Branch Points of Algebraic Functions and the Beginnings of Modern Knot Theory

Abstract

Many of the key ideas which formed modern topology grew out of "normal research" in one of the mainstream fields of 19th-century mathematical thinking, the theory of complex algebraic functions. These ideas were eventually divorced from their original context. The present study discusses an example illustrating this process. During the years 1895-1905, the Austrian mathematician, Wilhelm Wirtinger, tried to generalize Felix Klein's view of algebraic functions to the case of several variables. An investigation of the monodromy behavior of such functions in the neighborhood of singular points led to the first computation of a knot group. Modern knot theory was then formed after a shift in mathematical perspective took place regarding the types of problems investigated by Wirtinger, resulting in an elimination of the context of algebraic functions. This shift, clearly visible in Max Dehn's pioneering work on knot theory, was related to a deeper change in the normative horizon of mathematical practice which brought about mathematical modernity. © 1996 Academic Press, Inc.Viele der Ideen, die die moderne Topologie geprigt haben, stammen aus 'normaler Forschung' in einer der Hauptstr6mungen des mathematischen Denkens des 19. Jahrhunderts, der Theorie komplexer algebraischer Funktionen, und wurden erst allmihlich yon ihrem urspringlichen Kontext abgelOst. Ein Beispiel dieser Entwicklung wird diskutiert. In den Jahren [1895][1896][1897][1898][1899][1900][1901][1902][1903][1904][1905] INTRODUCTIONDue to the rapid development and application of new knot invariants in mathematics and physics following Vaughan Jones' discovery of a new knot polynomial, knot theory has received growing attention within and even outside the mathematical community. 2 In this context, it has often been asked why the knot problem--of all topological problems--was among the first to be studied by early topologists of our century such as Heinrich Tietze, Max Dehn, James W. Alexander, and Kurt Reidemeister. This question appears all the more puzzling since 19-century work on knots had certainly not been at the cutting edge of mainstream mathematical research--unlike, for instance, the topological problems that arose in connection with the theory of algebraic functions or algebraic geometry. In the following, an answer to this question will be given. Using hitherto unpublished correspondence between the Austrian mathematician, Wilhelm Wirtinger, and Felix Klein, it will be shown that modern knot theory did in fact originate from these latter fields. Furthermore, while they were familiar to the pioneers of modern topology, a series of events gradually left these roots forgotten by their followers.In considering the origins of modern knot theory, I will do more than merely retrace the technical developments. Rather, this formation of a new field of mathematical research illustrates a certain pattern which, in a nutshell, may be characterized as follows:Thesis. What appears, at first sight, to be the invention of a new mathematical discipline, turns o...

Knot Invariants in Vienna and Princeton during the 1920s: Epistemic Configurations of Mathematical Research

“…I owe an apology to all those who have heard me emphasizing the crucial role of this "object," or more precisely, of this imagination in the history of knot theory for about ten years now. A comparison of the present arguments with my first discussion of it in print(Epple 1995) will show that it played quite different roles in this history. For more information, see section 4 below.…”