# Quasipositive links and electromagnetism

## TL;DR

This paper constructs complex algebraic curves to embed any link into a quasipositive link, enabling the creation of stable, knotted electromagnetic null lines and analyzing their evolution as contactomorphisms, linking topology with electromagnetism.

## Contribution

It provides a method to embed any link into a quasipositive link via complex algebraic curves and connects electromagnetic field evolution with contact topology.

## Key findings

- Any link can be embedded into a quasipositive link as a satellite of the Hopf link.
- Constructed complex curves can produce arbitrarily knotted, stable electromagnetic null lines.
- Electromagnetic field evolution can be described as a family of contactomorphisms with knotted field lines.

## Abstract

For every link $L$ we construct a complex algebraic plane curve that intersects $S^3$ transversally in a link $\tilde{L}$ that contains $L$ as a sublink. This construction proves that every link $L$ is the sublink of a quasipositive link that is a satellite of the Hopf link. The explicit construction of the complex plane curve can be used to give upper bounds on its degree and to create arbitrarily knotted null lines in electromagnetic fields, sometimes referred to as vortex knots. Furthermore, these null lines are topologically stable for all time. We also show that the time evolution of electromagnetic fields as given by Bateman's construction and a choice of time-dependent stereographic projection can be understood as a continuous family of contactomorphisms with knotted field lines of the electric and magnetic fields corresponding to Legendrian knots.