Drawing Seifert Surfaces that Fiber the Figure-8 Knot Complement in S 3 Over S 1

Few areas of contemporary mathematics can boast of such an abundance of elementary and compelling examples as low dimensional topology. This is the study of curves, surfaces and 3-manifolds. The intense preoccupation of its practitioners with the concrete, tangible arrangement of plastic forms has led them to adopt a vividly pictorial genre of exposition. The process of inventing these apt examples, call it descriptive topology, appears to have its own visual logic and graphical rules. These also deserve serious study. As a pedagogical utility, descriptive topology should be straightforward and technically uncomplicated. It should never distract from the mathematics it means to illustrate. Above all, the topological design of its pictures should be memorable, ready to hand at the tip of a pencil or a piece of chalk. Apt, easy, unambiguous and memorable pictures are what this article is about. In order not to try the reader's patience or offend editorial hospitality, however, I shall avoid vague generalities and focus on a particular example. Technical details on drawing pictures were outlined in [5] and a primer for a general method is in preparation [6]. Here I shall tell a picture story about visualizing the fibration over the circle of the figure-8 knot complement. This exercise of the imagination consists of filling up the void of space, closed by a point at infinity to form the 3-sphere, S3, with a continuous succession of surfaces spanning the knot. That is, through each point not on the knot, K, there will pass a unique copy of a compact, connected, bordered, 2-sided surface, F, called a Seifert surface, whose boundary is the knot. The reasons why one should want to do this, and why it should be possible at all, belong to the fascinating biography of this remarkable knot. It is too long to tell about here. May this brief anecdote serve as an invitation to a rewarding treasure hunt in the literature. For a start, you should look at Rolfsen's masterpiece of descriptive topology [12] and then proceed to [9]. My own interest in it stems from a project to illustrate some ideas in [3] and [20]. This example is of particular interest to me because it belongs to an area of topology that already has a highly developed and effective graphical shorthand. This shorthand consists of "schematic" diagrams which are highly abstract, and often terse. Recognizable pictures of familiar shapes, arranged in space as dictated by the diagrams, help us see and remember the encoded information. Sometimes, a particularly well-designed picture can lead directly to an insight that is hard to decode from all the algebra generated in the service of precision. What follows is meant to be readable on an undergraduate level not much beyond advanced calculus. John Stillwell's clear and superbly illustrated text [17] covers the topological gaps. The more sophisticated reader, whose indulgence I beg for frequent abus de langage, is invited to read into my pictures many more ideas from the theory of fibered knots than can be explained at this level. At the end are "solutions" to some of these implied exercises.