Riemann Rolls On
Shifting from drawing trees with lines to drawing them with ribbons, we find that some alterations are permitted. A Y vertex with lines looks the same upward (product) as it does downward (coproduct). With ribbons, we might have diagrams like this which were considered by Kauffman et al in studying invariants for templates. An example of a template is the one for the Lorenz attractor, namely where the ribbons are permitted to curl around via cap elements. The study of templates and knots took off with the work of Joan Birman et al on the periodic orbits of the Lorenz system. Knots correspond to words in the 2 letters defined by the holes in the template, around which one draws knots. Robert Ghrist came up with a 4 holed template, which is universal for knots and links.