# Topology of three-dimensional active nematic turbulence confined to droplets

## TL;DR

This study investigates the topology and dynamics of three-dimensional active nematic turbulence within spherical droplets, revealing complex defect structures, transitions, and the influence of surface alignment on defect behavior.

## Contribution

It provides the first detailed analysis of 3D active nematic defects under confinement, highlighting their topological structures and dynamic transitions through numerical modeling.

## Key findings

- Defects form closed loops and surface-to-surface segments due to confinement.
- Active turbulence involves defect creation, annihilation, and transformation events.
- Surface alignment influences defect coupling and dynamics.

## Abstract

Active nematics contain topological defects which under sufficient activity move, create and annihilate in a chaotic quasi-steady state, called active turbulence. However, understanding active defects under confinement is an open challenge, especially in three-dimensions. Here, we demonstrate the topology of three-dimensional active nematic turbulence under the spherical confinement, using numerical modelling. In such spherical droplets, we show the three-dimensional structure of the topological defects, which due to closed confinement emerge in the form of closed loops or surface-to-surface spanning line segments. In the turbulent regime, the defects are shown to be strongly spatially and time varying, with ongoing transformations between positive winding, negative winding and twisted profiles, and with defect loops of zero and non-zero topological charge. The timeline of the active turbulence is characterised by four types of bulk topology-linked events --- breakup, annihilation, coalescence and cross-over of the defects --- which we discuss could be used for the analysis of the active turbulence in different three-dimensional geometries. The turbulent regime is separated by a first order structural transition from a low activity regime of a steady-state vortex structure and an offset single point defect. We also demonstrate coupling of surface and bulk topological defect dynamics by changing from strong perpendicular to inplane surface alignment. More generally, this work is aimed to provide insight into three-dimensional active turbulence, distinctly from the perspective of the topology of the emergent three-dimensional topological defects.