Knot Theory on Disentangled

Knots and links

Mon, 01 Jun 2026 16:04:48 +0200

The history of knot theory goes back to the nineteenth century, when Lord Kelvin proposed a description of atoms as knotted vortices in the ether. Although this physical theory was later falsified and ultimately abandoned, the mathematical framework it inspired evolved into a key component of low-dimensional topology. Knot-theoretic concepts can be found whenever linking, entanglement, and topological constraints affect physical behavior, including statistical mechanics, plasma physics, and molecular biology. The topology of magnetic field lines is crucial for understanding solar activity, and DNA knotting and supercoiling have emerged in research on biophysics. Indeed, according to research by De Witt Sumners, Nicholas Cozzarelli, and others, knot theory can be used to explain how enzymes alter DNA topology during transcription and replication [1] and [2].

Invariants

Tue, 02 Jun 2026 16:04:48 +0200

A central problem in knot theory is determining when two knots, or more generally two links, are equivalent. Since a single knot or link can be represented in many different ways, mathematicians have introduced the notion of invariants. These are numbers or polynomials associated to a knot or link that remain unchanged when the knot is deformed without being cut or passed through itself. In a sense, invariants function as fingerprints, allowing one to compare different representations of the same underlying topological object. If two links have different values of a given invariant, then they cannot possibly be equivalent.

A stochastic interpretation of knot invariants

Tue, 02 Jun 2026 16:04:48 +0200

What do knot invariants of the type introduced in invariant have to do with stochastic processes? As I shall discuss in this article, it turns out that a stochastic viewpoint of invariants is incredibly useful for understanding their properties, as well as deriving more general invariants.

Studying proteins and polymers using knot theory

Wed, 03 Jun 2026 16:04:48 +0200

Many kinds of data in science and engineering is expressed in terms of point sets; literally a table consisting of numbers. Yet, in the real-world, continuous structures make their appearance in many shapes and forms. In particular, the structure of proteins or polymers can be described in terms of curves in 3-dimensional space. Where one finds curves, it is to be expected that entanglement plays a role. Indeed, according to the second law of thermodynamics, we cannot possibly expect for polymers to be structured in an ordered way: the probability for that to happen is rather low. Instead long polymer chains thread through one another, and its entanglement determines its properties. Entanglement is the reason a melt of long polymers behaves like honey rather than water.