Orbits of asteroids, a braid, and the first link invariant
type: Item
instance of: scholarly article
title: Orbits of asteroids, a braid, and the first link invariant
zbMATH Open document ID: 0918.01014
DOI: 10.1007/BF03024400
author: Moritz Epple
published in: The Mathematical Intelligencer
publication date: 1999-08-04T00:00:00Z
review text: 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 The author's search in Gauss's mathematical notebooks in summer 1994 brought to light an unpublished page, which shows that Gauss was probably the first to think about braids and braid groups.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).
Mathematics Subject Classification ID: 01A55
Mathematics Subject Classification ID: 57-03
Mathematics Subject Classification ID: 20F36
zbMATH DE Number: 1233994
zbMATH Keywords: Gauss and topology
zbMATH Keywords: number of intertwinings
zbMATH Keywords: braid groups
zbMATH Keywords: linking double integral
zbMATH Keywords: braid
reviewed by: Ülo Lumiste
MaRDI profile type: Publication
cites work: Q3963208
cites work: Q4082024
cites work: Riemann, Betti and the birth of topology
full work available at URL: https://doi.org/10.1007/bf03024400
OpenAlex ID: W2324691963
description: scientific article; zbMATH DE number 1233994