Université Paris-Sud (Orsay), 6-10 July 2015 ]]>

I was thinking mainly about the work of Peter Roegen, using finite-type invariants to distinguish proteins (conceptually this makes a lot of sense):

e.g. http://www2.mat.dtu.dk/people/Peter.Roegen/poster2002.pdf

There was one effort going the other direction, using the known structures of known proteins to see which knots occur in nature. Alexander polynomials suffice to distinguish them, fwiw. http://knots.mit.edu/

]]>The thing to understand is that the things in the angular brackets are evaluations (complex numbers), while the graphs you see are occuring as local pieces of bigger diagrams. So for example, the local picture:

contains five 1-valent vertices. These are all connected (`concatenated’, we say) with 1-valent vertices in other graphs. So the image above would never occur in isolation- it would occur only as a subimage of a larger graph such as for example:

(ignore labels… I snipped these images out of different parts of Okazaki’s paper, so the labels happen not to match).

So the featured equation is living inside a skein module. Okazaki defined this skein module in Definition 2.1. Its elements are formal sums over the complex numbers of certain graphs. Relations allow you to replace any one of these graphs with a linear sum of other graphs, all of which coincide with the first graph except in a small part where they differ as pictured.

The analogy is to combinatorial group theory. In combinatorial group theory, a group is presented by generators (symbols) which we string together (concatenation), and then quotient the free group that we thus obtain by rewrite moves (relations), allowing us to replace one string by some other string whenever it occurs (*e.g* if your group is abelian).

The situation for skein modules is entirely analogous. We have `generating diagrams’, which can be concatenated *e.g.* by fusing a 1-valent vertex in one of them with a 1-valent vertex in another (if you think of each edge as representing a string (why not?) then your elements are sort-of graphs of strings which come together and split apart at vertices, and concatenation is exactly as mentioned). The free structure thus obtained is they quotiented by rewrite moves, which are the relations. The whole thing is occuring on a diagram in the plane. It’s a wonderfully natural and fruitful conception, actually!