Chemical Society Reviews

Knot Theory in Modern Chemistry

Journal: Chemical Society Reviews

Manuscript ID CS-TRV-06-2016-000448.R1

Article Type: Tutorial Review

Date Submitted by the Author: 10-Aug-2016

Complete List of Authors: Horner, Kate; Durham University, Department of Mathematical Sciences
Miller, Mark; Durham University, Department of Chemistry
Steed, Jonathan; Durham University, Department of Chemistry
Sutcliffe, Paul; Durham University, Department of Mathematical Sciences
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Journal Name RSCPublishing

ARTICLE

Knot Theory in Modern Chemistry

Cite this: DOI: 10.1039/x0xx00000x Kate E. Horner,*a Mark A. Miller,b Jonathan W. Steedb and Paul M. Sutcliffea

Knot theory is a branch of pure mathematics, but it is increasingly being applied in a variety of
sciences. Knots appear in chemistry, not only in synthetic molecular design, but also in an
Received 00th January 2012, array of materials and media, including some not traditionally associated with knots.
Accepted 00th January 2012
Mathematics and chemistry can now be used synergistically to identify, characterise and create
DOI: 10.1039/x0xx00000x knots, as well as to understand and predict their physical properties. This tutorial review
provides a brief introduction to the mathematics of knots and related topological concepts in
www.rsc.org/
the context of the chemical sciences. We then survey the broad range of applications of the
theory to contemporary research in the field.

Key Learning Points development of modern knot theory, which today is an area of
mathematics within the field of topology. In 1867, Sir William
• Some fundamentals of knot theory.
Thomson (later to become Lord Kelvin) proposed that atoms
• Knot theory and closely related ideas in topology can
are composed of knotted vortices of the aether.2 While this
be applied to modern chemistry.
hypothesis subsequently turned out to be incorrect, not least
• Knots can be formed in single molecules as well as in
because the existence of the aether was later disproved, it did
materials and biological fibres using a mixture of self-
spark a fascination with knots that has lasted for well over a
assembly, metal templating and optical manipulation.
century. In mathematical terms, a knot is defined as a non-self-
• The inclusion of knots in molecular structures can
intersecting closed curve in three-dimensional space.
alter chemical and physical properties.
Importantly, this means that for a knot in a piece of rope to be
• Knots are surprisingly ubiquitous in the chemical
considered a mathematical knot, the free ends must be joined.
sciences.
On a closed loop, the knot can be distorted but not removed or
fundamentally altered.
1. Introduction to Knot Theory The theory of abstract mathematical knots is concerned with the
The birth of mathematical knot theory can be traced back to the characteristics that are locked into a closed curve by the
work of Vandermonde (1771),1 who was a musician by presence of a given knot. These characteristics allow knots to
training, but in later life made contributions to both be classified and compared, and provide a basis for
mathematics and chemistry. However, it was physicists of the understanding the implications of knots when they arise in a
mid-19th century who provided the impetus for the physical system. Knots are being found to play a role in more
and more scientific contexts, and knot theory is therefore
gradually making its way into many fields of study, from
chemistry to physics and even anthropology.3
In this tutorial review we examine the application of knot
theory broadly across the chemical sciences, ranging from its
direct application to molecular and colloidal structures, to less
clear-cut systems such as tangled gel matrices, knotted proteins,
and interwoven polymers. In mature areas that have been
previously reviewed in more depth, such as molecular and
biological knots, we include just fundamental points, a few
recent developments and key references to more specialised
reviews. For more detail on the underlying theory of knots
outlined in this article, we recommend the highly accessible
introductory text by Adams.3

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representations of the trefoil knot by two Reidemeister moves

of type I and two of type III is shown in Figure 2b.
Reidemeister moves of type I and II change the number of
crossings in a diagram. A suitable sequence of all three types
of Reidemeister moves can therefore (in principle) be used to
simplify a given knot diagram until the number of crossings can
be reduced any further, and this number is then called the
crossing number of the knot. The crossing number is useful in
the taxonomy of knots. For example, the knots in Figure 1 are
labelled according to their crossing number using Alexander-
Briggs notation, where knots with the same crossing number
are grouped together and labelled with a subscript (the order of
labels for a given crossing number is arbitrary and so has no
real significance).
Converting a knot diagram to its simplest form may be difficult
1.1 Classification of Knots in practice if there is a large number of crossings.
Consequently, an important challenge in knot theory is to
An important aspect of knot theory is the classification of determine the knot to which a particular knot diagram
knots, which is designed to answer questions like: What kind of corresponds. Key tools in this endeavour are knot invariants,
knot is this? Are these two knots the same? Is this the simplest which input a knot and output a mathematical object, for
way to draw this knot? Is this knot actually just the unknot? The example a binary digit (True or False), an integer, a
unknot (see Figure 1), also called the trivial knot, is simply a polynomial, or something more complicated. We have already
closed curve which can be smoothly deformed into a circle. It is seen one example of a knot invariant, namely the crossing
necessary to include the unknot in knot theory for number, where the output is an integer. The crucial property of
completeness; defining the unknot is analogous to including a knot invariant is that any two diagrams of the same knot have
zero in the set of integers, so that when counting objects we are identical values of the invariant. In other words, smooth
able to deal with the case of not having any objects at all. deformations of a knot do not affect knot invariants. We see
Two knots are considered to be the same if one can be smoothly from Figure 1 that the unknot and the trefoil are not the same
deformed into the other, whilst avoiding self-intersections. To knot, as they have different values (zero and three respectively)
draw a knot, it is convenient to project the curve in three- for the invariant crossing number. On the other hand, the three
dimensional space onto a plane, to produce a knot diagram knots in Figure 1 that have crossing number six cannot be
(some examples are displayed in Figure 1) consisting of strands distinguished on the basis of this invariant alone.
with crossings, where the strand that goes under must be Some invariants output a polynomial, consisting of coefficients
distinguished from the strand that goes over. For a given knot, and powers of an arbitrary variable, t. For example, the so-
different knot diagrams can be obtained by changing the plane called Alexander polynomial of the trefoil is ∆    1 
of the projection. Furthermore, smooth deformations that keep  . Knot polynomials are rather abstract objects; in most
the knot topologically the same may change the number of applications, the variable t has no physical significance and a
crossings in an associated knot diagram. Any two diagrams of plot of ∆ against t also lacks a simple interpretation. For the
the same knot can be interconverted by a suitable sequence of purpose of identifying knots, the important features of the
just three types of elementary “Reidemeister moves”, shown in polynomial are the powers of t that appear in it (1, 0 and −1 for
Figure 2. For example, the interconversion of two equally valid the trefoil example) and their coefficients (1, −1 and 1). These

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derived from a given knot diagram. Knot classification is still a

daunting task, even for fairly small crossing numbers, as there
are more than 1.7 million different knots with sixteen or fewer
crossings. It is not known whether there exists a knot
polynomial that is a complete invariant, i.e., one that would
uniquely identify all knots.
An analogy can be made between calculating a knot invariant
and performing gas chromatography. In chromatography, the
goal is complete separation of all chemical components and
failure to achieve this results in co-elution (see Figure 3). This
means that two different eluates could mistakenly be identified
as being the same compound. The same is true of an incomplete
knot invariant; a given invariant may be unable to distinguish
between two knots that are actually different. An extreme
example is a knot invariant that takes only the values True or
powers and coefficients are determined by the order in which False, such as tricolourability. A knot is tricolourable if each of
the over- and under-crossings of the knot are encountered as the strands between adjacent undercrossings in a projection can
one moves along the curve and, crucially, the same set of be assigned one of three colours such that each crossing brings
powers and coefficients is obtained for any diagram of a given together either all three colours or just one colour (excluding
knot. Hence, the pattern of powers and coefficients of t are a the trivial possibility of assigning the same colour to all
“fingerprint” of a knot’s topology regardless of how it is drawn. strands). For example, the 61 knot is tricolourable whereas 41 is
Identifying a knot from the powers and coefficients of its not (see Figure 1). As this knot invariant separates all knots into
polynomial is analogous to the way that a chemist might only two types, it is clearly unable to identify a specific knot.
identify a molecule from the fragmentation pattern in its mass However, it can, for example, distinguish between the trefoil
spectrum. knot and the unknot, since the former returns True and the latter
Other common knot polynomials include the Jones and False. Stronger knot invariants, such as knot polynomials, allow
HOMFLY polynomials, which, like the Alexander polynomial, a better distinction between knots, and work effectively for all
are named after their discoverers. The polynomials can be knots below a certain level of complexity.
computed from a knot diagram and hence have an important
role in answering the central question of whether two diagrams 1.2 Links
represent the same knot. However, the Alexander polynomial The concept of a knot can be extended to include collections of non-
cannot distinguish between all the knots displayed in Figure 1, intersecting closed curves, which are known as links. A knot is then
because the left-handed and right-handed trefoils have the same the special case of a link with only one component. Much of the
Alexander polynomial. The two versions of the trefoil are treatment of knots, such as the polynomials, can be generalised
nevertheless distinct knots because one cannot be continuously to links. Just as the trivial knot is called the unknot, there are
deformed into the other. One way to tell the left- and right- trivial links or “unlinks”. The two-component unlink, which is
handed trefoils apart is by their respective Jones polynomials, just two distinct unknots, can be seen in Figure 4. The simplest
which (unlike the Alexander polynomials) do have different non-trivial two-component link is the Hopf link, also displayed
sets of coefficients and powers. Knots like these, which cannot in Figure 4, where two unknots are linked once, as exemplified
be smoothly deformed into their own mirror image, are called in chemical systems by [2]catenanes. The simplest link
topologically chiral. This concept of chirality is similar to that invariant is the number of components but another important
in chemistry, where a chiral molecule has a non- invariant is the linking number, which is an integer that
superimposable mirror image. However, it is important to measures how many times each component winds round the
remember that topological chirality is defined by the inability other. It does this by counting the number of crossings between
of a knot to be deformed into its mirror image, and not by the the strands of the two components, distinguishing between the
particular geometrical configuration of a knotted structure. two possible orientations of crossings, denoted positive and
The Jones polynomial, and its HOMFLY generalisation, can negative as illustrated in Figure 5. The linking number is half
distinguish between all the knots shown in Figure 1, including the difference between the number of positive and negative
the mirror images of the trefoil, but for knots with ten or more crossings. Reversing the orientation of either component
crossings there are examples of different knots that have the changes the sign of the linking number, but its absolute value is
same polynomial, even for some knots with different crossing independent of the choice of orientations. It is perhaps not
numbers. In fact, it is still unknown whether there is a non- surprising to find that (with appropriate choices of orientation)
trivial knot that has the same Jones polynomial as the unknot. the unlink, Hopf link and Solomon's link, presented in Figure 4,
Despite these limitations, knot polynomials are still amongst have linking number 0, 1 and 2, respectively. It is less obvious
the most useful tools for distinguishing between knots because that the Whitehead link, also shown in Figure 4, has linking
they do distinguish many knots and can be systematically

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an odd number of half-twists in the chain of p-π orbitals

necessitates a mismatched, anti-bonding, overlap at some point
in the chain. The mismatch is a manifestation of the fact that
Möbius strips are non-orientable. This means that it is
impossible to choose a unit normal vector consistently at all
points on the surface (see Figure 6). In Section 5 we will see
that the topological property of non-orentability can have
important physical consequences.

1.4 Braids
There is an intimate relationship between knots, links and
number zero, but this is a result of it having an equal number of braids. A braid is a set of intertwined strings that are fixed at
positive and negative crossings, yielding zero for the difference. the top and bottom and are always pointing downwards, so that
Turning to non-trivial links with three or more components, an no string ever turns back up. This is like plaited hair, where the
interesting family are the Brunnian links, which are defined by strands are fixed at the scalp and at the bottom by a hair band.
the property that the removal of any one component leaves only Braids are inherently related to knot theory since all knots and
unlinked unknots. The Borromean rings (Figure 4) are the links can be obtained as the closure of a braid by joining the
simplest example, and consist of three linked unknots, no pair ends (see Figure 7 for an example). However, the closure of
of which are directly threaded through one another. different braids can produce the same link, just as different knot
Just as for knots, Alexander-Briggs notation can be used for diagrams may correspond to the same knot. It is interesting to
links by adding a superscript to show the number of note that the Jones polynomial was originally defined as a braid
components in the link. For example, the two-component invariant that was shown to depend only on the type of the
unlink (shown in Figure 4) is denoted 0 while the Hopf link is closed braid.
2 and the Solomon’s link 4 .
2. Molecular Knots
1.3 Möbius Strips
Perhaps the most direct way to apply knot theory to chemistry
There is also an intrinsic connection between knot theory and
is to synthesise molecules with knotted topology. An obvious
the strange topology of Möbius strips. A physical Möbius strip
starting point is a knot with the smallest crossing number.
can be formed by half-twisting a strip of paper an odd number
Creating a knotted topology using indirect chemical synthesis
of times and then fixing the ends together. This has the
methods is a significant challenge and requires careful reaction
intriguing effect of giving the object only a single edge and a
design and the use of templating methods. The field is highly
single side. With one half-twist, the edge forms the unknot but
active and a wide range of creative techniques have been
with three half-twists the edge forms the trefoil knot.
employed to produce knotted and linked molecules. Some key
Certain characteristics of Möbius strips have been known in
illustrative advances are summarised in Figure 8 (see
molecular systems for some time.4 For example, Möbius
References 5–7 for extensive reviews).
aromaticity involves a twisted arrangement of the conjugated π
orbitals and requires 4n π-electrons, in contrast to the more 2.1 Simple Molecular Knots
common Hückel aromaticity, which requires 4n + 2 electrons.
The difference in the aromaticity rule arises from the fact that

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So far only three different knot types have been realised unknotted products. This is the case for the more recently
synthetically, namely the 31 trefoil knot, the 41 figure-of-eight prepared molecular pentafoil knot (Figure 8c and the knot
knot and the 51 pentafoil knot.7 The first well characterised labelled 51 in Figure 1) which is the most complex, non-DNA
molecular knot was a trefoil synthesised by Dietrich-Buchecker molecular knot prepared to date.9 Here, reversible metal-imine
and Sauvage in 1989 (Figure 8a).8 This landmark advancement bond coordination was used to allow correction of any
was achieved by using transition metal templating to form a unwanted bond formation. Moreover, anion templating with
helical structure which was then covalently cyclised to create a chloride and careful use of stereoelectronic effects, symmetry
permanent knot. Templating is an effective route to knot and linker length, were all needed in order to form this
synthesis because metal ions have a well-defined coordination complicated structure.
geometry and the strength of coordination strikes the right In very recent work,10 the pentafoil knot has been reached by a
balance between lability and stability to promote reliable different route, based on ring-closing olefin metathesis. This
formation of the desired structure. It is also possible for the approach has the advantage that, once the knot has been
metal ions to serve a dual purpose by catalysing the ring closure formed, the Fe(II) template in each of the pentafoil’s five lobes
chemistry that links the templated fragments.6 Trefoil knots and the halide ion in its central cavity can be removed, leaving
have also been produced by hydrogen-bonded templating the uncoordinated knotted ligand. Without the metal cations,
methods, as in amide-amide hydrogen bonding and by dynamic the ligand is flexible, but can readily be rigidified by
combinatorial chemistry (DCC) approaches which have given coordination with Zn(II) ions in place of the original Fe(II).
rise to a trefoil knot reported by Sanders and co-workers in This metallated form acts as an effective catalyst for carbon-
2012 (Figure 8b) based on a naphthalenediimide aqueous halogen bond cleavage because it is one of the strongest
disulfide dynamic combinatorial library. The knot assembly is noncovalent binding synthetic hosts of Cl– and Br– known. It
driven by hydrophobic effects. DCC methods have also resulted can efficiently catalyse the generation of a carbocation from the
in a figure-of-eight knot (Figure 8e) and Solomon’s link hydrolysis of bromodiphenylmethane for example, whereas the
reported in 2014.6 Interestingly, resolved chiral building blocks unknotted form is inactive. The knot structure is crucial in
give rise to a topologically achiral figure-of-eight knot while a achieving this chemical function because it restricts the
racemic mixture gives a different meso figure-of-eight knot. A conformations that the ligand may adopt, stabilising the active
Solomon’s link was also produced in 2013 by the Leigh group form. The metal-free knotted ligand is catalytically inactive,
based on metal templating via a tetrameric cyclic double providing a means for allosteric regulation of the catalysis;
helicate scaffold (Figure 8f).7 binding of metal ions at one set of locations (the lobes of the
For more complex knots, metal templating must be used pentafoil) affects binding of the halide at another point (the
alongside other self-assembly and synthetic techniques in order central cavity).
to obtain the desired structure rather than a complex mixture or

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There are several iconic links in mathematical knot theory (see

Figure 4) and these links have also become enticing synthetic
targets for chemists. However, unlike the trefoil, which was
only synthesised for the first time in 1989, the simplest link, the
Hopf link, has been known in synthetic chemistry for far
longer. In 1960, Wasserman was able to create small amounts
of two interlocked molecular rings using a statistical threading
approach.13 He named the product a catenane, from the Latin
“catena”, meaning chain. The preparation of these molecular
Hopf links (termed [2]catenanes) became far more efficient
with the use of self-assembly and templating techniques, in a
More generally, once molecular knots have been created, they
can then be derivatised, either to allow for the synthesis of
higher assemblies of knots (see Section 2.2) or to alter or study
the properties of the system itself. For example, an amide-based
trefoil knot has been mono, di- and tri-dendronised to form
molecules known as “dendroknots”.11 These have a knotted,
topologically chiral core with one, two or three dendritic side-
arms at the periphery. The chirality of the central knot has been
shown to have some effect on the preferred orientation of the
propeller-like dendritic substituents. Moreover, this type of
functionalisation has allowed simple molecular knots to be used
as nano-sized scaffolds for a wide range of potential
applications.5

2.2 Molecules Containing Multiple Knots

Once the synthesis of simple molecular knots was established,
it became possible to include knotted moieties into larger
molecular structures. In knot theory, non-trivial knots can be
added to make a composite knot, where knot addition means
placing the two knots side-by-side, removing a short segment
from each knot and joining the free ends of one knot to those of
the other. In a similar way, tied open-chain fragments have
been combined experimentally by Sauvage and co-workers to
make molecular composite knots.12 Knots that cannot be
constructed by such an addition are called prime knots, and it is
this type of knot that we have been considering so far in this
review.
In chemistry, multiple non-trivial knots can be appended to
simpler molecular structures, although this does not make them
composite knots in the mathematical sense. The steric bulk of
appended knots allows them to function as stoppers in a
rotaxane, for example, as in the trefoil-knot-stoppered rotaxane
shown in Figure 9 described as a ‘knotaxane’ by the authors.5
Small cyclic oligomers of up to four units in size have been
produced using trefoil knot motifs as building blocks.5 These
types of structure have been termed “knotanophanes” (see
Figure 9) and exist in a number of diastereomeric forms as a
result of the chirality of each knot (see Figure 1). However, the
topological isomers have similar physical properties making
separation and/or identification of such molecules an interesting
challenge.

2.3 Molecular Links

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similar way to the simple knots. In 1999, a method for

producing [2]catenanes by using two pre-formed rings was
reported and described as a molecular version of “magic
rings”.14 The trick for achieving this interlocking was the
introduction of metal ions into the ring backbone. Inclusion of
these metals led to the formation of reversible metal-ligand
bonds which allowed the pre-formed rings to split and reform
once they became entwined with another.
The less common Solomon’s link (sometimes less accurately
called a Solomon’s knot) can also be found in chemistry as a
doubly-interlocked [2]catenane. Solomon’s links, just like
trefoil knots, are topologically chiral. The first successful with a graph in three-dimensional space, consisting of a
synthesis of a molecular Solomon’s link was achieved by using collection of vertices together with a set of edges (the strands)
metal-templating to create a helical arrangement of molecular that connect them. In graph theory, the term “valency” (or
fragments which could then be cyclised to form the required “degree”) is used to refer to the number of edges connected to
doubly interlocked link.15 Unfortunately, this only yielded the vertex. An example of a graph with two trivalent vertices
approximately 2 % of the desired product. However, since then, and three edges is shown in Figure 11c. A graph may contain
several more efficient methods have been developed, such as cycles, which are closed loops obtained by following the edges
“all-in-one” syntheses using template-directed self-assembly of the graph from a starting vertex back to the same vertex
(Figures 8e and f).16 On a similar theme, the so-called without visiting any edge or intermediate vertex more than
“Solomon’s cube” was synthesised using a once. Graphs are perhaps most familiar in chemistry in the form
metallosupramolecular self-assembly process.17 The Solomon’s of the bonding networks in molecules, where the vertices of the
cube structure is essentially a Solomon’s link motif involving graph correspond to the atoms and the edges of the graph
four palladium ions and four tris(3-(3- represent the chemical bonds.
pyridyl)phenylester)cyclotriguaiacylene ligands, of which the A graph is called unknotted if it can be smoothly deformed
latter are able to intertwine producing the complex topology (whilst avoiding edge intersections) to lie entirely within a
(Figure 8g). plane. This definition for graphs is consistent with that for
One very recent example of synthetic advancement in this field knots, since the only knot that can be deformed to lie entirely
is the preparation of a triply-interlocked [2]catenane, one which within a plane is the unknot; the planar projection of any other
the authors called the Star of David catenane.18 This procedure knot requires crossings that take strands out of the plane. One
involved using six tris(2,2′-bipyridine) motifs which formed a way to recognise that a graph is knotted (although this is not a
helicate structure around six iron(II) cations. Ring-closing necessary condition) is to identify a cycle in the graph that is a
olefin metathesis then allowed for the final catenane formation. non-trivial knot. To go further and classify the specific
One of the key synthetic challenges in this development was the topology of a graph, the concept of polynomial invariants for
preparation of bipyridyl ligands which had restricted knots can be extended to graphs. For example, if two diagrams
conformational space to force the olefin metathesis step to form are merely different representations of the same graph, then
the desired product rather than oligomers or cross-linked they will have identical Yamada polynomials.22 A particularly
materials. Furthermore, the use of sulfate counterions was interesting family of graphs are those that are minimally
required to adjust the size of the circular helicate. knotted, which means that the graph is knotted even though all
The first molecular Borromean rings (see Links, Section 1.2) the cycles of the graph form only unknots or unlinks. One way
were reported by the Stoddart group in 2004 (Figure 8d).19 to produce a minimally knotted graph is to take an unknotted
Since the Borromean rings are Brunnian, they are topologically graph and replace a vertex that has valency n with a universal
distinct from triply-interlinked [3]catenanes like those in Figure n-ravel.21 The universal 3-ravel and 4-ravel are displayed in
10, produced by the lanthanide-directed synthesis of Figure 11a and 11b and both have the property that joining all
Gunnlaugsson and co-workers; removal of one ring from the the loose ends to an additional vertex results in a graph
latter structure results in a Hopf link rather than the unlink.20 (displayed in Figures 11c and 11d, respectively) that is knotted
As well as having fascinating mathematical properties, but contains only unknotted cycles and no links. These graphs
Borromean linkage motifs can also be found in a variety of are therefore simple examples of minimally knotted graphs.
materials (see Section 4.2). The crucial feature of a ravel is the mutual weaving of the
edges that emerge from the single vertex, rather than any
2.4 Ravels knotting or pairwise linking. In this respect, ravels are
The term ravel, defined in the chemistry literature in 2008,21 reminiscent of Brunnian links.
refers to a mathematical structure that has been studied for Recently, the first molecular ravel was synthesised by the
around forty years. It is a generalisation of a knot that allows Lindoy group which they described as like a “branched knot”.23
three or more strands to fuse at a given point, known as a Twelve bis-β-diketone derivatives form a triple helicate
vertex. From a mathematical perspective we are now dealing structure with the use of ferric chloride to promote

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same number of segments. For a sufficiently small number of

segments (6 or 8), it is even possible for the average shape of a
trefoil knot to be oblate. The non-trivial topology places severe
constraints on the conformations that can be explored by a
hexagon or octagon, but polygons with a larger number of
edges rapidly revert to being prolate on average. Figure 13
shows example conformations of a trefoil in a 50-segment
chain. The shapes of the chain have been characterised in the
images by their inertial ellipsoids, which show the extent to
which the polymer’s mass (carried by the vertices of the
polygon) is distributed in three orthogonal directions.
In the case of circular DNA strands, the average size and shape
has a strong influence on the molecules’ mobility in
electrophoresis. This technique uses a uniform electric field to
metallosupramolecular self-assembly. The resultant self-
pull the molecule through a gel against the friction caused by
interpenetrating architecture was shown (using X-ray
the gel network. Electrophoresis can separate topoisomers
diffraction) to be a universal 3-ravel (Figure 11c). It is proposed
(molecules that differ by their knotting, linking or supercoiling)
that non-covalent interactions such as π-π stacking contribute to
because they experience different frictional forces and therefore
the stability of the ravel structure along with efficient filling of
reach different drift velocities in a field of given strength. In
space, making it more stable than the competing helicate. The
low-density gels, the mobility of a given DNA strand increases
successful realisation of such an intricately intertwined
with increasing knot complexity. This is because a given
structure both represents a significant advance in synthetic
polymer becomes, on average, more compact with the
control of molecular topology and opens up a range of new
complexity of the knot and can move through the gel more
challenges for synthesis.
easily. However, in denser gels, the mobility initially decreases
with increasing knot complexity and only starts increasing for
3. Polymers still more complex knots.27 One possible explanation for this
The chain-like structure of a polymer makes it a potential host non-monotonic behaviour is that more complex knots are not
for knots. In this section, we examine the effects of knots on the only smaller, but also less compressible because more complex
physical properties of polymers. knots are able to access fewer geometrical conformations for a
given length of DNA. Hence, a less complex knot, though
larger at equilibrium, can more easily deform to pass through a
3.1 Ring polymers fine gel mesh. Sufficiently complex knots, however, would
The natural conformation of a stiff, unbranched, linear polymer eventually be compact enough for the mobility to start
would be rod-like. Such structures, which are non-spherical increasing again.27
and extend significantly further in one direction than in the two Simulations by Michieletto et al. suggest another possible
orthogonal directions, are called prolate. The opposite case, explanation for the electrophoresis results.28 Modelling the gel
where the object is notably shorter in one direction than in the as an imperfect cubic mesh, where some of the edges were cut,
other two, is an oblate, disc-like structure (see Figure 12). It is these authors examined the effect of knots becoming impaled
well established, but not intuitively obvious, that even a on the dangling ends of the gel. Although more complex knots
completely flexible linear polymer is more likely to adopt a were less likely to become impaled (due to their compactness),
prolate than an oblate conformation at a given instant.24 The their entanglement was more likely to be severe, resulting in a
origin of this result is entropic; there are more ways to arrange a longer delay before the molecule could resume its drift and
freely jointed chain such that it defines a prolate shape than an causing an overall decrease in mobility.
oblate one.
3.2 Open polymers
A very stiff ring polymer, on the other hand, would have to
adopt an oblate, circular configuration to minimise bending. Despite the fact that the mathematical definition of a knot only
Hence, it is even more surprising that, like linear polymers, a
sufficiently flexible ring polymer is prolate on average.25
However, a given ring polymer may only explore
conformations that do not alter its topology. Rawdon et al.
have investigated how the presence of a knot affects the
average shape of flexible ring polymers by modelling them as
freely jointed polygons of rigid struts.26 The general effect of a
knot is to make the conformations of the polymer more
compact and less prolate than an unknotted polymer with the

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strictly applies to a closed loop, in everyday life we often refer

to knots in open chains, such as shoelaces or electrical cables.
An apparent knot in the middle of an extended string can
intuitively be formalised by joining the ends of the string.
However, if the ends are buried within the tangle (as always
seems to be the case with Christmas tree lights) then there may
be several ways to join the ends, making the topology of the
knot ambiguous.
The same considerations come into play for knots in linear
polymers, and several loop closure schemes have been
developed.29 A method for locating knots in protein chains was
introduced by Taylor and led to the first detection (in protein
structures that were already known) of a deeply embedded
trefoil and the more complex figure-eight knot.30 Taylor’s
method was to keep the ends of the chain fixed while notionally broke at a strain energy more than 20% lower than the
tightening the structure until it either collapses into a straight
line (in the case of the unknot) or leaves a compact, well
defined knot away from the ends. The tightening works by
iteratively changing the position of each point on the chain to
lie more in line with its immediate neighbours either side, unknotted chain. The weakening of the chain is associated with
subject to segments not passing through each other. However, the high curvature in the knotted region. Stiff actin filaments
this approach does not always lead to sufficient simplification are dramatically weakened by the presence of a knot, as shown
because the tightening moves can become jammed. in the experiments of Arai et al.33 These authors tied knots in
An alternative method is to join the chain ends to arbitrary actin by attaching myosin-coated polystyrene bends to the ends
points that lie far outside the chain and to connect these of the filament and manipulating the beads using optical
extended end-points directly.29 The topology of the closed loop tweezers (see Figure 14). While a straight actin filament is able
can depend on the choice of external points, and statistical to withstand a tensile force of around 600 pN, the knotted
information can be gathered on how frequently a given knot is filaments broke (near the knotted region) at less than 1 pN.33
produced for random points. If no single result dominates this The stiffness of a chain also affects the thermodynamics of knot
sampling, one may draw the valid conclusion that the topology formation. A stiff molecule must pay a high enthalpic penalty
is ambiguous. The disadvantage of such stochastic closure to be bent into a knot. In contrast, for a very floppy chain, the
schemes is that repeated sampling of the knot type can be thermodynamic cost of a knot is largely entropic because the
computationally costly. segments cannot pass through each other and the fixed topology
An attractive method called the minimally interfering closure therefore places a restriction on the conformations that the
selects the path for connecting the chain ends depending on the molecule can access by thermal fluctuations. In fact, recent
structure itself.31 This scheme starts by constructing the simulations show that flexible chains can, on average, be less
structure’s convex hull, which is the smallest polyhedron that bent in the knotted region than their unknotted counterparts.34
contains the structure and has no concave region on its surface. The net result is that there is a non-zero optimal stiffness at
If the ends of the chain lie closer to each other than to the which the free energy of knot formation is minimised. This
surface of the convex hull then the chain is closed by joining optimum seems to be independent of the knot type, but is
the ends directly with a straight line. If, instead, the ends lie shifted to lower stiffness if the chain is confined, since
closer to the hull, then they are extended through the closest confinement reduces conformational entropy even in the
points on the hull and joined externally. The minimally absence of a knot.34
interfering closure usually returns the same result as the An important case of knot confinement concerns the DNA
dominant knot found by stochastic closure, providing a reliable contained in the protein shell of a virus. The virus must expel
classification relatively efficiently. genetic material in order to replicate, and the ejection process is
The presence of a knot in a chain-like molecule can reduce its strongly affected by the presence of knots. Simulations using a
mechanical strength. Just as stretching a string with a knot simple bead-and-spring model of DNA suggest that a knot acts
results in rupture near the point where the string enters the as a ratchet during ejection through a pore.35 The knot blocks
tightened knot, stretching a knotted polymer first concentrates ejection when it is pulled against the opening and the chain
the strain energy in bonds at the entrance and exit of the knotted must reptate through the knot to make progress. However, if
region and then leads to bond-breaking. This effect in the knot diffuses along the chain into the interior of the capsid
molecules was first shown computationally32 in a knotted then unhindered ejection may resume. Hence, the process
alkane with Car-Parrinello molecular dynamics simulations, occurs in jerks with more complex knots resulting in slower
which use density functional theory to calculate the energy of ejection. More recent simulations show that the hindrance
the system and allow for bond breaking. The knotted alkane

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caused by knots in the ejection process is greatly reduced if the 1-D coordination polymers can also exhibit braided type
tendency of DNA strands to align at a slight angle is accounted structures39 which, conceptually, can be derived from breaking
for.36 These “cholesteric” interactions favour spooled, rather the three discrete rings of a Borromean structure and extending
than randomly tangled conformations of the DNA within the the resulting linear threads to give a triple stranded braid. 2-D
virus. However, if the axis of the spool is not aligned with the networks can form interpenetrated, polycatenated or Borromean
opening in the shell then ejection will be delayed until the spool type entanglements.37 By far the most common are the
has rotated into a compatible orientation. polycatenated honeycomb and sql type (Shubnikov tetragonal
plane) nets. An example of the honeycomb network is
4. Network Materials [Ag(tricyanomethanide)] which adopts one of four
topologically different modes of parallel interpenetration, Fig.
4.1 Coordination and supramolecular networks 15c. 3-D networks can adopt a wide variety of complex
So far we have considered knots in isolated molecules. However, structure types, the most common being the diamondoid net,
knotted motifs can also be repeated to create a periodic, Fig. 15d, as exemplified by the twofold-interpenetrated
interwoven network. The field of topologically non-trivial [Zn(CN)2]. The analysis of a network topology has been greatly
coordination polymer networks is vast and has been facilitated in recent years by the ToposPro software which
comprehensively reviewed.37 All interpenetrating network enables automated analysis and structure type assignment.40
structures can be regarded as infinite, ordered polycatenanes or Network solids have a fascinating range of properties. For
polyrotaxanes and can be classified by the dimensionality (0-D example, in some cases interpenetration can enhance the
to 3-D) of the chemically bonded components that are linked surface area of metal-organic frameworks resulting in enhanced
together, and by the number, n, of independent interpenetrating gas sorption properties. In a different area, in 1993, a novel
networks. Trivially, a 0-D network consists of individual compound consisting of two fully interlocked graphite-like
molecules ("zero-dimensional" components) such as networks was found to exhibit magnetic properties below 22.5
macrocycles, which can be linked together, rather like paper K.41 Designing such molecular magnets is a key challenge in
chains, into a chain (1-D), sheet (2-D) or three-dimensional materials science. The compound has the formula
array. Such molecular linkages are not technically (rad)2Mn2[Cu(opba)]3(DMSO)2·H2O, where rad refers to 2-(4-
interpenetrated networks because the components are discrete. N-methylpyridinium)-4,4,5,5-tetramethylimidazoline-1-oxyl-3-
A recent example is an infinite [n]catenane comprising a oxide and opba to orthophenylenebis(oxamato). The
copper(II) chloride metallomacrocycle with bridging interlinking produces Hopf link type relationships between the
phenanthrene-based imidazole ligands (Figure 15a).38 1-D two networks.
coordination polymer chains can give rise to 1-D or 2-D
4.2 Gels
interpenetrated networks, but a 3-D network based on 1-D
components is not currently known. A 1-D polyrotaxane based Knots are not a concept traditionally associated with gel
on a silver(I) bis(imidazole) is shown schematically in Fig. 15b. structures. This is primarily because of the nature of the gel
network, which comprises an extended sample-spanning mesh
of fibres, few of which ever undergo closure. This means that
knots, by their strict mathematical definition, are rarely formed.
However, the fibres are susceptible to tangling, which locks
them into a particular topological state just as a knot is defined
by the crossings of a single closed curve. The tangling has a
huge impact on the gel properties and behaviour, not least in
supporting the permanent network that give gels their solid-like
elastic properties despite comprising generally around 99%

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(50 % L: 50% D) formed a flatter, more “woven” structure (see

Figure 16). Gelator units based on a 1:2 ratio of L:D or D:L
formed a fibrous assembly similar to the enantiomeric form but
with significant changes when studied by small-angle X-ray
scattering. These latter gels were also found to have reduced
macroscopic stability compared to the pure enantiomeric
version. The study also showed that the gelation behaviour of
the L, D, D and D, L, L organogels was based on both the fibre
helicity as well as their degree of entanglement.
Another example of stereochemistry affecting gel properties
appears in work using D-glucosamine-based supramolecular
hydrogels as a biomaterial to promote wound healing.45 Two
different gels were produced, one with Nap-L-Phe-D-
glucosamine (Gel I) as the gelator and the other with Nap-D-
Phe-D-glucosamine (Gel II). Both form stable hydrogels easily
but the transmission electron micrograph (TEM) images of each
show very different structures (shown in Figure 17). While Gel
I is a tangled network of small, irregular ribbon bundles (Figure
17 E), Gel II comprises small, rigid ribbons with much more
uniform widths and a far more ordered network (Figure 17 F).
The various physicochemical properties of the two gels were
studied and key differences between the circular dichroism
(CD) spectra and rheological properties were noted.
While knots by the mathematical definition are not currently
considered in gel materials, entanglement around nodes and
amid the fibres is key and has many of the same mathematical
characteristics as knots and braids.

5 Knots in Liquid Crystals

The knots described so far have all taken the form of particles
that are joined to make a self-contained physical structure.
fluid. It has been shown that the mechanical properties of
However, it is also possible to create knots by disrupting the
small-molecule gel systems do not match those of a colloidal
otherwise regular order of a background medium. Such knots
gel.42 Instead a more appropriate model is that of a cellular
are not a structure made out of particles that have been joined
solid which is formed from interconnected load-bearing struts.
together but, rather, a defect that traces a path through a host
The crosslinking of such structures allows the formation of
material.
tangled or knotted networks.
To visualise the concept, consider a nematic liquid crystal,
Ultra-strong chemical fibres are sometimes produced by
consisting of rod-like molecules that are, on average, aligned in
orientational crystallisation from a gel medium.43 Such fibres
a particular direction so that there is long-ranged orientational
are grown with a fibrillar crystallite structure which gives them
order but only short-ranged, liquid-like correlation of the
their strength. This requires the use of a supercooled, tangled
gel network. Molecular tangles within this gel have been shown
to be vital for the crystallisation process in this type of
technique.
The degree of gel network entanglement has also been shown to
be a key component in determining the morphology and
macroscopic behaviour of two-component dendritic peptide
gels.44 Diaminododecane formed dumbell-like supramolecular
complexes by hydrogen bonding with dendritic peptides
incorporating three chiral centres, which gel toluene.
Interestingly, the micro structures of the gels depended upon
the relative configurations of the asymmetric carbon atoms in
the peptides. Enantiomeric (L, L, L or D, D, D) gelator units were
found to form highly fibrous structures whereas racemic gels

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imposed by the presence of the colloid, the defect cannot vanish

altogether.
A pair of colloidal spheres can lower their combined free
energy by allowing the regions of distorted liquid crystal that
surround them to overlap, thereby reducing the overall
distortion. Hence, the topological defects induce an effective
attraction between the colloids, and the symmetry of the
interaction is the same as that between two quadrupolar charge
distributions.
It is also possible for the Saturn rings to join into a single
disclination loop that entangles both particles, providing a
mechanism for building structures out of the colloids, held
together by the topological defect (Figure 19). In principle,
defects may also become intertwined with themselves and with
each other. A breakthrough in the practical realisation of this
possibility came with the deployment of chiral nematic liquid
crystals.46 In a simple nematic, the molecules are, on average,
oriented in a particular direction (called the director), which is
the same at all points in the sample. In a chiral nematic, the
direction of molecular alignment gradually rotates as one
travels along an axis perpendicular to the director. This rotation
results in twisted orientational order that has helical chirality,
like a corkscrew. Disclination defects in a twisted nematic tend
to be longer than in a simple nematic because they no longer lie
in a plane, and this in turn encourages entanglement.
The topology of the defects can be manipulated using precisely
focused laser tweezers to break them and reconnect them in a
different way. This can be done at a “tangle”, which is the
region where two segments of the defect approach closely. The
two segments each have two ends, which form the four vertices
of a tetrahedron. There are three ways of directly connecting
pairs of vertices in a tetrahedron, and these arrangements can be
interconverted by cutting the segments with the laser and
molecules’ positions. The average direction of the molecular
reorienting the liquid crystal director field to re-knot the tangle.
axes can be represented by parallel lines running through the
In a trefoil knot, for example, any of the three crossings can be
material. Introducing a spherical colloidal particle disrupts this
rewired to reverse the orientation of the crossing (resulting in
alignment because the parallel lines cannot penetrate the
an unknot) or to bypass the crossing altogether (resulting in a
colloid. Furthermore, the colloid may have a preference for the
Hopf link). By a sequence of such operations, any desired
molecules of the liquid crystal to be anchored to its surface in a
topology can be made to order.46
particular orientation. Figure 18a shows a cross-section
The richness of defect structures is further enhanced if the
through a spherical colloid with homeotropic (perpendicular)
spherical colloid is replaced by a particle that itself has more
anchoring of the liquid crystal molecules. This local
complex topology. For example, a colloid in the shape of a
arrangement is incompatible with the background nematic
Möbius strip (with homeotropic anchoring on its broad surface)
ordering. One solution to the mismatch is the formation of a
introduces new constraints due to the non-orientability of the
disclination defect,46 consisting of a line at which three
surface (Figure 6). The disclination defect induced by such a
molecular alignments collide, as shown in Figure 18b. This
particle must thread the hole in the Möbius strip and cannot be
defect encircles the colloid, forming a closed loop like a Saturn
shrunk to a point.47 However, the form that a defect adopts is
ring. The defects are readily visible under optical microscopy
also influenced by its free energy in the given liquid crystal
due to the strong scattering of the polarised light by the
medium. In the case of a multiply-twisted Möbius colloid,
disordered regions of the liquid crystal, which makes them
multiple small disclination loops that thread the strip may be
appear dark between crossed polarisers.
more stable than a single defect that twists round the strip to
Being a defect in the nematic order, the disclination loop comes
make a torus knot47 (see Section 6 and Figure 23).
with a free energy cost. There is, therefore, a thermodynamic
In a recent development, micrometre-scale particles that
driving force to shorten the loop, and this causes the Saturn ring
themselves are knot-shaped have been used to induce defects in
to act like a stretched elastic band that seeks to reduce its length
nematic fields48 (Figure 20). The particles are polymeric tubes,
by contracting. Nevertheless, because of the fixed topology
created by a photopolymerisation process using spatially

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This driving force often involves a delicate interplay between

enthalpic and entropic contributions, which must originate from
the physical properties of the individual components, the
interactions between them and the influence of the surrounding
medium. Through the process of evolution, nature has
developed many remarkable examples of complex self-
assembly, including self-assembling knots in the form of
proteins with a knotted native conformation.7
A protein is a linear polymer of amino acids, of which there are
20 types. The sequence of amino acids ultimately determines
the compact folded structure that a protein adopts in its native
state, and this structure is essential for the protein to perform its
biological function. The most common protein knot is the
trefoil, but knots as complex as 61 exist. There are also many
examples of slipknots, i.e., a threaded loop that is untied (rather
than tightened) if the ends of the chain are pulled, like the bow
of a shoelace. The existence of knotted proteins raises a
number of interesting but difficult questions concerning the
effect of the knot on physical properties of the protein, the
evolutionary advantage of the knot, and the mechanism by
which the knot is tied. The current state of knowledge in all
these areas is included in the comprehensive recent review by
Lim and Jackson.7 Here we focus on ways that
macromolecular systems can be designed to self-assemble into
patterned femtosecond laser pulses. The surface chemistry of knots, and how some of the principles of self-assembly can be
these particles can be changed to switch the anchoring transferred to colloidal systems.
orientation of the liquid crystal between homeotropic Proteins are distinguished from homopolymers, and from each
(perpendicular) and tangential, which in turn changes the other, by having a particular sequence of amino acid
topology of the resulting defects. As for spherical colloids, the monomers. The addition of sequence information to the chain
defects induce tuneable interactions between the knot-shaped immediately provides a large degree of control over the
microparticles. Martinez et al. 48 point out that these systems of structures that the chain adopts. Even a simple “HP” lattice
interacting knots come rather close to a microscale realisation model of proteins allows the likelihood of a nontrivial knot to
of Kelvin’s vision of atoms as knots in the aether.2 be tuned from 10% to nearly 90% (see Figure 21).49 In this
In this introduction to knots in liquid crystals, we have model, each amino acid is represented by a single bead of just
concentrated on the topology of the defect lines. However, the
molecular alignment in the regions between the defects is also
important. In knot theory, the space around a knot (i.e.,
everything apart from the knot) is called the knot complement.
In liquid crystals, the topology of a knotted defect does not
completely determine the field of molecular alignment in the
complement, so it is possible for two knotted defects with the
same topology to have topologically different liquid crystalline
“textures” around them.

6 Self-assembly of macromolecular and colloidal

knots
We have now seen that molecular knots may arise by chance in
long polymers, be built by careful synthetic procedures, or can
even be tied mechanically using optical tweezers. However,
some of the synthetic knots described in Section 2 can be
regarded as forming by self-assembly in the sense that the
components organise into a specific topology and structure
without detailed external intervention. For any structure to self-
assemble spontaneously, there must be a free energetic driving
force from the unassembled components to the target structure.

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two possible types (hydrophobic or polar) and is confined to a bead alphabet that mimics the natural amino acids.51
cubic grid of points. For real proteins with the variety offered The alphabet of interactions in DNA molecules is much smaller
by all 20 amino acids, there is evidence that a protein is less than that in proteins, with just four different nucleotides.
likely to be knotted than a homopolymer of the same length and However, the highly specific interactions in adenine-thymine
flexibility, suggesting that evolution may have selected against and cytosine-guanine pairs make DNA a versatile
knots in native states.50 macromolecule for self-assembly. Kočar et al. have used a
Coluzza et al. have laid the groundwork for designing chains of combination of simulation and experiment to devise design
colloidal particles that fold into a well-defined knotted rules for the efficient self-assembly of highly knotted DNA
conformation by mimicking certain features of proteins.51 structures.52 These authors took as their target a hollow square-
Their proposed “colloidal polymer” exploits existing based pyramid, in which each edge is a hybridised double helix
technology for synthesising colloidal hard spheres with patchy of DNA. For a single DNA molecule to form such a structure,
interactions. Each patchy sphere has a single interaction site the strand must pass between the pyramid’s five vertices on a
(the patch) that can form a reversible bond with that on another route that travels along each edge exactly twice and in opposite
particle, imitating the interactions between amino acids due, for directions (see Figure 22a). Although this underlying path is
example, to hydrophobicity or to hydrogen-bonding. Chains of not knotted in itself, the final double helical structure requires
such particles can be designed to self-assemble into a selected the DNA strand to twist around itself. Once the initial edges of
knot in three stages. First, the structure and sequence of the the pyramid have formed, this twisting can only be achieved by
chain must be explored together in a Monte Carlo simulation to the termini of the strand repeatedly threading through the loops
identify plausible knotted structures. For a structure to be in the partially completed structure, generating a knot of around
“designable” it must have a low energy for a large number of 30 crossings. For efficient folding, a “free-end rule” must be
bead sequences. In the second stage, the chosen structure is observed, where a free end is a segment of the DNA strand with
held fixed while the sequence is optimised (by random no hybridised bases so far lying between it and the terminus.
mutations) to lower the energy. Finally, the designed sequence The rule states that each step in the self-assembly must involve
is tested by allowing the chain to fold in an unbiased at least one free end, which enables the strand to thread the
simulation. Reinforcing the conclusion of Wüst, Reith and existing loops relatively easily. Assembly is also assisted if
Virnau,49 knotted ground states can be designed to fold hybridisation starts in the middle of the strand, and the long
reproducibly using only two types of bead (neutral and chains to either side of this section are both threaded early in
attractive), but the design protocol also works well with a 20- the assembly process.
An alternative to building complex DNA knots that avoids
repeatedly threading a chain end through a loop deploys
enzymes to alter the topology of the assembly near crossings.
Type I topoisomerases are naturally occurring enzymes that
alter the linking number of double-stranded DNA complexes by
cleaving one of the strands and reconnecting it on the opposite
side of the other strand. In recent work, Seeman and coworkers
have used Topoisomerase I from Escherichia coli to create
knots and links from DNA.53 Two complementary sections of
single-stranded DNA are first brought together by “paranemic”
cohesion i.e. where the strands lie side-by-side without being
linked in a double helix. This partially frustrated arrangement
allows some, but not all of the complementary pairs to cohere.
The topoisomerase then acts at the kissing loops to cut one of
the strands and reseal it so that they are linked, as shown in
Figure 22b. The resulting double-helical structure enables full
base-pairing between the complementary strands.
Both protein and DNA knots start with a chain that is
eventually woven into a non-trivial topology. However,
computational studies suggest that knots can also be self-
assembled from fragments, with the underlying chain emerging
at the same time as the knot. For example, knots have
unexpectedly been found to be the ground state structures of
certain Stockmayer clusters.54 Stockmayer particles interact via
a Lennard-Jones potential plus a point dipole and are an
archetypal model for dipolar molecules or colloids that also
have van der Waals attraction. The Lennard-Jones (van der
Waals) interactions favour highly coordinated, compact

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assembled knots found in both the Stockmayer clusters54 and

the clusters of helical fragments.55

7 Conclusions
The demise of Kelvin’s vortex theory of atoms in the late
nineteenth century shifted interest in knots away from science
and into the realm of pure mathematics. Over a century later,
the resulting abstract theory of knots is being reunited with the
physical sciences in ways that Kelvin could never have
foreseen. The mathematical framework has been essential not
only for identifying and classifying knots but, even more
importantly, for helping to understand the physical
consequences of knotted topology.
This tutorial review has touched on knot-related phenomena in
the chemical sciences that are at different stages of their
development. While basic catenane structures have been
known for more than fifty years, it is only in the present decade
that knotted defects in chiral nematics have been used to bind
colloidal particles. Computer simulation is playing an
increasingly important role in many of the fields, allowing the
properties of knots to be tested systematically and predicted at a
level of detail that is sometimes, but not always, experimentally
accessible.
The present knot-related challenges in the chemical sciences are
diverse but largely centre on understanding and then controlling
the processes by which knots form. Such problems are often
dynamic in character, so that knots are no longer the static,
idealised drawings found in tables of knot topologies. We fully
expect that knot theory and the chemical sciences will continue
to intertwine in fascinating and sometimes unpredictable ways.
structures for small clusters, usually with icosahedral
symmetry. In contrast, dipolar interactions favour a head-to-tail
chain-like arrangement. The optimal compromise between
these competing effects is sometimes a knot. The knot’s Acknowledgements
underlying chain allows the dipole-dipole interactions to be This work was supported by a Leverhulme Trust Research
largely satisfied, but twisting the chain and threading it through Programme Grant RP2013-K-009, Scientific Properties of
itself gives each particle more contacts than just its two Complex Knots (SPOCK).
neighbours in the chain, thereby satisfying the van der Waals
attraction. Figure 23b shows an 819 cluster that is the Notes and references
a
energetically optimal state of a 38-particle Stockmayer cluster. SPOCK Group, Department of Mathematical Sciences, Durham
The 819 knot was also found in self-assembly simulations of University, South Road, Durham, DH1 3LE, UK. Email:
colloidal helical fragments that only interact through attractive kate.horner2@durham.ac.uk; Tel: +44 (0)191 334 1510
b
sites at their tips, shown in Figure 23c.55 The structures formed Department of Chemistry, Durham University, South Road, Durham,
by such fragments can be tuned by adjusting their arc length DH1 3LE, UK.
and pitch. There is a remarkable coincidence between the knots
found in this system and those that self-assembled from the 1 A. T. Vandermonde, Memoires de l’Academie Royale des Sciences
Stockmayer particles.54 The knots that emerge from the vast (Paris), 1771.
range of possible topologies are almost all torus knots. A torus 2 W. Thomson, Philos. Mag. Ser. 4, 1867, 34, 15–24.
knot is one that can be drawn by spiralling round a traditional 3 C. C. Adams, The Knot Book, American Mathematical Society,
holed doughnut two or more times before returning to the 2004.
starting point, as illustrated in Figure 23a, the simplest, non- 4 G. R. Schaller and R. Herges, Chem. Commun., 2013, 49, 1254–
trivial torus knot being the trefoil. As the figure shows, the fact 1260.
that torus knots can be drawn in such a regular way makes them 5 O. Lukin and F. Vögtle, Angew. Chem. Int. Ed., 2005, 44, 1456–
rather special. Regularity can lead to efficient packing and 1477.
interactions, in turn giving rise to the highly symmetrical self- 6 J.-F. Ayme, J. E. Beves, C. J. Campbell and D. A. Leigh, Chem.

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