Coloring link diagrams with a continuous palette

The well-known technique of n-coloring a diagram of an oriented link l is generalized using elements of the circle T for colors. For any positive integer r, the more general notion of a (T, r)-coloring is defined by labeling the arcs of a diagram D with elements of the torus Tr−1. The set of (T, r)-colorings of D is an abelian group, and its quotient by the connected component of the identity is isomorphic to the torsion subgroup of H1(Mr(l); Z). Here Mr(l) denotes the r-fold cyclic cover of S3 branched over the link l. Results about braid entropy are obtained using techniques of symbolic dynamical systems.