Genus of Alternating Link Types

A link projection' is said to be alternating iff it is connected and, as one follows along any component of the link, undercrossings and overcrossings alternate. A projection is trivial iff it is connected and has no crossings; otherwise it is non-trivial. We include trivial projections as alternating. An alternating link type is one which has an alternating projection. The principal result of this paper is Theorem 3.5 which contains the assertion that the degree of the reduced Alexander polynomial of an alternating link type plus one equals twice its genus plus its multiplicity. A second result, obtained as an immediate corollary of the same method which ultimately yields (3.5), is Theorem (2.13): The reduced Alexander polynomial of an alternating link type is an alternating polynomial. This theorem provides the simplest proof of the existence of non-alternating types. The image P of a connected, non-trivial projection has a natural decomposition as a graph. The vertices are the crossings, i.e., the images of the undercrossings, and the edges are the open arcs into which the crossings subdivide P. Since we shall have no reason to distinguish a point at infinity, we regard P as a spherical rather than a planar graph. The results of this paper are obtained by studying the image graph of a non-