A problem of Ulam about magnetic fields generated by knotted wires

TL;DR

This paper explores the relationship between the topology of knotted wires and the magnetic lines they generate, demonstrating that wires of certain knot types can produce magnetic lines of the same or different knot types, addressing a question posed by Ulam.

Contribution

It establishes that generic knotted wires have magnetic lines of the same knot type and that wires can be constructed to produce magnetic lines of any specified knot type, linking topology and magnetic field behavior.

Findings

A generic knotted wire has a magnetic line of the same knot type.

Any pair of knots can be realized with a wire and magnetic line of the second knot type.

Addresses Ulam's 1935 question about magnetic fields and knot topology.

Abstract

In the context of magnetic fields generated by wires, we study the connection between the topology of the wire and the topology of the magnetic lines. We show that a generic knotted wire has a magnetic line of the same knot type, but that given any pair of knots there is a wire isotopic to the first knot having a magnetic line isotopic to the second. These questions can be traced back to Ulam in 1935.