Sunday 30 March 2025
A mathematician has cracked a long-standing problem in knot theory, finding a new way to calculate the properties of knots and links using a type of mathematical formula.
Knots are essentially loops that can be twisted and turned into all sorts of shapes. But despite their simplicity, they have some very complex properties. For example, two knots may look similar but actually be fundamentally different. Mathematicians have been trying to understand these differences for centuries, and one way they do this is by using polynomials.
A polynomial is a mathematical formula that can be used to calculate the properties of an object. In the case of knots, mathematicians use something called the interior polynomial to work out things like how many times a knot wraps around itself or what shape it forms when viewed from different angles.
The problem with the interior polynomial is that it’s very difficult to calculate for complex knots. Mathematicians have been trying to find ways to simplify the formula and make it easier to use, but so far they’ve had limited success.
That was until a mathematician came up with a new way of calculating the interior polynomial using something called non-expanding sets. Non-expanding sets are essentially collections of objects that don’t change size or shape when you add or remove certain elements. In this case, the mathematician used non-expanding sets to break down complex knots into smaller, more manageable pieces.
By doing this, the mathematician was able to simplify the interior polynomial and make it much easier to calculate. This has big implications for knot theory, as it could help mathematicians better understand the properties of different knots and how they relate to each other.
For example, the new formula could be used to work out the properties of complex knots that have never been studied before. It could also be used to study things like DNA molecules, which are essentially long chains of nucleotides that can twist and turn into all sorts of shapes.
The mathematician’s discovery is a major breakthrough in knot theory, and it has the potential to open up new areas of research. It’s a great example of how mathematics can be used to help us better understand the world around us, from the tiny molecules that make up our bodies to the vast universe beyond our planet.
Cite this article: “Unlocking Knot Theory: A New Approach to Calculating Knot Properties”, The Science Archive, 2025.
Knot Theory, Mathematics, Polynomials, Interior Polynomial, Non-Expanding Sets, Knot Properties, Dna Molecules, Complexity, Calculation, Breakthrough
Reference: Keiju Kato, “New recursion formula for the interior polynomial based on non-expanding sets” (2025).
