Bordism invariants of colored links and topologically protected tricolorings

TL;DR

This paper introduces new bordism invariants for colored links, revealing topologically protected vortex knots and classifying their transformations under local surgeries, advancing understanding of topological stability in vortex structures.

Contribution

It constructs equivariant bordism invariants for colored links and applies them to identify topologically protected vortex knots and their transformation classes.

Findings

First examples of topologically protected vortex knots.

Classification of tricolored links under local surgeries.

Demonstration that links decay into simple loops or form trefoil knots.

Abstract

We construct invariants of colored links using equivariant bordism groups of Conner and Floyd. We employ this bordism invariant to find the first examples of topological vortex knots, the knot structure of which is protected from decaying via topologically allowed local surgeries, i.e., by reconnections and strand crossings permitted by the topology of the vortex-supporting medium. Moreover, we show that, up to the aforementioned local surgeries, each tricolored link either decays into unlinked simple loops, or can be transformed into either a left-handed or a right-handed tricolored trefoil knot.