> Eg Conway’s notation for knots I'm not sure how well-known this is, but Conway notation can be thought of as taking a Tait graph of the knot (i.e., a checker

> Eg Conway’s notation for knots I'm not sure how well-known this is, but Conway notation can be thought of as taking a Tait graph of the knot (i.e., a checkerboard graph) then reducing all the series-parallel networks until you get some irreducible graph. He noticed that certain kinds of series-parallel networks can be represented by an extended rational number (includes infinity), which can dramatically simplify the analysis of small knots -- to the point he was able to classify all of them in an afternoon! So, Conway notation corresponds to an irreducible planar graph with an expansion of each edge by a series-parallel network, with each edge in the expansion labeled by a rational number. You can even work out a set of moves such that if two such diagrams correspond to the same knot then there is a sequence of such moves connecting them. I do wonder if there's a similar analysis for lambda calculus represented as graphs... It might be that certain expressions play the role of rational tangles. (Though very likely not :-) )