Uncategorized – Francis Bischoff

The Stasheff Associahedra are a family of polytopes that seem to be ubiquitous, but in particular are strongly linked to associativity. Here are the first few:

The vertices of the $n$ -dimensional associahedron $K^n$ are in bijection with bracketings of the word $x_0 x_1 ... x_{n+1}$ , or equivalently, with the set of planar binary trees with $n+2$ leaves. For example, there is a single way of bracketing the word $x_0 x_1$ , and therefore, $K^0$ is a single point. On the other hand, there are two ways of bracketing the word $x_0 x_1 x_2$ . They are $(x_0 x_1)x_2$ and $x_0 (x_1 x_2)$ . Correspondingly, $K^1$ is an interval. Here are the first three associahedra with the labelling of their vertices by trees.

Loday pointed out a mysterious relationship between the trefoil knot and the $3$ -dimensional associahedron $K^3$ . Namely, he showed that there is natural way in which the trefoil knot can be drawn on the surface of $K^3$ . Here is a picture: