Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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



Higher-dimensional linking integrals

Authors: Clayton Shonkwiler and David Shea Vela-Vick
Journal: Proc. Amer. Math. Soc. 139 (2011), 1511-1519
MSC (2010): Primary 57Q45; Secondary 57M25, 53C20
Published electronically: October 1, 2010
MathSciNet review: 2748445
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [DG08a] 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. MR 2392864 (2008m:53183)
  • [DG08b] -, Linking integrals in the $ n$-sphere, Mat. Contemp. 34 (2008), 233-249. MR 2588613
  • [Epp98] Moritz Epple, Orbits of asteroids, a braid, and the first link invariant, Math. Intelligencer 20 (1998), no. 1, 45-52. MR 1601835 (99c:01013)
  • [Gau33] 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.
  • [Kup08] Greg Kuperberg, From the Mahler conjecture to Gauss linking forms, Geom. Funct. Anal. 18 (2008), no. 3, 870-892. MR 2438998 (2009i:52005)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 57Q45, 57M25, 53C20

Retrieve articles in all journals with MSC (2010): 57Q45, 57M25, 53C20

Additional Information

Clayton Shonkwiler
Affiliation: Department of Mathematics, Haverford College, Haverford, Pennsylvania 19041

David Shea Vela-Vick
Affiliation: Department of Mathematics, Columbia University, New York, New York 10027

Keywords: Gauss linking integral, linking number
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
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society