigation reveals that barrel 1 has a stronger binding affinity towards C₇₀ over C₆₀ , resulting in the predominant extraction of C₇₀ from a mixture of the two. Finally, the fullerene encapsulated barrels C₆₀ ⊂1 and C₇₀ ⊂1 were found to be efficient for visible-light-induced singlet oxygen generation. Such preferential binding of C₇₀ and photosensitizing ability of C₆₀ ⊂1 and C₇₀ ⊂1 are noteworthy.

Controlled Self-Assembly and Multistimuli-Responsive Interconversions of Three Conjoined Twin-Cages

Stimuli-responsive structural transformations between discrete coordination supramolecular architectures not only are essential to construct smart functional materials but also provide a versatile molecular-level platform to mimic the biological transformation process. We report here the controlled self-assembly of three topologically unprecedented conjoined twin-cages, i.e., one stapled interlocked Pd₁₂L₆ cage (2) and two helically isomeric Pd₆L₃ cages (3 and 4) made from the same cis-blocked palladium corners and a new bis-bidentate ligand (1). While cage 2 features three mechanically coupled cavities, cages 3 and 4 are topologically isomeric helicate-based twin-cages based on the same metal/ligand stoichiometry. Sole formation of cage 2 or a dynamic mixture of cages 3 and 4 can be controlled by changing the solvents employed during the self-assembly. Structural conversions between cages 3 and 4 can be triggered by changes in both temperature/solvent and induced-fit guest encapsulations. Well-controlled interconversion between such topologically complex superstructures may lay a solid foundation for achieving a variety of functions within a switchable system.

Self-assembly and guest-induced disassembly of triply interlocked [2]catenanes

Two triply interlocked [2]catenanes and one simple metallacage were constructed by tuning the widths of the organometallic dinuclear building blocks, and the interlocked architectures were disassembled by large aromatic molecules.

Efficient one-pot synthesis of [3]catenanes based on Pt(ii) metallacycles with a flexible building block

Three [3]catenanes were fabricated in high efficiency through the self-assembly of a 90° platinum(ii) receptor, a flexible bis(4,4′-bipyridinium) donor and a crown ether (DB24C8 or DB30C10).

Coordination assembly and host–guest chemistry of a triply interlocked [2]catenane

Triply catenated systems composed of two or more discrete coordination-metal cages through mechanical bonds exhibit excellent host–guest behaviors, which can be potentially applied in drug delivery systems.

Nucleation of Tiny Silver Nanoparticles by Using a Tetrafacial Organic Molecular Barrel: Potential Use in Visible-Light-Triggered Photocatalysis

Coordination-driven self-assembly of discrete molecular architectures of diverse shapes and sizes has been well studied in the last three decades. Use of dynamic imine bonds for designing analogous metal-free architectures has become a growing challenge recently. This article reports an organic molecular barrel (OB4^R ) as a potential template for nucleation and stabilization of very tiny (<1.5 nm) Ag nanoparticles (AgNPs). Imine bond condensation of a rigid tetra-aldehyde with a flexible diamine followed by imine-bond reduction yielded the discrete tetragonal organic barrel (OB4^R ). The presence of a molecular pocket ornamented with eight diamine moieties gives the potential for encapsulation of silver(I). The organic barrel was finally used as a molecular vessel for the controlled nucleation of silver nanoparticles (AgNPs) with fine size tuning through binding of Ag^I ions in the confined space of the barrel followed by reduction. Transmission electron microscopy (TEM) analysis of the Ag⁰ @OB4^R composite revealed that the mean particle size is 1.44±0.16 nm. The composite material has approximately 52 wt % silver loading. The barrel-supported ultrafine AgNPs [Ag⁰ @OB4^R ] are found to be an efficient photocatalyst for facile Ullmann-type aryl-amination coupling of haloarenes at ambient temperature without using any additives. The catalyst was stable for several cycles of reuse without any agglomeration. The new composite Ag⁰ @OB4^R represents the first example of discrete organic barrel-supported AgNPs employed as a photocatalyst in Ullmann-type coupling reactions at room temperature.

A hierarchical assembly strategy for near-infrared photothermal conversion: unconventional heterogeneous metalla[2]catenanes

Herein, we report a hierarchical assembly strategy for constructing heterogeneous half-sandwich organometallic D-A (D = π-donor, A = π-acceptor) interlocked structures, and their application in near-infrared (NIR) photothermal conversion. Thienothiophene and diketopyrrolopyrrole groups were selected as the D and A units, leading to two homogeneous metalla[2]catenanes with D-D-D-D and A-A-A-A stacks, respectively. By the ordered secondary assembly of homogeneous metalla[2]catenanes, two unprecedented heterogeneous D-A metalla[2]catenanes comprising an unusual mixed D-A-D-D and unconventional D-A-A-A stacks were realized by the combination of multiple noncovalent interactions, as all demonstrated by a detailed X-ray crystallographic study. Benefiting from the mixed D-A stacking modes, NIR absorption of heterogeneous D-A metalla[2]catenanes is significantly enhanced in contrast to homogeneous metalla[2]catenanes. Thanks to the enhanced NIR absorption and the fluorescence quenching effect from half-sandwich organometallic fragments, heterogeneous D-A metalla[2]catenanes displayed high-performance NIR photothermal conversion properties (η = 27.3%).

Self-assembly of metalla[3]catenanes, Borromean rings and ring-in-ring complexes using a simple π-donor unit

Despite extensive research and several stunning breakthroughs in the synthesis of interlocked molecular species, [3]catenanes, Borromean rings and ring-in-ring complexes are exceedingly rare and their targeted synthesis remains a formidable challenge. Herein, a series of Cp^*Rh-based homogeneous and heterogeneous interlocked structures have been prepared by coordination-driven self-assembly, not only including metalla[2]catenanes and molecular Borromean rings, but also linear metalla[3]catenanes and ring-in-ring complexes. The interlocked structures are all based on bithiophenyl groups. The bithiophenyl groups effectively enhance the strength of the inter-ring interactions and play a crucial role in the formation of these interlocked structures. By taking advantage of the strong interaction between π-donor (D) and π-acceptor (A) groups, the electron-deficient methylviologen cation was introduced into a cationic metallarectangle based on bithiophenyl groups. Taking inspiration from these results, a cationic metallarectangle based on A units was threaded into a metallarectangle based on D units, leading to a heterogeneous D–A ring-in-ring structure.

Selective synthesis and structural transformation between a molecular ring-in-ring architecture and an abnormal trefoil knot

The synthesis of complicated supramolecular architectures and the study of their reversible structural transformations remains a fascinating challenge in the field of supramolecular chemistry. Herein, two types of novel coordination compounds, a non-intertwined ring-in-ring assembly and an abnormal trefoil knot were constructed from a strategically selected Cp*Rh building block and a semi-rigid N,N'-bis(4-pyridylmethyl)diphthalic diimide ligand via coordination-driven self-assembly. Remarkably, the reversible transformation between the abnormal trefoil knot and the ring-in-ring assembly or the corresponding tetranuclear macrocycle could be achieved by the synergistic effects of Ag+ ion coordination and alteration of the solvent. Single-crystal X-ray crystallographic data and NMR spectroscopic experiments support the structural assignments.

Coordination-Directed Construction of Molecular Links

Since the emergence of the concept of chemical topology, interlocked molecular assemblies have graduated from academic curiosities and poorly defined species to become synthetic realities. Coordination-directed synthesis provides powerful, diverse, and increasingly sophisticated protocols for accessing interlocked molecules. Originally, metal ions were employed solely as templates to gather and position building blocks in entwined or threaded arrangements. Recently, metal centers have increasingly featured within the backbones of the integral structural elements, which in turn use noncovalent interactions to self-assemble into intricate topologies. By outlining ingenious recent examples as well as seminal classic cases, this Review focuses on the role of metal-ligand paradigms in assembling molecular links. In addition, the ever-evolving approaches to efficient assembly, the structural features of the resulting architectures, and their prospects for the future are also presented.

Temperature Controls Guest Uptake and Release from Zn4L4 Tetrahedra

We report the preparation of triazatruxene-faced tetrahedral cage 1, which exhibits two diastereomeric configurations (T1 and T2) that differ in the handedness of the ligand faces relative to that of the octahedrally coordinated metal centers. At lower temperatures, T1 is favored, whereas T2 predominates at higher temperatures. Host–guest studies show that T1 binds small aliphatic guests, whereas T2 binds larger aromatic molecules, with these changes in binding preference resulting from differences in cavity size and degree of enclosure. Thus, by a change in temperature the cage system can be triggered to eject one bound guest and take up another.

Investigation of the Anticancer Activity of Coordination-Driven Self-AssembledTwo-Dimensional Ruthenium Metalla-Rectangle

Coordination-driven self-assembly is an effective synthetic tool for the construction of spatially and electronically tunable supramolecular coordination complexes (SCCs), which are useful in various applications. Herein, we report the synthesis of a two-dimensional discrete metalla-rectangle [(η6-p-cymene)4Ru4(C6H2O4)2(2)2](CF3SO3)4 (3) by the reaction of a dinuclear half-sandwich ruthenium (II) complex [Ru2(η6-p-cymene)2(C6H2O4)Cl2] (1) and bis-pyridyl amide linker (2) in the presence of AgO3SCF3. This cationic ruthenium metalla-rectangle (3) has been isolated as its triflate salt and characterized by analytical techniques including elemental analysis, Fourier-transform infrared spectroscopy (FT-IR), proton nuclear magnetic resonance spectroscopy (1H-NMR), carbon nuclear magnetic resonance spectroscopy (13C-NMR), 1H-1H correlation spectroscopy (COSY), 1H-1H nuclear Overhauser effect spectroscopy (NOESY), diffusion ordered spectroscopy (DOSY), and high-resolution electrospray ionization mass spectrometry (HR-ESI-MS). Significantly, the 2D cationic ruthenium metalla-rectangle showed better anticancer activity towards three different cell lines (A549, Caki-1 and Lovo) as compared with the parent ruthenium complex (1) and the commercially used drug, cisplatin.

Coordination-driven self-assembly of a molecular figure-eight knot and other topologically complex architectures

Over the past decades, molecular knots and links have captivated the chemical community due to their promising mimicry properties in molecular machines and biomolecules and are being realized with increasing frequency with small molecules. Herein, we describe how to utilize stacking interactions and hydrogen-bonding patterns to form trefoil knots, figure-eight knots and [2]catenanes. A transformation can occur between the unique trefoil knot and its isomeric boat-shaped tetranuclear macrocycle by the complementary concentration effect. Remarkably, the realization and authentication of the molecular figure-eight knot with four crossings fills the blank about 4₁ knot in knot tables. The [2]catenane topology is obtained because the selective naphthalenediimide (NDI)-based ligand, which can engender favorable aromatic donor-acceptor π interactions due to its planar, electron-deficient aromatic surface. The stacking interactions and hydrogen-bond interactions play important roles in these self-assembly processes. The advantages provide an avenue for the generation of structurally and topologically complex supramolecular architectures.

Molecular knots and links continue to fascinate synthetic chemists. Here, the authors use stacking and hydrogen-bonding interactions between a set of similar building blocks to construct several complex molecular topologies, including a figure-eight knot and a trefoil knot.