# Knot spectrum of turbulence

## TL;DR

This paper investigates the topological complexity of vortex lines in quantum turbulence, using knot invariants to quantify the distribution and dynamics of vortex knots in superfluid helium.

## Contribution

It introduces a novel approach to quantify vortex topology in quantum turbulence through the knot spectrum and reveals scaling laws for knotting probability.

## Key findings

- Quantum vortex tangles contain large-degree knots that form, vanish, and reform.
- Knotting probability increases with vortex length and saturates at a characteristic length.
- The distribution of vortex topologies follows a specific scaling law.

## Abstract

Streamlines, vortex lines and magnetic flux tubes in turbulent fluids and plasmas display a great amount of coiling, twisting and linking, raising the question as to whether their topological complexity (continually created and destroyed by reconnections) can be quantified. In superfluid helium, the discrete (quantized) nature of vorticity can be exploited to associate to each vortex loop a knot invariant called the Alexander polynomial whose degree characterizes the topology of that vortex loop. By numerically simulating the dynamics of a tangle of quantum vortex lines, we find that this quantum turbulence always contains vortex knots of very large degree which keep forming, vanishing and reforming, creating a distribution of topologies which we quantify in terms of a knot spectrum and its scaling law. We also find results analogous to those in the wider literature, demonstrating that the knotting probability of the vortex tangle grows with the vortex length, as for macromolecules, and saturates above a characteristic length, as found for tumbled strings.