I'~.lt--~_.,:,.-- Jeremy Gray I Orbits of Asteroids, a Braid, and the First Link Invariant Moritz Epple O n 22 January 1833, Carl Friedrich Gauss wrote a short passage in one of his mathematical notebooks which was to become widely known among mathematicians and physicists soon after it was fn'st published in 1867: Of the geometria situs, which Leibniz foresaw and into which only a pair of geometers (Euler and Vandermonde) were granted the privilege of taking a faint glance, we know and have, after a century and a half, little more than nothing. A centralproblem in the overlapping area of geometria situs and geometria magnitudinis will be to count the in- tertwinings [Umschlingungen] of two closed or infinite curves. Let the coordinates of an undeter- mined point of the first curve be x, y, z; of the second x', y ', z ;" and let HI(x, _ x)2 + @, _ y)2 + (z' - z)2] [(x' - x)(dydz' - dzdy') + (y' - y) dzdx' - dxdz') + (z' - z) (dxdy' - dydx')] = V; then this integral taken along both curves is = 4m~r, m being the number ofintertwinings. The value is reciprocal, i.e., it re- mains the same if the curves are in- terchanged. 1 The elusive science ofgeometria si- tus which Gauss was referring to was soon afterward given the modem name of Topologie--topology--by one of Gauss's students, Johann Benedikt Listing. 2 Geometria magnitudinis, on the other hand, denoted the kind of an- alytical geometry which the 18th cen- tury had elaborated so impressively, based on the 17th-century ideas of Descartes, Newton, and others. The beautiful formula Gauss wrote down connected the geometry of magnitude with that of position: A linking number, dependent only on the relative positions of two curves in the topological sense, was calculated by an integral involving the coordinates of points on these curves; topological information was ex- tracted from analytical information. The text of Gauss's fragment poses several historical riddles. As in many other passages of his notebooks, Gauss gave no indication of any proof or argument for his claim, nor did he give any reasons which had led him to consider the linking of space curves at all. Without further information, we cannot even be sure how his claim should be interpreted mathematically: Is it a definition; i.e., did Gauss want to say that the possible values of the double integral on the left side of his formula are integer multiples of 47r, and that, therefore, the integer ap- pearing on the right side could be de- fmed as the linking number of the two curves involved? Or is Gauss's formula a theorem, computing an indepen- dently defined numerical invariant of intertwined curves by analytical means? We thus have the following four ques- tions: 1. When, and how, did Gauss find the integral? 2. How did he know that the values of this integral were integer multiples of 4~r?. Column Editor's address: Faculty of Mathematics, The Open University, Milton Keynes, MK7 6AA, England lWerke, Vol. V, p. 605. All emphasis in this and the following quotations is in the originals. Square brackets are used to indicate my omissions or additions. 2First in a letter of 1836, then in Listing's essay Vorstudien zur Topologie, published in 1847. The name geome- trfa situs, or analysis situs, however, was retained by Riemann and later Poincar& Only in the first decades of the 20th centuw, topology gradually replaced analysis situs. Gauss's reference to Euler is to the latter's Solutio problematis ad geometriam situs pertinentis of 1736, dealing with the K6nigsberg bridges; the reference to Vandermonde is to a paper entitled Remarques sur les problemes de situation of 1771, in which Vandermonde studied various weaving patterns and their symmetries, along with the problem of circuits of knight's moves. Both papers are reprinted in English translation in (Biggs, et aL 1976). 9 1998SPRINGER-VERLAG NEW YORK, VOLUME 20, NUMBER 1, 1998 45