Molecular Knots

Abstract The first synthetic molecular trefoil knot was prepared in the late 1980s. However, it is only in the last few years that more complex small‐molecule knot topologies have been realized through chemical synthesis. The steric restrictions imposed on molecular strands by knotting can impart significant physical and chemical properties, including chirality, strong and selective ion binding, and catalytic activity. As the number and complexity of accessible molecular knot topologies increases, it will become increasingly useful for chemists to adopt the knot terminology employed by other disciplines. Here we give an overview of synthetic strategies towards molecular knots and outline the principles of knot, braid, and tangle theory appropriate to chemistry and molecular structure.


Introduction
Knots are basic elements of structure that are exploited in tools,m aterials,a rchitecture,a nd construction. [1] In prehistory,t he ability to tie knots had am ajor impact on human development, enabling early man to make useful implements such as bolas,a xes with blades tied to handles,a nd fishing nets, [2] and eventually to weave fabrics.Knots have even been used by some civilizations to store and pass on information [3] ( Figure 1).
Humans are not the only species to use knots.O ther primates have been observed to tie knots to make tools ( Figure 2a). [4] Some birds,as pectacular example being the weaver bird, incorporate knots into their nests [5,6] (Figure 2b), and hagfish and some eels tie themselves into knots as The first synthetic molecular trefoil knot was prepared in the late 1980s.However, it is only in the last few years that more complex small-molecule knot topologies have been realized through chemical synthesis.The steric restrictions imposed on molecular strands by knotting can impart significant physical and chemical properties, including chirality,s trong and selective ion binding, and catalytic activity.A sthe number and complexity of accessible molecular knot topologies increases,itwill become increasingly useful for chemists to adopt the knot terminology employed by other disciplines.H ere we give an overview of synthetic strategies towards molecular knots and outline the principles of knot, braid, and tangle theory appropriate to chemistry and molecular structure. Glossary 11190 Figure 1. The impact of knotting on technology:a)Spherical stones thought to be bola weights, which would need to be tied together for hunting, date back 500 000 years. [1] b) The oldest surviving man-made knots are those of the Antrea net, afishing net made of willow with a6cm mesh dating to 8300 BC. [2] c) Knots, in the form of quipu, have been used to record and communicate information, with the earliest examples possibly predating the invention of the written word. [3] Image (a) "Stone ball from aset of Paleolithic bolas" reproduced from https://goo.gl/vyAh85 (downloaded 5M ay 2017) under awikimedia creative commons license. Image (b) "Pieces of the Antrea net" reproduced from https://goo.gl/y0026E (downloaded 5M ay 2017) under awikimedia creative commons license. Image (c) "Quipu from the Inca Empire" reproduced from https://goo.gl/tqZyPW (downloaded 5M ay 2017) under aw ikimediacreative commons license. Knotting exploited by animals:a)W attana the orangutan tying aknot. [4] b) The African weaver bird uses knots to tie its nest securely. [5,6] c) Hagfish knot their bodies to generate force when pulling flesh off acarcass. [7] Image (a) reproduced from Ref, [4] with permission from the University of Chicago Press. Image (b) "southern masked weaver by wim de groot" reproduced from https://goo.gl/ ZpD09h (downloaded 5May 2017) under aw ikimediacreative commons license. Image (c) reproduced from Ref. [7] with permission from Springer Nature.
am echanism to generate leverage when tugging at flesh [7] ( Figure 2c). Thep hysical significance of knotting is increasingly becoming apparent in fields as varied as colloids, [8] liquid crystals, [9] optical beams, [10] soap films, [11] superfluids, [12] and the origins of the early universe. [13] In molecular terms,knots are found in circular DNA [14] and approximately 1% of proteins, [15] and they form spontaneously in polymer chains of sufficient length and flexibility. [16] As every sailor,m ountaineer, and scout knows,d ifferent types of knots have different characteristics that make them more or less suited for ap articular task:" bend knots" give the strongest binding between two lengths of rope;"hitches" are best for tying rope around an object;a nd "loop knots", or "nooses", allow degrees of movement between the components they connect. [17] There is no reason to suppose that different types of knots will not be just as important, versatile,and useful in the molecular world. However,s cientists will not be able to investigate that hypothesis until they are able to access asignificant range of different molecular knot topologies.
Ther igorous mathematical study of knots began in the 19th century as an attempt to explain atomic theory.P eter Guthrie Taitsinitial forays into knot theory were carried out at Lord Kelvinss uggestion that in doing so he might find evidence to support the theory that atoms were knotted vortices in the "lumniferous aether", with each element corresponding to ad ifferent knot. [18,19] The" knotted aether" theory was short-lived, [20] but interest in the classification and mathematical properties of knots continued. With the discovery,and ultimately the synthesis,ofmolecular knots in the latter part of the 20th century,k not theory and chemistry share ac lose relationship once again. Here we give an overview of the current state-of-the-art in the synthesis and properties of molecular knots and how their structures relate to broader knot theory.

Knot Theory
Aknot is mathematically defined as acircle embedded in 3D space.D ifferent knots are,t herefore,d ifferent entanglements in aclosed loop,rather than in the open strings in which we find knots in our everyday world. The" closed-loop" definition is necessary from at opological standpoint as any entanglement in al inear strand with two ends can be untied by deformation (for example,t he untying of shoelaces). The following section gives an overview of knot theory and terminology relevant to chemistry and molecular structure.

Representations of Knots
Them inimum number of crossings,w here one string passes over or under another, is one of ak nots" invariants" (that is,a ni ntrinsic property of ap articular knot). The simplest representation of ak not, often referred to as the "reduced representation", is one depicting the fewest number of crossings.F urther crossings can be introduced by twisting the knot, ac onformational change in molecular structural terms.S uch twists are called "nugatory crossings". [21] Each knot can be represented in an infinite number of different representations by adding nugatory crossings to the reduced representation. Forexample in Figure 3c,anugatory crossing has been added to the reduced form of the trefoil knot (3 1 ) that has three crossings in Figure 3b.
Ak not can be oriented, which means that an arbitrary point of the knot is chosen and the entire loop of the knot is traversed in ag iven direction. In an oriented knot, positive and negative crossings can be distinguished (Figure 3a). Positive crossings describe ar ight-handed helix, negative crossings aleft-handed helix. This allows another property to be quantified, the writhe (Wr) of ak not diagram. Writhe is the sum of positive and negative crossings,w here ap ositive crossing has avalue of + 1and anegative crossing has avalue  . Reduced diagrams and writhe:a )Definition of an egative and ap ositive crossing. b) Aknot can be oriented by followingi ts loop in an arbitrary direction. In the depicted orientation,t he shown trefoil knot has aw rithe (Wr) of 3, as all three crossings are positive. The orientation can be reversed by rotation along the indicated C 2 -axis. c) In atrefoil knot with nugatory crossings, the writhe can take any value. Knot diagrams without nugatory crossings are referred to as "reduced". of À1. Writhe is not ap roperty of the knot, but of the knot diagram (in other words,aparticular conformation of amolecular knot). From atopological standpoint, writhe can take any value, as any number of positive or negative nugatory crossings can be added;i n practice the restrictions on conformations that can be adopted for as mallmolecule knot will limit writhe (nugatory crossings will generally add conformational strain and be entropically unfavorable unless stabilized in some way). Formost knots the writhe of the reduced representation is independent of the reduced representation chosen (an exception being the so-called "Perko pair", see Section 2.9).
To move between any different diagrams of ak not, ac ombination of just three different types of manipulations are necessary,k nown as the Reidemeister moves (independently discovered by Reidemeister [22] as well as Alexander and Briggs [23] in the 1920s). These moves are only applied to as mall section of the knot and are characterized by the number of strands involved ( Figure 4). TheR eidemeister Im ove describes the addition of an ugatory crossing to as ingle strand, and is the only Reidemeister move which changes the overall writhe of the knot (Figure 4a). Achemical example would be introducing atwist in amacrocycle.T he Reidemeister II move moves one loop entirely over another, thereby creating two crossing points ( Figure 4b)a nd corresponding in chemical terms to moving one macrocycle over another, for example.T he Reidemeister III move involves three strands: one is moved over the crossing of two others,atransformation represented by conformational changes within am olecular trefoil knot (Figure 4c).
In Figure 5a Reidemeister moves are used to transition between the D 2 -symmetric and D 3 -symmetric representations of at refoil knot;F igure 5b shows the same process for am olecular knot, whereby the conformation in which Sauvageso riginal molecular trefoil knot [27] is synthesized is transformed through the linear helicate approach (Section 3.1.1) to the conformation in which molecular trefoil knots are synthesized using single metal-ion templates (Section 3.1.2).

Classification of Knots
Ac onvenient way to classify knots is using the Alexander-Briggs notation, [23] commonly used for knots with up to 10 crossings and used throughout this Review.I n this notation, every knot is denoted in the form X Y ,where X corresponds to the number of crossings and Y is an index that distinguishes the knot from others with the same number of crossings. Y was originally determined by sorting the knots for ag iven number of crossings by increasing torsion number.
Steffen Woltering was born in Münster (Germany). He studied chemistry at the Georg-August-UniversitätGçttingen (Germany) and the University of Edinburgh (UK), and obtained his MSc from the former in 2012. He recently completed his PhD in Prof. David Leigh's group in Manchestero nthe template synthesis of interlocked molecules. They are named after the number of components involved in the movement. a) Reidemeister I refers to the creation or removal of an ugatory crossing, the number of crossings/writhe changes by AE 1. It is equivalent to twisting amacrocycle to form an additionall oop. [24] b) ReidemeisterII passes one string over another,the number of crossings changes by AE 2b ut the writhe remains the same. It is equivalent to moving one moleculars trand over another. [25] c) Reidemeister III refers to passing astring over acrossing. The number of crossings and writhe are unchanged. Note the nugatory crossing in the left-hand triketone trefoil knot structure, which is necessary for aReidemeister III move for any alternating knot. [26] Thetorsion number is amathematical property of aknot that was popular in the first half of the 20th century. [28] Themost basic characteristic of aknot is whether or not it has crossings,that is,whether aknot is trivial or nontrivial. A trivial knot has no crossings and can be deformed to atorus.A trivial knot is sometimes referred to as the "unknot", denoted 0 1 ,a nd in molecular terms corresponds to amacrocycle.

Prime and Composite Knots
Nontrivial knots have at least three crossings and are either prime or composite.Prime knots cannot be constructed by combining simpler knots,w hilst composite knots can, by performing the so-called "knot sum". [29] This is analogous to numbers:p rime numbers can only be divided by themselves and one,composite numbers are the product of smaller prime factors.C omposite knots are written using the hash symbol (#) to connect the Alexander-Briggs notation of the constit- Figure 6. Chirality in knots. a) The trefoil knot 3 1 is topologically chiral, as it cannot be deformed to its mirror image form 3 1 *w ithout the strand passing through itself. b) The square knot 3 1 #3 1 *, acomposite knot obtained by connectingtwo trefoil knots of opposite handedness, is achiral, as the mirror plane s projects it onto itself. c) The figureeight knot 4 1 can be transformed into its mirror image. By flipping the part shown in red over the part shown in blue, an upside-down version of the mirror image is obtained after deformation, thereby making it topologically achiral. Fort he sake of brevity,n ot every Reidemeister move is shown here for this transformation. The same process (corresponding to conformational changes in amoleculars tructure) applied to Sauvage's trefoil knot [27] (in this case molecular D 3 -symmetry cannot be achieved as one loop is chemicallyd ifferent to the other two). uent prime knots (using + /À to indicate the handedness of each prime knot if the handedness is defined, or *toindicate the opposite handedness of ap rime knot compared to the others if only relative handedness is relevant, as in Figure 6).
Then umber of knots with the same number of crossings increases dramatically with an increasing number of crossings:t here are 2p rime knots with 5c rossings,1 65 with 10 crossings,and more than 250 000 with 15 crossings.There is no known formula to calculate the number of possible prime knots for agiven number of crossings. [21]

Chirality in Knot Theory
Ac hemical compound is chiral if its structure cannot be projected on to itself by ar otary reflection and cannot be deformed to its enantiomer due to as ufficiently high inversion barrier. This is sometimes referred to as Euclidean chirality.M athematically,h owever, knot projections can be deformed as much as desired (without the strand passing through itself) and knots are only topologically chiral if they cannot be deformed continuously to superimpose with their mirror image.T hus,o bjects with Euclidean chirality are not necessarily topologically chiral (e.g. molecules with asymmetric carbon centers or the mirror images of the reduced representation of the figure-eight knot 4 1 shown in Figure 6c). Thetrefoil knot 3 1 is topologically chiral, as one mirror-image form cannot be continuously deformed into the other (Figure 6a). Although this is clearly true from simple observation, the topological chirality of at refoil knot was only proven mathematically in the early 20th century. [30] Most knots are topologically chiral;o ft he more than 1.7 million prime knots with up to 16 crossings,f ewer than 2000 are achiral. Achiral knots were initially referred to as "amphichiral" or "amphicheiral" (a term introduced by Tait) by mathematicians,b ut over the time the term "achiral", so familiar to chemists,has also become common in topology. [31] An example of at opologically achiral knot is the composite square knot, which is formed from the knot sum of two trefoil knots with opposing handedness,a nd has aplane of symmetry where the knots are joined (Figure 6b). Ther epresentation of the topologically achiral 4 1 knot in Figure 6c appears to be chiral at first sight, as it is not immediately apparent how the knot can be deformed to its mirror image.H owever,t hrough as eries of Reidemeister moves,o ne mirror-image form can be converted into the other, thereby demonstrating that the 4 1 knot is topologically achiral.
Thea bsence of chirality in the 4 1 knot is easier to see if amore symmetrical representation [32] is used. Figure 7ashows the 4 1 knot in its reduced form with four crossings,w hile am ore spherical form with eight crossings is shown in Figure 7b.T he spherical representation is formed by adding four nugatory crossings to the reduced representation and has arotary inversion axis (S 4 ), which means that arotation of 908 8 converts this representation into its mirror image,t hereby making it achiral. Figure 7c shows acoordination complex [33] with the same spatial arrangement of ligand strands (see Scheme 15 for its synthesis and chemical structure).

Invertible Knots
Some knots are invertible,w hich means that they can be continuously deformed to give ar eversed orientation of the closed loop.Ifthe trefoil knot is oriented as in Figure 3b,the orientation changes when the knot is rotated along one of its C 2 axes.T he existence of non-invertible knots was only discovered in 1963, as most knots with low crossing numbers are invertible (the simplest non-invertible knot is 8 17 ). [21] For higher numbers of crossings,t he fraction of non-invertible knots rises dramatically.I nt opology,c hiral invertible knots are termed "reversible", chiral non-invertible knots are termed "fully chiral", and achiral invertible knots are referred to as "fully achiral". [21] 2.6. Alternating and Non-Alternating Knots As triking feature of knots with fewer than 8c rossings is that they can all be represented in forms in which overpasses and underpasses alternate when the strand is traversed (Figure 8a). Knots that can be represented this way are said  to be alternating.I tw as originally thought that all knots can be drawn in an alternating pattern, but the existence of nonalternating knots was demonstrated by Little [34] and proven during the 20th century. [35][36][37] Figure 8b shows one of the three simplest non-alternating knots,8 19 .A ll alternating achiral knots have an even number of crossings. [21] In addition, all reduced representations of an alternating knot have aconstant writhe.T his is not necessarily the case for non-alternating knots,w hich historically led to some duplications in knot tables. [38,39] If an alternating pattern is achieved for ak not, it can generally be easily determined whether the knot is prime and distinguished from others,t his is harder for nonalternating knots.A lthough alternating knots are dominant for knots with low crossing numbers,the fraction of alternating knots appears to tend exponentially towards zero when the crossing number is increased. [21]

Torus Knots
Torus knots are afamily of knots that can be drawn on the surface of at orus (donut shape) without the closed loop intersecting itself.T hey can be abbreviated by the symbol T(p,q), where p and q are integers that describe how many times the torus is passed in the poloidal and toroidal [40] directions,r espectively,b efore the two ends are joined. A torus knot is obtained if p and q are co-prime. [41] Switching p and q gives the same torus knot with adifferent geometry,as shown for the trefoil knot in Figure 9. All torus knots are topologically chiral (except if p or q = 1, which yields the unknot). Forknots with an odd number of crossings,the knot of lowest order in the Alexander-Briggs notation X 1 is always atorus knot. Torus knots are amenable to chemical synthesis by linear (Section 3.1.1) and circular (Section 3.3.2) double helicate approaches.

Twist Knots
Tw ist knots are another family of knots that are generated by adefined pattern:two strands are twisted n times and the open ends linked before closure.T his process is illustrated in Figure 10 b. Foraneven number of crossings,X,the twist knot is X 1 in Alexander-Briggs notation. Foro dd numbers of crossings,the twist knot is X 2 (X 1 is in this case atorus knot). Especially for low numbers of crossings,t wist knots are very abundant, three of the four simplest knots are twist knots (the trefoil knot 3 1 ,t he figure-eight knot 4 1 ,a nd the three-twist knot 5 2 ). Topoisomerases tend to form predominantly twist knots in DNA, as these topologies result when the topoisomerase breaks aD NA duplex at an ode and allows the crossing duplex to pass through the gap before resealing the broken DNA. [42] Tw ist knots are asubset of alarger group of "clasp knots" ( Figure 10). Theg eneral structure of ac lasp knot C(p,o)i s shown in Figure 10 c. Fortwist knots p = 2. [43]

Knot Tables
Thes ystematic tabulation of knots started in the 19th century, [18,19] one of the most commonly used versions today is the Rolfsen knot table. [44] All knots (alternating and nonalternating) with up to 16 crossings have likely been found, [21] with the number standing at slightly more than 1.7 million prime knots.The last pair of duplicates (two structures in knot tables that are actually the same knot) to be discovered, was Figure 9. Torus knots. At orus knot T(p,q)r uns p times in the poloidal direction (i.e. through the cavity) and q times in the toroidal direction (i.e. around the cavity) around the surface of atorus without the strand intersecting. Swapping p and q results in the same knot. a) T(2,3) is the trefoil knot 3 1 .b )T(3,2) is also the trefoil knot. Twist knots with an even number of crossings are denoted as X 1 ,t hose with an odd number as X 2 (in this case X 1 is atorus knot). c) Twist knots are atype of clasp knot C(p,o). Twist knots are C(2,o)clasp knots. d) The generation pattern for (q,r,s)p retzel knots. Pretzel knots consist of left or right-handed helices connected together (see Section 2.11).
two representations of a1 0-crossing knot called the "Perko pair" in the 1970s. [39] Thet abulation of alternating knots has been extended up to 22 crossings,w ith more than 6billion found to date. [29] Figure 11 shows ak not table with all prime knots having up to eight crossings.

Braid Representations
Abraid is aset of discrete strands that cross each other in ad efined pattern. To create ac orresponding closed-loop knot, the ends of the strands are connected so that no additional crossings are generated, as illustrated in Figure 12. Every knot can be represented as ab raid and, therefore,f or chemists,abraid indicates ap otential synthetic pathway to any given molecular knot topology.T he pattern for the simplest torus knots in Figure 10 ac onsists of two strands twisted about each other with the ends connected. Braid representations of higher order torus knots are shown in Figure 12 a,b. Figure 12 c,ds how braid representations for several achiral knots.N ote that the braids have an inversion center (indicated with ad ot, i,i nt he figure). Such braids are called reverse rotated palindromes (RRP), [45] and if ak not can be represented by an RRP then it must be achiral. Theb raid in Figure 12 cf orms the achiral 4 1 knot (n = 1), repeating the recurring unit (n = 2) gives the achiral 6 3 knot and repeating it once more (n = 3) leads to the achiral 8 9 knot. Theb raid Figure 11. Knot table of all prime knots having up to eight crossings including the unknot 0 1 .T orus knots are depicted in red, achiral knots in black, non-invertible knots in white, and non-alternating knots in green. shown in Figure 12 disanRRP for any value of n and is also a" Brunnian braid", as removal of any one strand leaves the other two unconnected. [46,47] Just as ak not has an infinite number of diagrams,aknot can also be represented by an infinite number of different braids.H owever,e very knot has am inimum braid representative,analogous to the reduced representation of aknot. The minimum braid is the one with the fewest number of crossings and strands.A na dvantage of braid representations over traditional knot diagrams is that they can be conveniently stored in computer-readable form. [48] 2.11. Tangle Representations [49] Knots can be broken down into smaller key fragments,socalled "tangles", which were introduced by Conway [50] and have proved useful in describing the behavior, properties,and transformations of local entanglements.
At angle is ar egion of ak not that can be surrounded by ac ircle so that the knot crosses the circle exactly four times.  Figure 5) the two tangles are not (the strands run either from NW to SE or from NW to SW).
Tangles can be constructed from basic building blocks, some of which are shown in Figure 13 b. At angle with two parallel strands running from NW to SW and NE to SE is called the 1 tangle.T wo parallel strands running from NW to NE and SW to SE, gives the 0t angle.Atangle containing ap ositive crossing is a1tangle,a nd one with an egative crossing is a À1t angle.T hese building blocks can be combined (multiplied) to give rational tangles,a sillustrated in Figure 13 c. Starting with a À2t angle (two negative crossings), the tangle is first reflected along the NW-SE axis. This is then joined to asecond tangle,which in this case is a À1 tangle,t hereby resulting in a À2 À1t angle.T oa dd at hird tangle,t he original tangle is reflected along the NW-SE axis and then the additional tangle added (in this case a3tangle) to obtain a À2 À13tangle.Ifthe ends of arational tangle are connected, ar ational knot (or link) is obtained, such as for Figure 13 a, where atrefoil knot is formed.
This process can be further generalized with tangle multiplication:n ot only integer tangles (such as the À1 tangle,2tangle,e tc.) can be connected in this way,s oc an more complicated ones.T his is illustrated in Figure 13 d. Furthermore,t here is the operation of tangle addition, as shown in Figure 13 d. These operations are not necessarily commutative or associative.T he resulting tangles are called algebraic tangles and can be closed to form algebraic knots.
Tangle addition leads to ad iagram that is sometimes referred to as a"pretzel knot" (q,r,s), as shown in Figure 10 d. Thei ntegers q, r,a nd s define either right-handed (positive values) or left-handed helices.F or example,t he trefoil knot can be described as the (1,1,1) pretzel knot. Changing the sign of all descriptors yields the other enantiomer of the same knot. [51] Thef irst non-invertible knots that were discovered belonged to the class of pretzel knots. [51] Tangles are aconvenient way to describe and classify even complex knots in terms of how the knot is structured locally.
Thek notted regions in the knots shown in Figure 10 can be described as tangles,a nd every torus knot is obtained by closing an m tangle (Figure 10 a), while every twist knot is obtained by closing an n 2tangle (Figure 10 b). Just as braids can be seen as astrategic blueprint for synthetic chemists for constructing different knots,tangles provide away of thinking about synthons for crossings that need to be assembled in aparticular way.

Interconversions of Knots
Knots can be transformed into other knots by inverting (i.e.r emoving or adding) crossings.O ne characteristic of knots is their "unknotting number", which refers to the minimum number of crossings that have to be inverted to give the unknot from ag iven knot. Theu nknotting number for some knots can be easily determined:the unknotting number of at wist knot is always 1 ( Figure 14 a) and for at orus knot T(p,q)i s 1 = 2 (pÀ1)(qÀ1) (Figure 14 b). [52] Theu nknotting number is often more difficult to determine for other Figure 13. Tangle representations. a) Asection of aknot (indicated by ad ashed circle) can be split into tangles. The fixed entry points of the string are named after the cardinal directions.b )Some basic tangles for the construction of more complex tangles. c) Forthe construction of arational tangle, the starting tangle is reflected along the NW-SE axis and the new tangle added to the NE and SE crossing points. The resulting tangle can be further extended by the same procedure. d) Generalization of tangle multiplication. T 2 is not restricted to being an integer tangle such as in (c). e) Ad ifferent way to connect tangles is by addition. The sum is formed by connectingthe NE and SE crossing point of the first tangle T 1 to the NW and SW crossing point of the second tangle T 2 ,r espectively. knots, [49] but it is always less than half of the total number of crossings for ag iven knot. [53] One way of transforming ak not is to change its tangles (Section 2.11). Another possibility is to use so-called kmoves. [49] A k-move describes the introduction of k positive (or Àk negative) crossings into as et of two strands (Figure 15 a). This is related to the way chemists introduce crossing points into molecules by,f or example,u sing metalion coordination to twist, orientate,orentwine ligand strands upon binding (Figure 15 b).
How closely two knots are related can be expressed by the "Gordian distance", that is,t he number of crossing changes needed to interconvert the knots. [54] It can be seen in Figure 14 bt hat inverting one crossing of the pentafoil (5 1 ) knot yields the trefoil (3 1 )k not. TheG ordian distance between the two knots is thus 1.

Synthesis of Molecular Knots
Thes ynthesis of molecular knots requires mechanical restriction of the relative positions of molecular components in as imilar manner to that needed to construct other mechanically bonded molecular architectures,n amely links (catenanes) and threaded molecular rings (rotaxanes). [55,56] It is,t herefore,u nsurprising that many of the advances in the synthesis of small-molecule knots have come from groups also active in catenane and rotaxane synthesis.S uccessful syntheses of small-molecule knots have been reported since the late-1980s. [56][57][58][59][60] In the following section the most significant methods and strategies for the synthesis of small-molecule knots developed to date are discussed. (3 1 ) Thetrefoil knot (3 1 )isthe simplest nontrivial knot and the most amenable to chemical synthesis.M any different methods to synthesize trefoil knots have been reported. Thefirst of these was Jean-Pierre Sauvageslinear metal helicate strategy, an extension of the method his group employed in the metaltemplate synthesis of [2]catenane (Hopf link) [61] Cu I 3 (Scheme 1). Thet etrahedral Cu I ion holds the two bidentate ligands in am utually orthogonal arrangement such that the curvature of the ligands creates two crossing points.The metal Figure 14. Unknotting numbers. a) Any twist knot has an unknotting number of 1, as inverting one crossing is sufficientt ogive the unknot 0 1 .b)The unknotting number of atorus knot T(p,q)i s 1 = 2 (pÀ1)(qÀ1), In this example, the pentafoil knot 5 1 is converted into atrefoil knot 3 1 by inverting one of its crossings. Changing asecond crossing gives the unknot 0 1 .S othe unknotting numberis2( = 1 = 2 (2À1)(5À1)). Figure 15. Changinge ntanglement using k-moves. a) A k-move introduces k-positive crossingsi naset of two strings, a Àk-move introduces k-negative crossings. b) As upramolecular 3-move induced by Cu I ions forms the scaffoldf or the synthesis of amolecular trefoil knot. [27] Scheme 1. Sauvage's synthesis of a [ 2]catenane (Cu I 3)byp assive [62] metal-template synthesis. The phenanthroline-Cu I system formed the basis for the synthesis of several other mechanically interlocked moleculart ypes (rotaxanes, trefoil knot, Solomon link). [61] All of the cap-and-stick structures shown in this Review are X-ray crystal structures produced from coordinates taken from the Cambridge Structural Database (CSD).

Molecular Trefoil Knots
ion can subsequently be removed after macrocyclization to yield metal-free [2]catenane 3.T he synthesis can be carried out using one preformed macrocycle (2)o r having both ligands (1)macrocyclize around the template (Scheme 1).

Linear Double Helicates
Sauvage realized that this metal-template approach could be extended to form linear helicates that produce more complicated interlocked structures on closure ( Figure 16).
Such linear double helicates can be considered ac hemical system based on the braid for the synthesis of T(x,2) torus knots shown in Figure 10 a. Then ext topology to be synthesized after the [2]catenane was the trefoil knot 3 1 ,o btained from ad inuclear Cu I complex. [27] Them olecular topology was determined unambiguously by X-ray crystallography [63] (Scheme 2c). In initial designs, the phenanthroline units were connected by short alkyl chains and the resulting yields of the trefoil knot were low (< 10 %). Introduction of a mphenylene unit increased the preorganization of the helicate, and designs based on ligand 4a gave trefoil knot 5a in almost 30 %yield [64] (Scheme 2a). Ring closing metathesis (RCM) as amethod of covalent capture increased the yield of the knot to 74 %( 5b;S cheme 2b). [65] Thet emplate system could also be varied:t he use of octahedral Fe II and two tridentate ligands 6,i nstead of Cu I complexes of bidentate ligands, yielded molecular trefoil knot 7 [66] after RCM (20 %y ield).
As noted in Section 2.4, trefoil knots are chiral. The enantiomers of 5a could be separated by cocrystallization with ac hiral phosphate anion. [67] Theu se of enantiopure ligands allowed the synthesis of asingle enantiomer of trefoil knot 8,t hereby demonstrating the influence of geometric chirality on topological chirality for the first time in amolecular knot. [68]

Single Metal Ion Templates
Thei dea of using transition metal ion templates to assemble catenanes and knots actually predates the first Figure 16. The linear helicate approach to simple knots and links. Metal ions induce the twisting of the ligand strands to form adouble helix (k-moves, Section 2.12). If the number of crossingsi sodd, amolecularknot is created upon connectinga/a' and b/b'. Foran even number of crossings, links ([2]catenanes) are produced. The linear helicate approach was successfullyd emonstrated by Sauvage for the first three in this series (Hopf link, trefoil knot, and Solomon link), but fails for higher order topologies such as the pentafoil knot 5 1 (Section 3.3.1).
Scheme2. Molecular trefoil knots prepared from al inear helicate strategy.a)Synthesis of trefoil knot Cu 2 5a after covalent capture of linear double helicate Cu 2 4a 2 by Williamson ether synthesis. [64] b) The yield of the trefoil knot was significantly increasedb yusing RCM for the macrocyclization reactions. [65] c) X-ray structure of the related trefoil knot Cu 2 5c.T his early design was obtained in lower yield, as the alkyl chain connecting the phenanthroline units is less preorganized than the m-phenylene unit used in later designs. [63] d) Ar elated approach using Fe II and terpyridine derivatives instead of Cu I and phenanthroline ligands. [66] e) Enantioselective synthesis of atrefoil knot by the linear helicate approach. [68] Sauvage catenane by ad ecade.I n1 973 Sokolov proposed [69] that octahedral metal ions might be used to position three ligands in mutually orthogonal orientations suitable for the template synthesis of at refoil knot ( Figure 17). Nearly 30 years later Hunter and co-workers prepared an "open knot" by wrapping as ingle ligand strand containing three bipyridine units around an octahedral Zn II ion with bisphenol Zd erivatives as ab end-inducing linker (Scheme 3). [70] The" overhand" knot Zn9 could then be closed by RCM to give trefoil knot 12 or 13,depending on the length of the chain employed. [71] Aparticularly short and efficient synthesis of trefoil knots can be achieved using lanthanide ions as the template (Scheme 4). Ae uropium or lutetium trication was used to assemble acircular trimeric helicate from three 2,6-diamido-pyridyl ligands (Scheme 4a). Joining the ligand ends by RCM afforded trefoil knot 15 a in 55-62 %y ield. [72] Introducing chiral centers into the ligand strands gave atrefoil knot (15 b) of single handedness [73] (Scheme 4a). TheX -ray crystal structure of the enantiopure knot is shown in Scheme 4b. Theu se of as ingle ligand strand 16 incorporating three 2,6diamidopyridyl units reduces the number of closures required to from the knot from three to one,thereby enabling knot 17 to be obtained in up to 90 %yield (Scheme 5). [74] 3.1.3. Active Metal Template Synthesis Active template synthesis [55] is as trategy for forming mechanically interlocked molecules,whereby metal ions play an active role in catalyzing the bond-forming reactions that covalently capture the final product as well as organizing the building blocks in the manner of ac onventional "passive" template.The approach was originally introduced to facilitate the synthesis of rotaxanes [75,76] and catenanes. [77] However,the concept has been successfully extended to the synthesis of at refoil knot (Scheme 6). [78] Ligand 18,w hich possesses one pyridyl and two bipyridyl units,b inds aC u I ion between the two bipyridyl units to create ac rossing point. Ac oppercatalyzed alkyne-azide cycloaddition (CuAAC) reaction of the azide and alkyne termini through the resulting loop by asecond Cu I ion coordinated to the pyridine group generates the other two crossings required for the trefoil knot. 1 HNMR Figure 17. Sokolov's proposed route for the synthesis of amolecular trefoil knot templated by the octahedral coordination sphere of atransition metal. Modified from Ref. [69] with permission from the Royal Society of Chemistry. Scheme 3. Hunter'ss ynthesis of atrefoil knot using asingle metal ion template, via open knot Zn9.F unctionalizing the ends of the open knot with alkene units enabled closure to the trefoil knot by RCM. The structure of the open knot was determined by X-ray crystallography. [70,71] Scheme 4. Synthesis of atrefoil knot by the circular helicate approach.a )Asingle lanthanide ion entwines three 2,6-diamidopyridyl ligand strands 14 in its coordination sphere, thereby forming atrefoil knot upon connection of the ligand end groups. The achiral precursor 14 a yields aracemic mixture of the two enantiomers of trefoil knot Ln15 a.The use of C 2 -symmetric ligand 14 b gives enantiopure trefoil knot Ln15 b.b )X-ray crystal structure of enantiopure trefoil knot Ln15 b. [72,73] and drift tube ion mobility mass spectrometry (DT IM-MS) studies demonstrated that the reaction product 19 had the topology of at refoil knot.

Directing Trefoil Knot Formation through p-Interactions and/or Hydrogen Bonding
TheS toddart group utilized p-p interactions to try to direct the assembly of at refoil knot through the macrocyclization of as ingle strand, although the putative knot was isolated in < 1% yield and proved difficult to fully characterize. [79] Hydrogen bonding is the directing influence for the formation of several trefoil knots formed as unexpected reaction products.T he first of these was ac ompound first isolated by Hunter,who isolated acompound expected from previous work [80] to be an amide [2]catenane 23 with two rings of different size [81] (Scheme 7, bottom). TheV çgtle group repeated the synthesis several years later and obtained an Xray crystal structure that showed that this compound was actually at refoil knot (22;S cheme 7, top). [82] It later proved possible to separate the two knot enantiomers. [83] This episode illustrates the important role that X-ray crystallography can play in unambiguously identifying ap articular molecular topology;itisall too easy to misinterpret 1 HNMR spectroscopy and mass spectrometry data with complex, often highly symmetrical, knot and link architectures.
Hydrogen bonding was the driving force behind the assembly of another organic trefoil knot based on steroidderived building blocks,s erendipitously discovered by Feigel [84] (Scheme 8). In contrast to 22,w hich is rather unsymmetrical in its X-ray crystal structure because of the hydrogen-bonding network (Scheme 7), trefoil knot 25 shows almost perfect C 3 symmetry in the solid state.A saconsequence of the chirality of the building blocks,the synthesis of 25 is enantioselective and yields only at refoil knot of Dhandedness.

Dynamic Combinatorial Libraries
Dynamic combinatorial chemistry (DCC) is ap owerful tool for generating interchanging mixtures of compounds,socalled dynamic combinatorial libraries (DCLs). [85,86] In recent years,k nots have been found in DCLs that generate cyclic oligomers (sometimes also including catenanes) of various sizes.Sanders and co-workers discovered that trefoil knot 27 Scheme 6. Active template synthesis of trefoil knot 19.One crossing is generated by Cu I coordination to the bipyridine groups, which forms aloop. ACuAAC reaction of the azide and alkyne terminii sdirected through the loop by the second coordinated Cu I ion, thereby forming the trefoil knot. [78] Scheme 7. The condensation product of 20 and 21 was originally proposed by Hunter to be [2]catenane 23. [81] Several years later,X -ray crystallography by the Vçgtle group showed that the product was actually trefoil knot 22. [82] An etwork of hydrogen bonds responsible for directing the assemblyo fthe knot is shown by dashed lines. Scheme 8. At refoil knot 25 obtained by ring closure of steroid trimer 24.A nextended network of hydrogen bonds is visible in the solid-state structure, which has almost perfect C 3 -symmetry( the hydrogen bonds are indicated by dashed black bonds in the crystal structure and as red dashed lines in one of the subunits in the diagram). [84] Scheme 9. Amoleculart refoil knot 27 discovered in adynamic covalent library.T he interlocked structure minimizes the exposure of the hydrophobic surface area to the solvent by burying part of the molecule in the central cavity. [87] was formed in aD CL built from building blocks of the trimeric electron-poor p-system 26 (Scheme 9). [87] By using water as the solvent and dynamic disulfide exchange to establish the DCL, trefoil knot 27 could be formed in high yield. Thed riving force for knot formation is likely the minimization of the hydrophobic surface area in the knotted structure.
Tr abolsi and co-workers also discovered at refoil knot unexpectedly formed in ad ynamic mixture of building blocks, [88] similar to as ystem used by the Stoddart group to Scheme 10. Synthesis of trefoil knot 28 based on imine exchange. [88] During the crystallization process,t wo bromide anions are incorporated in the central cavity,one above the other. [90] Attempts at crystallization in the absence of Br À were unsuccessful. Scheme 11. a) Schill's approach towards amolecular trefoil knot using acovalent scaffold. Trimerization of acrowded quinone 31 was projected to give amoleculart refoil knot after cyclization and hydrolysis. [92] b) Walba's Mçbius strip approach towards molecularknots. Introducing ahalf twist in compound 32 before connecting the ends yields molecularMçbius strip 33.Statistical twisting was too disfavored to yield trefoil knot 34 after ozonolysis. [93] Angewandte Chemie Reviews Scheme 12. Towards the synthesis of molecular knots by using covalent scaffolds. a) Ah ybrid approach towards amolecular trefoil knot by using Cu I and a1,3,5-substituted benzene as the template. The interlocked structure of 35 after cyclization was confirmed by X-ray data, but it was not possible to remove the central benzene template to yield aknot. [96] b) At refoil knot assembled around abenzene-1,3,5-tricarboxylic acid template. The template could be removed after cyclization, but the formation of 36 could not be confirmed experimentally. [26] Scheme 13. Synthesis of ametalla-trefoil knot by trimerization of ethylene glycol bridged quinolines 37 with Ag I .C oordination bonds from oxygen to silver are omitted for clarity. [33] Scheme 14. Synthesis of molecular figure-eightk not 39 by disulfide exchange in adynamic covalent library.Knotting likely results from the hydrophobic effect. [99] assemble Borromean rings. [89] Dynamic imine bond formation between diformylpyridine 29 and bipyridine 28 in the presence of Zn II to template the assembly process produced trefoil knot 30 along with aHopf link catenane and Solomon link (Scheme 10). [88] Thes olid-state structure of 30 features two Br À ions,o ne above the other,i nt he trefoil knot cavity (Scheme 10). [90] Thei mine groups of the knot could be subsequently reduced with NaBH 4 if Cd II ions were used as the template in the knot-forming reaction. [91] 3.1.6. Covalent Scaffolds and Statistical Approaches One of the first proposed synthetic approaches towards molecular trefoil knots was suggested by Schill and Tafelmair through the use of ac rowded quinone 31 (Scheme 11). Tr imerization of such aq uinone and subsequent cyclization could yield at refoil knot upon hydrolysis (Scheme 11 a). In practice,the synthetic route was too long to be realized. [92] It should be noted that it is crucial to connect quinones of the same handedness,o therwise the crossings can be removed through aReidemeister II move.
An alternative approach by Walba et al. used ethylene bridges between two glycol chains (Scheme 11 b). [93] It was hoped that such ag lycol chain of sufficient length 32 would statistically twist around its own axis,thereby giving amolecular knot 34 after cyclization and ozonolysis of the alkene rungs.Although it was possible to induce one half-twist by this approach, thus producing am olecular Mçbius band [94] 33, multiple twists were too disfavored to yield knotted products.
Other covalent scaffold approaches toward trefoil knots include Siegelsh ybrid approach in which a1 ,3,5-trisubstituted benzene together with three Cu I ions acts as atemplate for the synthesis of an interlocked species 35 (Scheme 12 a). Removal of the central benzene unit from the structure would generate atrefoil knot. [95,96] Fenlon assembled ap olyethylene trefoil knot around abenzene-1,3,5-tricarboxylic acid template and used RCM to close the knot. Although the template could be removed, insufficient characterization data was obtained to confirm the formation of trefoil knot 36 (Scheme 12 b). [26]

Metallaknots
Hosseini and co-workers have described the synthesis of several knotted molecular structures with metal centers as integral parts of the topology,termed metallaknots. [33] Strictly speaking, these structures are not true knots,a st he metal center is coordinated to other parts of the molecule and such branching is not within the definition of ak not as ac losed loop.Other examples of branched knotted molecular systems include ravels [97] and knotted cages. [98] Scheme 13 shows the synthesis of at refoil metallaknot based on Ag I coordination to al igand consisting of two quinoline units bridged by an ethylene glycol oligomer.

Molecular Figure-Eight Knots 4 1
Although as ignificant number of synthetic routes,b oth accidental and designed, have now been established for the simplest knot (trefoil 3 1 ), examples of the neighboring entry in knot tables (Figure 11), the figure-eight knot (4 1 ), are scarce. So far, the reduced representation with four crossings has not been realized as as mall molecule.H owever, the eightcrossing representation of the 4 1 knot with an S 4 -axis ( Figure 7) has likely been discovered in aD CL (its structure determination based largely on symmetry and NMR data). [99] Similar to the related trefoil knot (Scheme 9), the driving force for the formation of the 4 1 knot (39)isminimization of the hydrophobic surface area, as the synthesis was carried out in an aqueous buffer (Scheme 14). Although af igure-eight knot is topologically achiral (see Section 2.4), knot 39 is chiral due to the cysteine moieties present in the chain. A meso form of compound 39 was also prepared.
Currently,t he only other example of am olecular figureeight knot is am etallaknot described by Hosseini and coworkers (Scheme 15). [33] Thec rystal structure features an intricate array of p-interactions and the coordination geometry of the Ag ions with the glycol units and quinoline moieties is responsible for the formation of the metallaknot. The4 1 metallaknot also adopts the eight-crossing S 4 -symmetrical knot representation.

Molecular Pentafoil Knots 5 1 3.3.1. Linear Helicate Approach
Thefigure-eight knot (4 1 )isfollowed by the pentafoil knot (5 1 )inknot tables ( Figure 11). Similar to the trefoil knot, the pentafoil knot is atorus knot (Section 2.8). This suggests that the linear double helicate strategy ( Figure 16) might be suitable to form such aknot. TheSauvage group was able to form at rinuclear linear helicate and close it to the corresponding Solomon link. [100] However, all attempts to synthesize apentafoil knot from atetranuclear linear helicate Li 4 41 2 failed (Scheme 16). [101] There are likely several reasons for the failure of this strategy.F irstly,a st he helicate becomes longer, the distance increases between the strands of the braid that need to be closed to give the desired product, so incorrect closures become more likely.Inaddition, the center of longer helicates can be significantly strained, so it is likely that some mismatched helices also form ( Figure 18).

Circular Helicate Approach
Theinherent problems of linear helicates for the synthesis of knots (and links) can be overcome by bringing the ends of the helicate closer to each other through ab ent or fully circular design (Figure 19). Theh igh symmetry of ac ircular helicate also means that simpler ligands can be used, thus making the chemical synthesis easier,a sf ewer recognition motifs per ligand are required. However,the number of new bonds that need to be generated to form the closed-loop knot increases,w hich suggests that reversible bond-forming reac-tions that can "error check" the assembly process could be advantageous.
Circular helicate systems of the form Fe II x L x with x = 4, 5, and 6w ere serendipitously discovered by Lehn and coworkers in the 1990s. [102][103][104] Thev alue of x is affected by the ligand structure and an anion template effect. Thel igands form ad ouble helix woven around the metal centers,w hich means that, in principle,s uitably modified ligand strands could be used to form torus knots and links.T he first knot to be synthesized from acircular helicate was pentafoil knot (5 1 ) 44,w hich was assembled by formation of imine bonds [105] (Scheme 17 a). Thed iamine building block 43 contains two oxygen atoms that allow the required folding of the glycol chain because of the gauche effect. [106] Pentafoil knot 44 could not be demetalated due to the lability of the imine groups when not coordinated to ametal center. Therelated pentafoil knot 46 (formed from alkene-terminated ligand 45)w as covalently captured by RCM in 98 %yield (Scheme 17 b), and could be readily demetalated under basic conditions. [107] TheX-ray crystal structures of both 44 and 46 (the latter is shown in Scheme 17 c) feature achloride ion originating from the assembly process present in the central cavity.ASolomon link [108] (a doubly interlocked [2]catenane) and aS tar of David catenane [109] (a triply interlocked [2]catenane) have also been synthesized by using this approach through the use of tetrameric (x = 4) and hexameric (x = 6) circular helicates, respectively.

Higher Order Knots
Recently the circular helicate strategy was successfully extended from double to triple helicates.T his was possible Scheme 15. Synthesis of afigure-eight metalla-knot by tetramerization of ethylene glycol bridged quinolines 40 with Ag I .C oordinationb onds from oxygen to silver are omitted for clarity. [33] Scheme 16. Synthesis of alinear double helicate with five crossings. Attempts to ring-close Li 4 41 2 to the corresponding pentafoil knot (5 1 ) were unsuccessful. [97] Figure 19. Transition from alinear double helicate to acircular double helicate. a) One of the limiting factors of the linear helicate approach is the increasing distance between the ends. b) Bending the helix brings the ends closer together,b ut does not reduce the length (and complexity) of the ligand strands. c) In acircular helicate, an additional metal ion brings additional organization to the ends of the helix. The symmetry enables shorter (simpler) ligands to be used at the cost of requiring more reactions (five as opposed to two for the linear helicate in the case of a5 1 knot) to achieve closure of the loop. because the Fe II ions used to assemble the circular helicates are octahedral, and so can organize three strands containing bidentate groups,r ather than only two.I nb raid representations,this means changing the braid from the one depicted in Figure 10 at ot he one in Figure 12 a, as shown in Figure 20. Amolecular 8 19 knot was prepared by using this approach (Scheme 18);t he resulting structure is the tightest knot reported to date,with 24 atoms per crossing. [110] Thereaction of ligand 47 with FeCl 2 generated ac ircular triple helicate, which was closed to the 8 19 knot through RCM. Steric restraints made sure that the closures could only take place between strands coordinated to neighboring iron centers, thereby affording the non-alternating molecular 8 19 knot 48. This method for connecting strands that are not bound to the same metal center should be applicable to ar ange of higher order knots and links. Scheme 17. Synthesis of molecular pentafoil knots (5 1 )v ia circular double helicates. a) Fe 5 44 is obtained by formation of an imine bond between ligand 42 and diamine 43 in the presence of Fe II .T he size (pentamer) of the circular helicate is determined by achloride anion template. [105] b) The yield of the pentafoil knot is increasedb yusing ligand 45,w hich allows for covalent capture of the closed-loop knot by RCM. In contrast to Fe 5 44,F e 5 46 does not decompose upon demetalation. [107] c) Solid-state structure of Fe 5 46 (the structure of Fe 5 44 was also determined by X-ray crystallography). Figure 20. The complexityo fknots accessible from helicates increases from a) al inear double helicate to b) acircular double helicate to c) a circular triple helicate. [110] Angewandte Chemie Reviews

Composite Knots
Small-molecule composite knots (see Section 2.3) have yet to be synthesized as discrete entities.T heir synthesis is rendered difficult by the fact that the combination of two chiral knots can give multiple products ( Figure 21). Just as the dimerization of ar acemic chiral molecule can give two enantiomeric chiral dimers (RR and SS)and an achiral meso compound (RS), the same is true for knots (Figure 21 a): The connected sum of ac hiral knot with itself can form two enantiomeric knots (such as when two left-handed or two right-handed trefoil knots are connected to form granny knots) and one achiral knot (such as when two trefoil knots of opposite handedness are connected to form asquare knot). If two different chiral knots are connected, four different combinations are possible (Figure 21 b). Thek not sum of two achiral knots always yields an achiral composite knot. [111,112] To date,p rogress on the synthesis of small-molecule composite knots is limited to the report by Sauvage and coworkers of the low-yielding synthesis of am ixture of composite knots.T he dimerization of racemic open trefoil knot precursor 49 by Glaser coupling gave trace amounts of ap roduct that was assigned to be am ixture of granny and square knots 50 and 51 (Scheme 19). [113]

Properties and Applications of Small-Molecule Knots
Knotting am olecular backbone significantly restricts the conformations am olecule can adopt, effectively preorganizing the structure through mechanical constraints.I tc an induce chirality,i rrespective of the presence of classical Euclidean stereochemical elements.Although the number of synthesized knots is still small, small-molecule knots have already been shown to exhibit strong and selective anion binding,chirality,and catalytic activity,including asymmetric catalysis and allosteric catalysis.

Knot Dynamics
Thedynamic properties of catenanes and rotaxanes have been explored as part of the development of artificial molecular machines. [114,115] Thed ynamic behavior of the mechanically constrained backbones of molecular knots is, as yet, relatively unexplored. In their seminal paper on molecular trefoil knots, [27] Dietrich-Buchecker and Sauvage reported that demetalation of the knot leads to broadening of the aromatic region of the 1 Hspectrum, thus suggesting slow reptation (snake-like movement) of the knot chain. This effect was not observed for non-interlocked side products. Similar broadening was observed in the recently synthesized 8 19 molecular knot, which is particularly tightly knotted. [110] It was subsequently shown that removing just asingle metal ion from Sauvagest refoil knots results in ac onformational change which, depending on the spacer used in the helicate, rendered removal of the remaining metal ion either faster (with an alkyl linker) or slower (with a m-phenylene linker). [116] Lukin and Vçgtle reported that the dynamic behavior of his hydrogen-bonded trefoil knots is solventdependent;i ns olvents other than DMSO,t he knots were found to undergo slow dynamic motion (indicated by broad signals in the 1 HNMR spectra). [117]

Chirality
Some examples of topological chirality in small-molecule knots have been studied, as it is possible to either carry out the synthesis of chiral knots asymmetrically [68,73,74,84,87,99] or separate the enantiomers of knots produced through ar acemic synthesis. [64,67,83,107,110] Thee nantiomers of chiral knots have Scheme 18. Synthesis of an 8 19 knot based on the circular triple helicate approach. Tetramer Fe 4 47 4 is formed by the reaction of ligand 47 with FeCl 2 .C ovalent capture by RCM yields knot Fe 4 48. [110] . . This is analogous to forming dimers of aracemic compound:ameso diastereomer( combining R and S)i sobtained as well as chiral diastereomers (combining R and R or S and S). b) Forming the knot sum of two different chiral knots gives four distinguishable knots, analogous to joining two different chiral centers in amolecule.
Scheme 19. Synthesis of amixture of molecularcomposite knots. The dimerization of two open trefoil knots 49 leads to the formation of molecularg ranny knot 50 and molecular square knot 51 among other products, as deduced by MS and NMR data. Which knot is formed is determined by the handedness of the two open trefoil precursors. If two complexeso fthe same handedness are combined, agranny knot is obtained, while the combination of two complexeso fopposing handedness yields asquare knot. [113] Angewandte Chemie Reviews been studied by circular dichroism (Figure 22), and the spectrum of enantiopure trefoil knot 15 b shows ag reater ellipticity than the corresponding topologically isomeric macrocycle.This finding suggests that the topological chirality of the knot has as ignificant effect on the asymmetry of the chromophore environment. [73] Enantiopure trefoil knot 17,w hilst encapsulating europium, was found to catalyze the asymmetric Mukaiyama aldol reaction with up to 66 % ee ( Figure 23). [74] On the basis of luminescence decay lifetime measurements,itwas postulated that the mechanism of the catalyzed reaction involved coordination of the aldehyde to the knot-bound lanthanide ion whilst the knot maintained ac hiral environment in the vicinity of the aldehyde.

Host-Guest Chemistry and Catalysis
Va rious small-molecule knots have been found to act as host molecules that strongly bind to guest metal ions (which often facilitate their synthesis), organic molecules,o ra nions. Lukin and Vçgtle found that athin layer of an organic trefoil knot could adsorb octane. [117] Thetrefoil knots of Tr abolsi and co-workers (Scheme 10) can be transmetalated and the anion within the central cavity changed (binding of I À ,N 3 À ,SCN À , and NO 3 À reported). [90,91] Chloride anions are used to template the assembly of the pentameric circular helicates used to assemble 5 1 knots 44 and 46 in Section 3.3.2. The resulting pentafoil knots bind chloride anions in the central cavity with K % 10 10 m À1 in MeCN,thus making them amongst the strongest chloride-binding synthetic molecules known and with an affinity to chloride comparable to that of silver salts. [118] Tr ansmetalation of pentafoil knot 46 with Zn 2+ allowed aderivative of the knot to be used for allosteric regulation of Lewis acid carbocation catalysis of Diels-Alder and Michael reactions ( Figure 24). [107] With the Zn 2+ coordinated to the knot, abromide ion could be abstracted from trityl bromide to yield ac atalytically active trityl cation. No catalytic activity was observed when the knot was not present or was demetalated. In addition, the metalated knot could catalyze the Ritter reaction, whereby bromide was abstracted from bromodiphenylmethane to give the benzhydryl cation, which subsequently reacted with acetonitrile.T he bromide ion was Figure 22. Circular dichroism (CD) spectra of the two enantiomers of a) Sauvage's trefoil knot 5a [67] and b) pentafoil knot 46. [107] Neither knot has elements of Euclidean chirality.Reproducedf rom Refs. [67] and [107] with permission from Wiley-VCH and the AmericanA ssociation for the Advancement of Science, respectively. gives the metalated knot with an empty central cavity.b )The metalated knot can remove bromide from trityl bromide to give the catalytically active trityl cation. c) Subsequent demetalation regenerates the organic knot ligand 46,thereby shutting down the catalytic activity. [107] removed from the knot cavity by reaction with methyl triflate, thus regenerating the empty central cavity and allowing the catalyst to turn over.
In many of these early examples of properties and putative applications,t he knotted architecture of the molecules plays ac rucial role.W ith no elements of Euclidean chirality,i ti st he topology of the knot that leads to the CD response of 5a and 46 in Figure 22. Similarly,t he knotted topology of ligand 46 is crucial for the allosteric catalysis shown in Figure 24;unknotted ligand strands coordinate with Zn 2+ ions to form triple helicates and linear oligomers which do not bind anions nor have catalytic activity.

Knots in Synthetic Polymers
Just as earphone cables and spaghetti have at endencyto become entangled, polymer chains of sufficient length and flexibility also undergo spontaneous knotting at the molecular level. [119] Theoccurrence of knots in polymers was statistically modeled by Vologodskii and co-workers,w ho predicted that knot formation was likely to occur in polymers where the length of the monomer unit was significantly longer than the thickness of the chain, such as DNA. [16] Further studies determined that ever more complicated knots are likely to be formed as the length of ap olymer increases,a nd that these knots are increasingly likely to be composite (see Section 2.3) rather than prime. [120] Simulations have also shown that some knots are favored structures that could self-assemble from solutions of helical building blocks with sticky ends. [121] The location of knots in non-uniform polymers has also been investigated. [122][123][124] Recent investigations have allowed polymeric knots to be visualized directly by atomic force microscopy (AFM;F igure 25) [125] and for their controlled assembly by metal-template synthesis and high dilution cyclization. [126,127] It is well known that knotting astring (or rope) decreases its strength, and when pulled at either end such as tring will break at the entrance to the knot. Mountaineers and fishermen know to use knots with ahigh efficiency (i.e.aknot that decreases the strength of as trand the least) to maintain the strength of ar ope or line. [17] Thew eakening effect of tying aknot in amolecular strand has also been considered;itwas shown theoretically that tightening ak notted polyethylene chain should cause strain energy to be located at the CÀC bonds at the entrance to the knot. [128] Further tightening of the knot might, therefore,r esult in the breaking of one of these CÀCbonds at alower dissociation energy than an unknotted chain. An experimental demonstration of such aphenomenon has been reported:tying aknot in an actin filament by using molecular tweezers reduced its tensile strength by approximately two orders of magnitude,a nd pulling of the polymer chain caused breakage where the knot was located ( Figure 26). [129] Additionally,i ntermolecular entanglement of multiple polymer chains can also result in knotting, thereby affecting the morphology and mechanical properties of polymeric materials. [130] Ar elated 3D interwoven material has recently been reported, which also displays high elasticity when demetalated. [131]

Knots in DNA
Examples of circular DNAc ontaining knots were first found in 1976, [132] nearly ad ecade after the discovery of naturally occurring DNAl inks. [133,134] Thet opology of the DNAwas imaged by electron microscopy.D NA knotting is mediated by topoisomerase enzymes,w hich either allow the passage of as ingle strand through the nick in the complementary strand (Type I) or the passage of asegment of duplex DNAthrough adouble-stranded break (Type II). [135] Incubation of DNAwith topoisomerase Ifrom E. coli. gave amixture of all possible knots up to at least seven crossing points (with no stereocontrol, Figure 27). [136] Topoisomerases were originally thought to act under thermodynamic control;h owever,t opoisomerase II was shown to use ATPh ydrolysis to reduce DNAk notting below the equilibrium value. [137] Thep resence of knots in DNAh inders transcription and replication and can lead to mutations. [138][139][140] Therefore,t opoisomerases are required to maintain cell function by unknotting DNAa nd reducing supercoiling. [141,142] Am olecule of circular DNAc annot be unknotted if topoisomerase enzymes are not present, unless its backbone is broken. Therefore,t he knot invariant (see Section 2.1) of DNAdoes not change in routine manipulation (e.g. in isolation and analysis) and provides auseful handle for studying DNAinvitro. [14] Figure 25. Knotted and interlocked cyclic polymers imaged by AFM. [125] Adapted from Ref. [125] with permission from Wiley-VCH. Figure 26. An actin filamentt ied in aknot by moleculartweezers. [129] Tightening of the knot eventually causes the filament to break at the location of the knot. Adapted from Ref. [129] with permission from Springer Nature.
Supercoiled DNAc an become knotted and form links during site-specific recombination when genome rearrangement is performed by the recombinase enzymes. [143] Tangle theory (Section 2.11) has proved useful in understanding the topological implications of the actions of such recombinases. [144] Linear DNAi sd ensely packed and highly confined in phage capsids,w hich results in ah igh writhe value (see Section 2.1). [145] This manifests itself in av ery high level of DNAknotting (ca. 95 %ofthe molecules), with apreference for torus knots (with ah igh writhe value) and only trace amounts of the achiral figure-eight knot (with av ery low writhe value) being observed. [146] Perhaps surprisingly,n aturally knotted RNAh as not yet been reported, with the underlying reasons for its absence unclear. [147] Artificial DNAa nd RNAk nots have also been reported, through pioneering work by the Seeman group. [148]

Knots in Proteins
As most proteins consist of abackbone with two termini, they do not form the closed loops required for mathematically defined knots.However,imaginary connecting of the termini without generating additional crossing points provides af ramework that allows entanglements ("knotting") within proteins to be analyzed. In addition, cross-linking of aprotein backbone by disulfide bonds or prosthetic groups can lead to interlocked and knotted structures within proteins (e.g. "cysteine knots"). [149][150][151] Thefirst protein to be identified with aknotted backbone was carbonic anhydrase. [152] This protein is tied in al oose trefoil knot, but only afew residues need to be removed from one terminus to unknot the protein, so it was suggested that this protein only forms an "incipient" knot. [153] It was suggested that the mechanisms of folding prevented reptation of the protein chain required for knot formation within the protein core.A nother loosely knotted protein was reported shortly afterwards,also containing atrefoil knot, where up to 10 residues could be removed before unknotting the protein chain ( Figure 28). [154] In 2000 Taylor introduced an ew computational method for probing knots in proteins. [155] This allowed entanglements buried in the core of proteins (known as deeply knotted proteins) to be discovered for the first time.A nalysis of the Protein Data Bank (PDB) revealed as eries of proteins that formed deeply knotted left-and right-handed trefoil knots (3 1 )and two proteins that formed amore complicated figureeight knot (4 1 ). One of the latter proteins,acetohydroxy acid isomeroreductase,required the removal of over 300 residues to unknot the protein. Many other knots in proteins have been identified, with the most complicated to date being aS tevedore knot (6 1 )i nD ehI, an a-haloacid dehalogenase ( Figure 29). [156] Ar ecent analysis of the PDB,h owever,s uggests that the proportion of proteins that are knotted is small (ca. 1%). [157] This proportion is lower than what would be expected for similar heteropolymers,which implies that nature has,for the most part, specifically avoided protein knotting. [158] Simulations have suggested that local ordering within the hydrophobic core of proteins disfavors entanglement and knot formation, which itself is determined by the protein Figure 27. Knotted DNA (with conventional representations of knots) produced on incubationo fcircular DNA with atopoisomerase I enzyme and imaged by electron microscopy. 136 Adapted from Ref. [136] with permission from the American Society for Biochemistry and Molecular Biology. Figure 28. Schematic representation of aloosely knotted protein. [154] A trefoil knot is formed by passage of the B9 b-strand leading to the C-terminus through aloop formed by the sequenceB 1![central domain]!B5!H3!B6. Adapted from Ref. [154] with permission from the American Chemical Society.
sequence. [159] Interestingly,i th as also been found that many knotted proteins have loop segments,w hich unknotted proteins with similar structures or sequences lack, and so may be required to promote knotting. [160] This suggests that the knotting of proteins may be at least partially encoded in protein primary structure.H owever,astudy of the first synthetic knotted protein, whose sequence was derived by modifying an unknotted dimeric protein, showed it could fold in seconds without the need to incorporate knot-promoting segments. [161] Although proteins may not necessarily contain aknot for aspecific purpose,itisworth noting that ubiquitin hydrolase needs to be particularly resistant to unfolding. It folds to give a5 2 knot, and it has been shown that complex knotting grants kinetic stability to proteins. [162,163] As most known knotted proteins are enzymes,a nd the knot is usually located in the catalytic domain, knots may have an important effect on enzymatic activity. [164,165] Thep resence of ac ysteinek not within the core of ap rotein has been shown to confer exceptional stability. [166]

Conclusions and Outlook
Thep ast 30 years have seen the invention and serendipitous discovery of ar ange of synthetic strategies for the construction of molecular trefoil knots,enabled by progress in the control of reactivity,c onformation, and supramolecular structure.H owever,t he synthesis of other molecular knots remains an almost entirely unconquered challenge for synthetic chemistry.O ft he six billion prime knots tabulated to date, [29] only four-the trefoil, figure-eight, pentafoil, and 8 19 knot-have been synthesized thus far using small-molecule building blocks.T hat is av ast volume of completely unexplored molecular space.A st he number of molecular knot topologies that become accessible increases,c hemists will start to develop af uller picture of their properties (at which point are entangled molecular strands prone to breaking?a nd if knotting weakens am olecular strand, can it be used to promote bond breaking?) and will discover which knots have properties best suited for ap articular purpose. Once we can make molecular knots and understand their properties,knotting may start to have an impact on functional molecule and material design in the same way that tying knots proved so important for advancing the technology of our earliest ancestors.

Glossary
An explanation of knot, braid, and tangle terminology used in this Review that may be unfamiliar to chemists: Achiral/amphichiral/amphicheiral knot Ak not that can be deformed continuously into its mirror image.

Alexander-Briggs notation
Notation of the form X Y used to distinguish aknot from others. X refers to the number of crossings, Y is av ariable used to differentiate knots with the same number of crossings.
Alternating knot Ak not that can be represented in away that over-and underpasses of the strand alternate.

Braid representations
Every knot can be represented as abraid of n strands.T he closed loop knot is obtained by connecting the strands at the braid ends (see Section 2.10).

Chiral knot
Ak not that cannot be continuously deformed into its mirror image.
Composite knot Ak not that can be described as the combination of two or more prime knots.
Crossing Apoint in which the projection of aknot crosses itself;crossings can be positive or negative (see Section 2.1).
Gordian distance Then umber of crossing changes needed to interconvert two knots (see Section 2.12).

Invariant
An intrinsic property of aparticular knot, for example its minimum number of crossings.

Invertible knot
Ak not that can be continuously deformed into areversed orientation of itself (see Section 2.5). Figure 29. X-ray crystal structure and reduced schematic diagram of DehI, the most complicated knotted protein known to date, which contains aS tevedore knot (6 1 ). [156] Adapted from Ref. [156] with permission from PLOS.

k-moves
Thei ntroduction of k positive crossings into aset of two strings.

Non-alternating knot
Ak not that cannot be represented in away where over-a nd underpasses alternate.
Nugatory crossing Ac rossing that can be removed by twisting.

Pretzel knots
Knots obtained by connecting left-and right-handed helices to form closed loops (see Section 2.11).
Prime knot Ak not that cannot be described as the combination of simpler knots.
Reduced representation Depiction of aknot with its minimum number of crossings.

Reidemeister moves
As et of string manipulations that transform different representations of the same knot into each other (see Section 2.1).

Tangles
Building blocks of entanglements from which knots can be created.

Torus knot
Ak not that can be drawn on the surface of atorus without intersecting (see Section 2.7).
Tw ist knot Ak not created by twisting two strands n times and interlocking the open ends before closure (see Section 2.8).

Unknot (trivial knot)
Ak not that can be deformed into arepresentation without any crossings.

Unknotting number
Them inimum number of crossings of aknot that have to be flipped to yield the unknot.

Writhe
Thes um of positive and negative crossings in the representation of ak not (see Section 2.1).