Knotted structures in high-energy Beltrami fields on the torus and the sphere

TL;DR

This paper demonstrates that for large odd integers, one can construct Beltrami fields on the sphere and torus with vortex structures matching any prescribed knotted and linked curves, up to a smooth deformation.

Contribution

It proves the existence of Beltrami fields with prescribed knotted vortex structures on the sphere and torus, extending the understanding of vortex topology in high-energy fields.

Findings

Existence of Beltrami fields with specified knotted vortex lines.

Construction of vortex structures matching arbitrary links and knots.

Applicability to both sphere and torus geometries.

Abstract

Let S be a finite union of (pairwise disjoint but possibly knotted and linked) closed curves and tubes in the round sphere S^3 or in the flat torus T^3. In the case of the torus, S is further assumed to be contained in a contractible subset of T^3. In this paper we show that for any sufficiently large odd integer \lambda there exists a Beltrami field on S^3 or T^3 satisfying curl u = \lambda u and with a collection of vortex lines and vortex tubes given by S, up to an ambient diffeomorphism.