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)].

### MSC:

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

### Citations:

Zbl 1195.57056; Zbl 1169.52004
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### References:

 [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
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