Planar algebra

In mathematics, planar algebras first appeared in the work of Vaughan Jones on the standard invariant of a II1 subfactor [1]. They also provide an appropriate algebraic framework for many knot invariants (in particular the Jones polynomial), and have been used in describing the properties of Khovanov homology with respect to tangle composition [2] [3].

Given a label set $I$ with an involution, and a fixed set of words $W$ in the elements of the label set, a planar algebra consists of a collection of modules $V_\omega$, one for each element $\omega$ in $W$, together with an action of the operad of tangles labelled by $I$.

Definition

In more detail, given a list of words $w_1, w_2, \ldots, w_k$, and a single word $w_0$, we define a tangle from $(w_1, w_2, \dots, w_k)$ to $w_0$ to be a disk D in the plane, with points around its circumference labelled in order by the letters of $w_0$, with $k$ 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 $w_i$, 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 $I$ and boundaries labelled by $W$ forms an operad.

This operad acts on the modules $V_\omega$ as follows. For each tangle $T$ from $(w_1, w_2, \ldots, w_k)$ to $w_0$, we need a module homomorphism $Z_T : V_{w_1} \otimes V_{w_2} \otimes \cdots \otimes V_{w_k} \longrightarrow V_{w_0}$. Further, for a composition of tangles, we must get the corresponding composition of module homomorphisms.

Examples

Temperley–Lieb

The Temperley–Lieb algebras can be retrofitted as a planar algebra.

Fix an element $\delta \in R$ in the ground ring. Take a one element label set, and allow words of even length. (Thus the words correspond exactly to nonnegative even integers.) For each even integer $2n$, let $V_{2n}$ be the free module generated by (isotopy classes of) diagrams consisting of n non-intersecting arcs drawn in a disk, with the endpoints of the arcs lying on the boundary of the disk. The action of tangles is simply by gluing the appropriate disks into the tangle, removing any closed arcs, replacing each with a factor of $\delta$.

We can generalise this to allow more complicated label and word sets (including for example, the planar algebra version of the Fuss–Catalan algebras). For each label $i \in I$, fix $\delta_i \in R$ in the ground ring. For a word $\omega \in W$, the module $V_\omega$ is generated by (again, isotopy classes) of diagrams consisting on non-intersecting arcs drawn in a disk, labelled by elements of $I$ with endpoints on the boundary of disk, such that the induced labels on these points, when read in order, give $\omega$. The action of tangles is defined as before, with closed arcs labelled by $i$ being replaced by a factor of $\delta_i$.

Tangles

The (oriented) tangle planar algebra is a planar algebra with a two element label set, the nontrivial involution on it, and balanced even length words. It is generated, as a planar algebra, by the diagrams of the positive and negative crossings in knot theory, living in V++−−. Knot polynomials satisfying skein relations can be succinctly described as quotient maps from this planar algebra, which are rank 1 on $V_\emptyset$.

Subfactors

The planar algebras used in describing II1 subfactors have a two element label set, with the nontrivial involution, and the allowed words are the finite-length alternating words in these two elements. The labels on the tangles are typically illustrated by shading alternately the regions between the strands; the two types of strands are then distinguished by either having a shaded region on their right or on their left.