A Detailed Study: Using Parametric Mathematical Modeling to Develop a Geometric and Topological Intuition for Molecular Knots

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 nonunique;
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. 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.