Chapter 9 - Thin Position in the Theory of Classical Knots

Abstract

This chapter discusses the role that the notion of “thin position” has played in the theory of classical knots and to the understanding of knotted graphs in 3-space. The extension of thin position to graphs, beyond being of interest in its own right, also is shown to have applications in knot theory. The first knot invariant that is suggested by knot projection is the crossing number of the knot. There is another invariant, much like the crossing number and only slightly more difficult to describe, that is in fact remarkably well behaved under knot sum. The chapter presents a simple exercise that illustrates how thin position interacts with geometric properties of a knot, in particular with an essential surface in the knot complement. Trivalent graphs; presumably more general graphs, can be treated similarly but so far there seems to be no notable application to higher valence graphs. Surfaces that lie in a knot complement have the pleasant feature that boundary components are either horizontal or the height function on the boundary circles roughly follow that of the knot or the height function on.