Abstract
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.
Highlights
The study of knots started from the 1860s with William Thompson (Lord Kelvin) and his vortex model of the atom
Since parameterizations are non-unique; there is more than one set of parametric equations to specify the same molecular knots
There are many synthetic molecular knots that remain as a challenge in chemistry
Summary
The study of knots started from the 1860s with William Thompson (Lord Kelvin) and his vortex model of the atom. Sauvage winner of Nobel Prize, conducted a research group which leads to make the first synthetic molecular knot. They created a knot called a trefoil, which consists of two polymer strands intertwined at three cross points. 2 display three twisted rings were joined at a single atom, and mathematically are known as the Trefoil Knot. 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. Parametric equations are convenient and helpful to explore more complicated structure of molecular knots. Simplicity of this type of modeling will help biochemists and biophysicists to discover even complicated synthetic molecular knots
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