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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Higher-dimensional linking integrals
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by Clayton Shonkwiler and David Shea Vela-Vick PDF
Proc. Amer. Math. Soc. 139 (2011), 1511-1519 Request permission

Abstract:

We derive an integral formula for the linking number of two submanifolds of the $n$-sphere $S^n$, of the product $S^n \times \mathbb {R}^m$, and of other manifolds which appear as “nice” hypersurfaces in Euclidean space. The formulas are geometrically meaningful in that they are invariant under the action of the special orthogonal group on the ambient space.
References
  • Dennis DeTurck and Herman Gluck, Electrodynamics and the Gauss linking integral on the 3-sphere and in hyperbolic 3-space, J. Math. Phys. 49 (2008), no. 2, 023504, 35. MR 2392864, DOI 10.1063/1.2827467
  • D. DeTurck and H. Gluck, Linking integrals in the $n$-sphere, Mat. Contemp. 34 (2008), 239–249. MR 2588613
  • Moritz Epple, Orbits of asteroids, a braid, and the first link invariant, Math. Intelligencer 20 (1998), no. 1, 45–52. MR 1601835, DOI 10.1007/BF03024400
  • Carl Friedrich Gauss, Integral formula for linking number, Zur mathematischen theorie der electrodynamische wirkungen (Collected Works, Vol. 5), Koniglichen Gesellschaft des Wissenschaften, Göttingen, 2nd ed., 1833, p. 605.
  • Greg Kuperberg, From the Mahler conjecture to Gauss linking integrals, Geom. Funct. Anal. 18 (2008), no. 3, 870–892. MR 2438998, DOI 10.1007/s00039-008-0669-4
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Additional Information
  • Clayton Shonkwiler
  • Affiliation: Department of Mathematics, Haverford College, Haverford, Pennsylvania 19041
  • MR Author ID: 887567
  • ORCID: 0000-0002-4811-8409
  • Email: cshonkwi@haverford.edu
  • David Shea Vela-Vick
  • Affiliation: Department of Mathematics, Columbia University, New York, New York 10027
  • Email: shea@math.columbia.edu
  • Received by editor(s): September 8, 2009
  • Received by editor(s) in revised form: April 29, 2010
  • Published electronically: October 1, 2010
  • Communicated by: Daniel Ruberman
  • © Copyright 2010 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 139 (2011), 1511-1519
  • MSC (2010): Primary 57Q45; Secondary 57M25, 53C20
  • DOI: https://doi.org/10.1090/S0002-9939-2010-10603-2
  • MathSciNet review: 2748445