In this talk, I discuss replacing axioms in a Frobenius algebra with diagrams and constructing glyphs to represent those diagrams. The ideas are extended to considering isotopy classes of knots as a 4-category. Then we discuss braids, braided manifolds, braid movies, charts, chart movies, curtains, and curtain movies as methods of braiding simple branched covers in codimension 2. As usual, there are lots of diagrams.
This talk is based upon joint work with Seiichi Kamada. Abstract tensor notation for multi-linear maps uses boxes with in-coming and out-going strings to represent structure constants. A judicious choice of variables for such constants leads to the diagrammatic representation: boxes are replaced by glyphs. One of the most simple examples of a multi-category is the diagrammatic representation of embedded surfaces in 3-space that we all grew up learning. The standard drawing of a torus (an oval, a smile and a moustache) is a representation based upon drawings of surface singularities. We start from a two object category with a pair of arrows that relate them. Cups and caps are constructed easily. From these, births, deaths, saddles, forks, and cusps are created as triple arrows. At the top level, there are critical cancellations, lips, beak-to-beak, horizontal cusps, and swallow-tails. Interestingly, the structure extends inductively to describe many relations about handles in higher dimensions.
CKVK* Recommendation: The talk has many beautiful figures that are best viewed on Youtube in HD.