A problem of Ulam about magnetic fields generated by knotted wires

Open access version at https://arxiv.org/pdf/1602.03142.pdfIn 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.The authors are supported by the ERC Starting Grants 633152 (A.E.) and 335079 (D.P.-S.). This work is supported in part by the ICMAT–Severo Ochoa grant SEV-2015-0554