Knots and Symmetries in Mathematics

Courtesy: University of Tennessee Knoxville

Hey Folks,

We have an intriguing application of Machine Learning (ML) today: Knots and Symmetries in Mathematics! ML, as you're probably aware, works when we have a whole lot of data. In fact, the more data we have, the more likely we are to use ML or its more sophisticated brethren Deep Learning (DL). So without further ado, let's dive in!

What are knots and symmetries in mathematics?

A knot in mathematics is, as you would expect, inspired from real-life knots but isn't so exactly. Simply put a knot is a 2-D closed loop that exists in 3-D space. This concept could of course be extended to n-dimensional loops in m-dimensional space. Knots can be described mathematically in various ways. Sometimes two seemingly different descriptions are actually of the same knot. This can be determined by comparing properties of knots known as invariants. One of the questions surrounding invariants is: Are two invariants related? To answer this question requires combing through vast amounts of data and this is where AI can help.

A symmetry is a kind of invariance. When some transformation is applied on a mathematical object (such as polynomials or matrices) and the resulting object is unchanged from the original, the two objects are said to be symmetrical. Symmetries have been studied using graphs (networks of nodes) as well as with polynomials. Researchers have long suspected that polynomials can be calculated from graphs in order to find symmetries but figuring it out is often a hopeless task. Again AI steps in.

How does AI help?

ML (or DL) helps with recognising patterns in data. For example, in Computer Vision, ML can be used to classify images into categories. This classification, done by recognising features and then sorting them based on patterns, is probabilistic. So how does this help mathematicians?

Large and copious amounts of data is a good thing. Analysing that data is time-consuming and combing through it all can seem impossible. ML can be used to "narrow the field" for mathematicians so that they focus only on those patterns which show the most promise or have the highest probability. 

So in short, ML explores the vast amounts of data to pick out the most promising leads. The mathematicians then take those leads, turn them into conjectures and then either prove or disprove them. This approach has been shown to be fruitful when applied to knots and symmetries.

More details on this research advancement can be found in this article from Nature.