Planar Algebra - Definition

Definition

In more detail, given a list of words, and a single word, we define a tangle from to to be a disk D in the plane, with points around its circumference labelled in order by the letters of, with internal disks removed, indexed 1 through k, with the i-th internal disk having points around its circumference labelled in order by the letters of, and finally, with a collection of oriented non-intersecting curves lying in the remaining portion of the disk, with each component being labelled by an element of the label set, such that the set of end points of these curves coincide exactly with the labelled points on the internal and external circumferences, and at the initial points of the curves, the label on the curves coincides with the label on the circumference, while at the final points, the label on the curve coincides with the involute of the label on the circumference.

While this sounds complicated, an illustrated example does wonders!

Such tangles can be composed. With this notion of composition, the collection of tangles with labels in and boundaries labelled by forms an operad.

This operad acts on the modules as follows. For each tangle from to, we need a module homomorphism . Further, for a composition of tangles, we must get the corresponding composition of module homomorphisms.