https://www.sciencenews.org/article/molecules-knots-periodic-table
Remarkably, Knot theory is an unsolved mathematical problem. Knot theory arises naturally in topology, but it only applies to three dimensions. There's been breakthroughs, but the complexity just goes up. The 1980s found an abundance of algebraic invariants that helped classify many knots, but it didn't solve the problem. Since then, there's been a number of generalisations
Re-reading a little bit - the Jones polynomials allows the distinguishing of chiral, or mirror knots. But, Knot theorists found all kinds of knot properties long before, and the Jones polynomials(comes from a matrix representation of group theory of braids - kind of half knots).
A mathematiician, Vasil'ev came up with a knot theory based on singularities(and much homology and cohomology theory of these singulariites). Singularities are like intersections of planes or even curves. This is obvious thing to do intuitively, actually. But, Vasil'ev showed how to find many of these knot theory properties from invariants from his theory.
I should perhaps note that invariants are like algebraic expressions that remain the same after algebraic manipulation(in modern mathematics, that would be a group theory manipulation).
Mathematicians have derived the Jones polynomials from Vasil-ev's theory. And, Knot theory is still not solved. Today, there's been created a kind of virtual knots. I keep forgetting the formal name. Their kind like the imaginary number version of knots. It's a kind of subset theory of knots.
Well the above is a science and even technological application of knot theory.
