Topologically protected vortex knots and links

TL;DR

This paper introduces a new class of topologically protected vortex knots and links based on non-Abelian vortices, demonstrating their immunity to decay and classifying them using a novel invariant in physical systems like spin-2 Bose-Einstein condensates.

Contribution

It proposes the concept of topologically protected vortex structures from non-Abelian vortices and proves the existence and classification of $Q_8$-colored links in relevant physical systems.

Findings

Existence of topologically protected $Q_8$-colored links.

Development of the $Q$-invariant for classifying these links.

Application to physical systems like spin-2 Bose-Einstein condensates.

Abstract

We propose a class of tangled vortex structures, tied from non-Abelian topological vortices, which are immune against decaying through local reconnections and strand crossings that are allowed by the system. We refer to such structures as being topologically protected. We then turn our attention to topological vortices classified by the quaternion group $Q_8$ ($Q_8$-colored links), which are realizable in systems consisting either of the biaxial nematic or the cyclic phase of a spin-2 Bose--Einstein condensate, or of biaxial nematic liquid crystal, and prove the existence of topologically protected $Q_8$-colored links. Remarkably, the strongest invariant we construct, the $Q$-invariant of $Q_8$-colored links, can be used to classify $Q_8$-colored links up to allowed local surgeries on the vortex cores.