Hostname: page-component-7b7567c8c-7pwfn Total loading time: 0 Render date: 2026-11-14T10:13:14.093Z Has data issue: false hasContentIssue false
  • English
  • Français

A problem of Ulam about magnetic fields generated by knotted wires

Published online by Cambridge University Press:  27 November 2017

ALBERTO ENCISO and
DANIEL PERALTA-SALAS
ALBERTO ENCISO
Affiliation:
Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas, 28049 Madrid, Spain email aenciso@icmat.es, dperalta@icmat.es
DANIEL PERALTA-SALAS
Affiliation:
Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas, 28049 Madrid, Spain email aenciso@icmat.es, dperalta@icmat.es
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

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.

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 8 , August 2019, pp. 2048 - 2070
Copyright
© Cambridge University Press, 2017 

References

Arnold, V. I.. Sur la topologie des écoulements stationnaires des fluides parfaits. C. R. Acad. Sci. Paris 261 (1965), 17–20.Google Scholar
Bruce, J. W. and Giblin, P. J.. Curves and Singularities. Cambridge University Press, Cambridge, 1984.Google Scholar
Dennis, M. R., King, R. P., Jack, B., O’Holleran, K. and Padgett, M. J.. Isolated optical vortex knots. Nature Phys. 6 (2010), 118–121.Google Scholar
Enciso, A. and Peralta-Salas, D.. Knots and links in steady solutions of the Euler equation. Ann. of Math. 175 (2012), 345–367.Google Scholar
Enciso, A. and Peralta-Salas, D.. Existence of knotted vortex tubes in steady Euler flows. Acta Math. 214 (2015), 61–134.Google Scholar
Godbillon, C.. Dynamical Systems on Surfaces. Springer, Berlin, 1983.Google Scholar
Hirsch, M. W., Pugh, C. C. and Shub, M.. Invariant Manifolds. Springer, New York, 1977.Google Scholar
Hosoda, M., Miyaguchi, T., Imagawa, K. and Nakamura, K.. Ubiquity of chaotic magnetic-field lines generated by three-dimensionally crossed wires in modern electric circuits. Phys. Rev. E 80 (2009), 067202.Google Scholar
Hudson, S. R., Startsev, E. and Feibush, E.. A new class of magnetic confinement device in the shape of a knot. Phys. Plasmas 21 (2014), 010705.Google Scholar
Irvine, W. T. M. and Bouwmeester, D.. Linked and knotted beams of light. Nature Phys. 4 (2008), 716–720.Google Scholar
Mauldin, R. D. (Ed). The Scottish Book. Birkhäuser, Boston, 1981.Google Scholar
Pohl, W. F.. The self-linking number of a closed space curve. J. Math. Mech. 17 (1967/1968), 975–985.Google Scholar
Thomson (Lord Kelvin), W.. On vortex atoms. Proc. Roy. Soc. Edinburgh Sec. A 6 (1867), 94–105 (reprinted in: Mathematical and Physical Papers IV, Cambridge University Press, Cambridge, 2011).Google Scholar
Tkalec, U., Ravnik, M., Copar, S., Zumer, S. and Musevic, I.. Reconfigurable knots and links in chiral nematic colloids. Science 333 (2011), 62–65.Google Scholar
Ulam, S. M.. Problems in Modern Mathematics. Wiley, New York, 1964.Google Scholar