IDEAS home Printed from https://ideas.repec.org/a/spr/pardea/v4y2023i2d10.1007_s42985-023-00225-3.html
   My bibliography  Save this article

Topological approaches to knotted electric charge distributions

Author

Listed:
  • Max Lipton

    (Cornell University)

Abstract

Consider a knot K in $$S^3$$ S 3 with uniformly distributed electric charge. Whilst solutions to the Laplace equation in terms of Dirichlet integrals are readily available, it is still of theoretical and physical interest to understand the qualitative behavior of the potential, particularly with respect to critical points and equipotential surfaces. In this paper, we demonstrate how techniques from geometric topology can yield novel insights from the perspective of electrostatics. Specifically, we show that when the knot is sufficiently close to a planar projection, we get a lower bound on the size of the critical set based on the projection’s crossings, improving a 2021 result of the author. We then classify the equipotential surfaces of a charged knot distribution by tracking how the topology of the knot complement restricts the Morse surgeries associated to the critical points of the potential.

Suggested Citation

  • Max Lipton, 2023. "Topological approaches to knotted electric charge distributions," Partial Differential Equations and Applications, Springer, vol. 4(2), pages 1-12, April.
    • Handle: RePEc:spr:pardea:v:4:y:2023:i:2:d:10.1007_s42985-023-00225-3
    DOI: 10.1007/s42985-023-00225-3
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s42985-023-00225-3
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s42985-023-00225-3?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:pardea:v:4:y:2023:i:2:d:10.1007_s42985-023-00225-3. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.