Orbits of asteroids, a braid, and the first link invariant
A discussion is given about a short passage from 1833 in one of C. F. Gauss's mathematical notebooks, containing an integral formula for the number of intertwinings of two closed or infinite curves [published later in Werke. Band V (Collected works. Vol. V) (1987), p. 605] among the fragments on electromagnetic induction. Already 1869 \textit{J. C. Maxwell} reported on it to the London Mathematical Society and worked out the idea in his Treatise on Electricity and Magnetism (1873), \S\S 409-422. But the author of this paper had found an earlier writing of Gauss from 1804, which allows to develop a second way of looking at this linking double integral. In this small treatise [see Werke, Vol. VI, pp. 106-118] Gauss determines the celestial region in which a given new asteroid, or planet, might possibly appear; Gauss calls it zodiacus. Interpreting this treatise in modern mathematical language the author comes to the integrand exterior 2-form. Based on a letter of Gauss to Olbers from 1825 [Werke, Vol. VIII, p. 398] a connection has been seen between this and the calculation of the total curvature of a surface in 3-space by means of the Gauss map. The author's search in Gauss's mathematical notebooks in summer 1994 brought to light an unpublished page, which shows that Gauss was probably the first to think about braids and braid groups. This page immediately precedes some pages with geodetical calculations, dated 1830. The author presents its reproduction and a detailed discussion, and, as a whole, makes clear why Gauss in the fragment of 1833 put very high value on the slowly emerging geometria situs (topology).
scientific article; zbMATH DE number 1233994