Knot theory from Vandermonde to Jones

Leibniz wrote in 1679: “I consider that we need yet another kind of analysis, : : : which deals directly with position.” He called it “geometry of position”(geometria situs). The first convincing example of geometria situs was Euler’s solution to the bridges of Königsberg problem (1735). The first mathematical paper which mentions knots was written by A. T. Vandermonde in 1771: “Remarques sur les problemes de situation”. We will sketch in this essay 1 the history of knot theory from Vandermonde to Jones stressing the combinatorial aspect of the theory that is so visible in Jones type invariants. “When Alexander reached Gordium, he was seized with a longing to ascend to the acropolis, where the palace of Gordius and his son Midas was situated, and to see Gordius’ wagon and the knot of the wagon’s yoke: : :. Over and above this there was a legend about the wagon, that anyone who untied the knot of the yoke would rule Asia. The knot was of cornel bark, and you could not see where it began or ended. Alexander was unable to find how to untie the knot but unwilling to leave it tied, in case this caused a disturbance among the masses; some say that he struck it with his sword, cut the knot, and said it was now untied but Aristobulus says that he took out the pole-pin, a bolt driven right through the pole, holding the knot together, and so removed the yoke from the pole. I cannot say with confidence what Alexander actually did about this knot, but he and his suite certainly left the wagon with the impression that the oracle about the undoing of the knot had been fulfilled, and in fact that night there was thunder and lightning, a further sign from heaven; so Alexander in thanksgiving offered sacrifice next day to whatever gods had shown the signs and the way to undo the knot.” [Lucius Flavius Arrianus, Anabasis Alexandri, Book II, c.150 A.D. ] 1This essay is an extended version of talks given at U.C. Riverside (April 18, 1991), at the Mexican National Congress of Mathematicians (November 1991) [93], and at the Minisemester on Knot Theory at Banach Center (July 18, 1995).