Knots, Links and Representation Shifts

Introduction. The group π1(S − k) of a knot k contains an extraordinary amount of information. From combined results of W. Whitten [Wh] and M. Culler, C. McA. Gordon, J. Luecke and P.B. Shalen [CuGoLuSh] it is known that there are at most two distinct unoriented prime knots with isomorphic groups. Unfortunately, knot groups are generally difficult to use. Knot groups are usually described by presentations, and there is no practical algorithm to decide whether or not two knot groups are isomorphic. In 1928 J.W. Alexander used homomorphisms (representations) of knot groups onto better understood groups in order to obtain topological invariants. Since then knot group representations have been used effectively by many others. The representations of a given knot group into a fixed finite group have the additional attraction that they are finite in number and so can be tabulated. R. Riley began such a program in [Ri]. We take a new approach, examining the representations of the commutator subgroup K = [π1(S − k), π1(S − k)] into a fixed finite group Σ. Although Hom(K,Σ) is often infinite – in fact, uncountable – it has a rich structure that we can understand via symbolic dynamics. In this dynamical system the representations of the knot group π1(S − k) appear (by restricting their domains) as special periodic points. However, the system contains other periodic points and often nonperiodic points, information that can be used to understand more about the structure of the knot exterior and its various covering spaces. The techniques, all algorithmic, apply equally well to links.