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Knot theory is a branch of topology in pure mathematics, however, it has been increasingly used in different sciences such as chemistry. Mathematically, a knot is a subset of three-dimensional space which is homeomorphic to a circle and it is only defined in a closed loop. In chemistry, knots have been applied to synthetic molecular design. Mathematics and chemistry together can work to determine, characterize and create knots which help to understand different molecular designs and then forecast their physical features. In this study, we provide an introduction to the knot theory and its topological concepts, and then we extend it to the context of chemistry. We present parametric representations for several synthetic knots. The main goal of this paper is to develop a geometric and topological intuition for molecular knots using parametric equations. Since parameterizations are non-unique; there is more than one set of parametric equations to specify the same molecular knots. This parametric representation can be used easily to express geometrically molecular knots and would be helpful to find out more complicated molecular models.

The study of knots started from the 1860s with William Thompson (Lord Kelvin) and his vortex model of the atom. According to Kelvin model, atoms and different chemical elements were formed by different knots. During the past two decades, different areas in chemistry have been affected by knot theory [

{ x = sin ( t ) + 2 sin ( 2 t ) ; y = cos ( t ) − 2 cos ( 2 t ) ; z = − sin ( 3 t ) ; (1.1)

and the parametric equations for the trefoil-negative knot has the form:

{ x = cos ( t ) − 2 cos ( 2 t ) ; y = sin ( t ) + 2 sin ( 2 t ) ; z = − sin ( 3 t ) ; (1.2)

The trefoil knot has the property of being a Möbius strip (see

{ x = [ r + μ cos ( t 1 2 ) ] cos ( t ) ; y = [ r + μ cos ( t 1 2 ) ] sin ( t ) ; z = μ sin ( t 1 2 ) ; (1.3)

for μ in [ − w , w ] and t in [ 0,2 π ) . In this parametrization, the Möbius strip is a cubic surface with equation

− r 2 y + x 2 y + y 3 − 2 r x z − 2 x 2 z − 2 y 2 z + y z 2 = 0 (1.4)

Chemists make tiny molecular knots in their labs using directed self-assembly techniques. They synthesize these knots and join the fragments rather than tied in a single continuous biomolecular string. Trying to build more complicated knots will assist researchers to better understanding of which materials are better to use for a specific purpose based on strength, flexibility and other features. It can be possible to develop woven knots and forming functional materials with catalyzing properties which are heat-resistant [

In this paper, we study a new way to model synthetic molecular knots. Parametric equations are convenient and helpful to explore more complicated structure of molecular knots. We will provide some mathematical and topological representations for some of the molecular knots. Simplicity of this type of modeling will help biochemists and biophysicists to discover even complicated synthetic molecular knots.

Although it has been proved that Kelvin’s theory to be of little physical use, it pushed mathematicians to begin studying of knots in detail. Since Kelvin’s theory, mathematicians have developed increasingly different methods for making sense of knots complexity. Mathematicians are interested to determine whether two knots in

{ x = ( 2 + cos ( 2 t ) ) cos ( 3 t ) ; y = ( 2 + cos ( 2 t ) ) sin ( 3 t ) ; z = 2 sin ( 4 t ) ; (2.1)

Knot theory is a branch of a larger field of pure mathematics called topology which studies knots and links. Topology studies the properties of geometric figures which are unchanged by elastic deformations such as stretching or twisting. To warm up, we present some mathematical definitions which help to understand the topology of the knots. In topology, a surface is connected if every two distinct points on the surface are connected by a path on the surface. A surface is said to be closed if there is no boundary. The sphere and torus are closed and connected. A pair of linked tori is an example of unconnected surface [_{1} in

circular direction. If we identify the opposite edges of a square with the same direction, we get a torus.

If we show the radius from the center of the hole to the center of the torus tube be ρ 1 , and the radius of the tube be ρ 2 . Then the equation in Cartesian coordinates for a torus symmetric about the z-axis is

( ρ 1 − x 2 + y 2 ) 2 + z 2 = ρ 2 2 (2.2)

and the parametric equations are

{ x = ( ρ 1 + ρ 2 cos ( ϕ ) ) cos ( θ ) ; y = ( ρ 1 + ρ 2 cos ( ϕ ) ) sin ( θ ) ; z = ρ 2 sin ( ϕ ) ; (2.3)

where, ϕ , θ ∈ [ 0,2 π ) .

A knot is a closed curve in space without self intersections means that a knot is a simple closed curve [_{1} or the trivial knot (_{1} with three crossings (

In addition to the Alexander polynomial, there are two more advanced algorithms, the Jones and HOMFLY polynomials, which can be used to distinguish between different complex knots [

The simplest chiral knot which we have already discussed about it is the trefoil knot (_{1} in

There is another family of knots called torus knots which can be drawn as closed curves on the surface of a torus and have been demonstrated in

One of the disadvantages of using polynomials to analyze the knots is that they are working well for simpler knots without many crossings, however, when we have complicated knots with many crossings, they cannot be computationally effective and cannot recognize complicated knots as simpler ones. To solve this problem, researchers have been developed an alternative smoothing algorithm, sometimes referred to as the KMT reduction, was developed such that complex knotted structures are simplified by omitting regions of the chain unnecessary for maintaining the knot [

One of the interesting topic for topologist about knot theory is how we can produce an unknot like

Mathematically, there are three common methods to study the knots. Algebraic methods are a part of the theory of the fundamental group, algebraic topology, and so on. Geometric methods are from arguments that are essentially rigorous visual proofs. Combinatorial proofs are mostly very hard to describe in topological point of view [

A table of the simplest knots and links has been demonstrated in

According to this table, the knots are given names like 5_{1}; 5_{2} which imply to the first 5-crossing knot and the second one. The table doesn’t list knots which are connect-sums of simpler ones and just prime knots are included. The table also does not list mirror images. Most knots are distinct from their mirror images. There are examples of amphichiral, which are equivalent to its mirror. In the table, there is asterisk or symbol which is placed next to the diagrams of the amphichiral knots to demonstrate their unusual property. We usually consider the crossing number as a measure of the complexity of a knot or link, simply implies that it is drawable using a small number of crossings.

The beginning of building synthetic molecular was in the late 1980s when synthetic molecular trefoil was made. In the last few years, many complex synthetic molecule knots have been build. By imposing steric restrictions on molecular strands result in knotting impart significant physical and chemical properties, including chirality, strong and selective ion binding, and catalytic activity [

Beside synthetic molecular, we have another well-known application of knot theory in biology, DNA, RNA and proteins. DNA, RNA and proteins are three major classes of biopolymers and play an important role in the structural, dynamical properties of the biological systems [

The DNA follows a complex structure and it has indispensable topology. In all organisms, each contains a family of naturally occurring enzymes that change cellular DNA in order to interfere the replication, transcription and recombination process of cellular life. In order to investigate enzyme binding and mechanism, molecular biologists have brought a topological approach to enzymology, which is a protocol obtained experimentally in which we do a reaction between small artificial circular DNA substrate molecules and purified enzyme in vitro [

_{19} which has been making through chemistry. The 8_{19} knot in

{ x = sin ( t ) + 2 sin ( 3 t ) ; y = cos ( t ) − 2 cos ( 3 t ) ; z = sin ( 2 t ) ; (3.1)

The knot 8_{19} has been made with polymer strands of carbon, hydrogen, nitrogen and oxygen. Afterward, these polymer strands would be composed with iron and chlorine ions in a solvent liquid which is used to foster chemical interactions. Then, this compound would be heated to 266 degrees Fahrenheit for 24 hours, which would make the iron ions to bind to the polymer strands in particular locations. Particularly, four iron ions which each one has been bound to three polymer strands near their intended cross points, hold the strands together so that the knot could form.

It is always possible to change the reaction conditions by adding in a new catalyst, modulating the temperature, or changing the solvent as needed. Applying each of these changes make the polymer strands to braid around the metal ions and become more possible forming the 8_{19} knot. These steps are sometimes longer than a day and finally synthesizing of a 8_{19} knot would be completed. 8_{19} knot has one chlorine ion in the center, and four iron ions at the polymer cross points. After 30-minute chemical reaction these metal ions would be removed and the result is a pure 819 knot consisting of only the three polymer strands. The final knot has only 192 atoms long and it is a hundred times smaller than a mammalian cell. So far, 8_{19} knot is the most complicated molecular knot which has been synthesized in the laboratory and researchers are looking for the techniques involved in its building to incorporate other molecular knots with three or more polymer strands.

The 5_{1} knot with 5 minimal crossings, in

{ x = cos ( 2 t ) − 2 cos ( 3 t ) ; y = sin ( 2 t ) + 2 sin ( 3 t ) ; z = sin ( 2 t ) ; (3.2)

Other molecular knot, 7_{1} with 7 minimal crossings, cyclic symmetric realisations of knots requires more templates than prime knots with 8 crossing such as 8_{19} [_{1} knot in

{ x = cos ( 3 t ) − 2 cos ( 4 t ) ; y = sin ( 3 t ) + 2 sin ( 4 t ) ; z = sin ( 2 t ) ; (3.3)

The 9_{1} knot with 9 minimal crossings in

{ x = cos ( 4 t ) − 2 cos ( 5 t ) ; y = sin ( 4 t ) + 2 sin ( 5 t ) ; z = sin ( 2 t ) ; (3.4)

The T ( 4,3 ) knot in

{ x = cos ( t ) − 3 cos ( 3 t ) ; y = sin ( t ) + 3 sin ( 3 t ) ; z = sin ( 2 t ) ; (3.5)

The T ( 5 , 4 ) knot in

{ x = cos ( t ) − 3 cos ( 4 t ) ; y = sin ( t ) + 3 sin ( 4 t ) ; z = sin ( 2 t ) ; (3.6)

During the past 3 decades, many progresses have been made by chemist for making molecular trefoil knots, using different synthetic processes. However, there are many synthetic molecular knots that remain as a challenge in chemistry. Increasing the number of molecular knot topologies helps chemists to discover the

properties of synthetic molecular knots. The rigorous theories of knot theory in mathematics can help to discover more about synthetic molecular knots. By combining different array of knots in the lab, it can be possible to explore some of the properties such as self-assembling and other synthesized strands properties. Then, biophysicists can understand about knots actions in DNA, proteins or other naturally made molecules. In this paper, we provided a mathematical and topological view of this complicated world. Using parametric modeling, we can study rigorously different synthetic molecular knots. We have represented different knots using parametric equations which help to visualize a variety of synthetic molecular knots. Mathematically, this parametric representation or parameterization can be used to express a geometric object such as a curve or surface. Since parameterizations are non-unique; there is more than one set of parametric equations to specify the same molecular knots. Parametric equations are convenient for describing different synthetic molecular knots and can be considered as manifolds and algebraic varieties of higher dimension. It is always possible to convert a set of parametric equations to a single implicit equation through implicitization which can be done easily by eliminating the variable t from the equations. This study brings more mathematical intuitions in studying the synthetic molecular knots and will help chemists and biophysicists to discover more complicated properties of them.

The authors declare no conflicts of interest regarding the publication of this paper.

Azizi, T. and Pichelmeyer, J. (2020) Using Parametric Mathematical Modeling to Develop a Geometric and Topological Intuition for Molecular Knots. Applied Mathematics, 11, 460-472. https://doi.org/10.4236/am.2020.116033