Higher-dimensional linking integrals. (English) Zbl 1221.57038

A hypersurface \(M\) in Euclidean space \({\mathbb R}^n\) is said to be visible if every ray from the origin either misses \(M\) or meets it just once transversally. For example the unit \(n\)-sphere, \(S^n\), is visible. The authors derive an integral formula for the linking number, \(Lk(K,L)\) of two closed, oriented, null-homologous submanifolds \(K\) and \(L\) of a visible hypersurface \(M\) in \({\mathbb R}^n\) such that \(\dim(K) + \dim(L) = n-1\). When \(M\) is taken to be \(S^n\), this formula agrees with the formulas obtained independently and by other methods by D. DeTurck and H. Gluck [Mat. Contemp. 34, 233–249 (2008; Zbl 1195.57056)] and by G. Kuperberg [Geom. Funct. Anal. 18, No. 3, 870–892 (2008; Zbl 1169.52004)].


57R45 Singularities of differentiable mappings in differential topology
53C20 Global Riemannian geometry, including pinching
57M25 Knots and links in the \(3\)-sphere (MSC2010)
Full Text: DOI arXiv


[1] 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. · Zbl 1153.81348
[2] D. DeTurck and H. Gluck, Linking integrals in the \?-sphere, Mat. Contemp. 34 (2008), 239 – 249. · Zbl 1195.57056
[3] Moritz Epple, Orbits of asteroids, a braid, and the first link invariant, Math. Intelligencer 20 (1998), no. 1, 45 – 52. · Zbl 0918.01014
[4] 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.
[5] Greg Kuperberg, From the Mahler conjecture to Gauss linking integrals, Geom. Funct. Anal. 18 (2008), no. 3, 870 – 892. · Zbl 1169.52004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.