Knot invariants from symbolic dynamical systems

If G is the group of an oriented knot k, then the set Hom(K, Sigma) of repr esentations of the commutator subgroup K = [G, G] into any finite group Sig ma has the structure of a shift of finite type Phi(Sigma), a special type o f dynamical system completely described by a finite directed graph. Invaria nts of Phi(Sigma), such as its topological entropy or the number of its per iodic points of a given period, determine invariants of the knot. When Sigm a is abelian, Phi(Sigma) gives information about the infinite cyclic cover and the various branched cyclic covers of k. Similar techniques are applied to oriented links.