#### by Stephen Bigelow

#### Context

Jones defined planar algebras in the 1990s [3]. In “The annular structure of subfactors” [6], he explains that the definition of planar algebras grew out of an attempt to solve the massive systems of linear equations that define the standard invariant of certain subfactors. Since then, they have provided a radically new diagrammatic approach to subfactors.

Jones’ breakthrough paper “Index of subfactors” [1] can be seen in retrospect as an application of diagrammatic algebra to subfactors. However the diagrammatic approach to the Temperley–Lieb algebra was only introduced later by Kauffman [e1]. The first use of truly diagrammatic methods to subfactors was [4], which gave certain conditions under which a subfactor has an intermediate subfactor.

The paper “The annular structure of subfactors” is mostly about
__\( ATL(\delta) \)__, the annular Temperley–Lieb algebroid, which captures
the simplest part of the algebraic structure of a planar algebra.
Jones uses this approach to give a new proof of the existence and
uniqueness of the __\( E_6 \)__ and __\( E_8 \)__ subfactors, which are the most
interesting subfactors of index less than 4. The same basic
method was later used to prove the existence of the extended Haagerup
subfactor, and continues to be used to construct and
study
new subfactors.

#### Planar algebras and annular tangles

Factors are the building blocks of von Neumann algebras. Any nonzero
morphism between factors must be an embedding, so the study of factors
naturally leads to the study of subfactors. Most of the focus has
been on subfactors of type II__\( _1 \)__ factors, which seem to be the most
interesting type in the classification given by
Murray
and
von Neumann.

A subfactor of a type II__\( _1 \)__ factor has three important invariants:
the *index* is a number __\( \delta^2 > 0 \)__, the *principal
graph* is a bipartite multigraph, and the *standard invariant*
is a pair of nested sequences of algebras.
Popa
[e5]
gave
a list of axioms that characterize the standard invariant. Jones
gave an alternative axiomatization in terms of planar algebras.

It takes some time and care to define planar algebras rigorously,
but the basic idea is not so difficult. A planar algebra consists
of a sequence of vector spaces __\( P_n \)__, together with a multilinear
action of *planar tangles*. A planar tangle is a diagram
consisting of disjoint curves, or “strands”, drawn in a disk with
holes. Usually, the vectors in the planar algebra are formal linear
combinations of some kind of diagrams drawn in a disk, and a planar
tangle acts on diagrams by gluing the diagrams into the holes.
Every boundary circle comes with a special distinguished region,
which determines the correct orientation for gluing.

We briefly mention two of the technical details in the definition
of a planar algebra. First, each vector space __\( P_n \)__ comes with a
conjugate-linear “star” operation, which commutes with the
reflection operation on planar tangles. This gives rise to an inner
product, which must be positive definite. Second, a planar tangle
comes with a checkerboard shading of the regions between strands,
alternating between shaded and unshaded regions. Due to the shading,
there are actually two vector spaces __\( P_+ \)__ and __\( P_- \)__ in place of
__\( P_0 \)__.

An *annular tangle* is a planar tangle with only one input. In
other words, it is a Temperley–Lieb diagram drawn in an annulus.
An annular __\( (m,k) \)__-tangle is an annular tangle that has __\( 2k \)__ marked
points on the inner circle and __\( 2m \)__ on the outer.

The *annular Temperley–Lieb algebroid* __\( ATL(\delta) \)__ consists of
formal linear combinations of annular __\( (m,k) \)__-tangles. If an annular
tangle has a closed loop that does not go around the central hole,
then that loop can be deleted in exchange for multiplying by the
parameter __\( \delta > 0 \)__. Multiplication in __\( ATL(\delta) \)__ comes from
the operation of placing one annular tangle inside the hole of
another.

Every subfactor planar algebra is a module over __\( ATL(\delta) \)__, where
__\( \delta \)__ is the index of the subfactor. Most of “The annular
structure of subfactors” is concerned with analyzing modules over
__\( ATL(\delta) \)__, and drawing conclusions about subfactors.

Jones completely classifies the Hilbert __\( TL(\delta) \)__-modules in the
case __\( \delta > 2 \)__. Just knowing the possible dimensions of Hilbert
__\( TL(\delta) \)__-modules is enough to prove that some graphs cannot be the
principal graph of a subfactor. Jones gives some examples of this,
including results previously proved in
[3]
and
[e4].

The case __\( \delta < 2 \)__ requires special treatment. Jones gives some
results in specific cases that will be needed for the __\( E_6 \)__ and
__\( E_8 \)__ subfactors. A more extensive study is postponed to a later
paper with
Reznikoff
[7].

The study of __\( ATL(\delta) \)__ is an example of the diagrammatic approach
providing not just a new way to prove theorems, but a new idea of
what questions to ask. The action of annular planar tangles is a
natural choice of the “simplest” part of the algebraic structure
of a planar algebra, leading to a direction of investigation might
not seem so natural otherwise.

#### Construction of __\( E_6 \)__ and __\( E_8 \)__ subfactors

The subfactors with principal graphs __\( E_6 \)__ and __\( E_8 \)__ are the most
interesting of the __\( ADE \)__ classification of subfactors of index less
than 4. They were first constructed in
[e2]
and
[e3].
Jones gives an alternative, diagrammatic construction.
Since they are finite depth, it is easy to construct the subfactor
from its corresponding planar algebra.

Most of the focus is on the more complicated __\( E_8 \)__ planar algebra,
which we will call __\( P \)__. Jones ultimately obtains a presentation
for __\( P \)__ with one generator __\( \psi \in P_5 \)__, and five relations listed in
([6], Appendix B).
This presentation is at first an educated guess.
Start by assuming that __\( P \)__ is the subfactor planar algebra of type
__\( E_8 \)__. Then the dimension of __\( P_n \)__ is the number of paths of length
__\( 2n \)__ in the __\( E_8 \)__ graph that start and end at the vertex farthest
from the trivalent vertex. We can deduce the existence of __\( \psi
\in P_5 \)__ that is perpendicular to the Temperley–Lieb algebra __\( TL_5 \)__.
We can then find relations that __\( \psi \)__ must satisfy using further
dimension arguments and positive definiteness.

Once we have defined __\( P \)__, the main task is to prove that the relations
are powerful enough to make __\( P \)__ finite-dimensional, but not so
powerful as to make it trivial. This is a common problem. As Jones
says: “Probably any set of skein relations causing collapse to
finite dimensions (but not to zero) should be considered interesting.”

To prove that the dimension of __\( P \)__ is not trivial, Jones finds a
copy of __\( P \)__ inside the *graph planar algebra* __\( P^{E_8} \)__, as
defined in
[5].
We could compare this situation to the
problem of proving the nontriviality of a group given by generators
and relations. One way to do this is to find a nontrivial
homomorphism to a general linear group. The graph planar algebra
plays the role of the general linear group. This method is widely
applicable, since every subfactor embeds in the graph planar algebra
of its principal graph
[8].

Instead of proving that every __\( P_n \)__ is finite-dimensional, it
suffices to prove to prove that __\( P_\pm \)__ has dimension one. The
fact that __\( P \)__ is the subfactor planar algebra with principal graph
__\( E_8 \)__ then follows from a process of elimination. This is due to
what Jones calls “the paucity of graphs with norms less than 2”.

A diagram in __\( P_\pm \)__ is called a *closed diagram*. Jones
describes an *evaluation algorithm*, which uses the relations
to reduce an arbitrary closed diagram to a scalar multiple of the
empty diagram. In keeping with the theme of the paper, the two
most important relations are relations between annular tangles
applied to __\( \psi \)__.

The first relation is that any annular __\( (4,5) \)__-tangle applied to __\( \psi \)__
gives zero. Nowadays we would say __\( \psi \)__ is *uncappable*, meaning
that a diagram is zero if any strand forms a “cap” connecting two
points on the same copy of __\( \psi \)__.

The second relation comes from Lemma 8.1, and says that a certain
linear combination __\( v \)__ of annular __\( (6,5) \)__-tangles applied to
__\( \psi \)__ is 0. We call this a *braiding relation*, since it
says a strand can slide over (but not under) a generator, where we
allow diagrams that contain crossings, which can be resolved by the
usual Kauffman skein relation. The braiding relation is shown in
Figure 1, where __\( \psi \)__ is a circle, and we have
omitted the distinguished regions, the shading, and the coefficients
in the linear combination.

Jones uses the braiding relation by applying it inside a larger
diagram that has two copies of __\( \psi \)__. Repeatedly doing this, he
is able to show that two copies of __\( \psi \)__ that are connected by two
or more parallel strands can be written as a linear combination of
diagrams that have only fewer copies of __\( \psi \)__.

By an Euler characteristic argument, any nonempty closed diagram
has either a strand that forms a closed loop, a cap attached to a
copy of __\( \psi \)__, or two parallel strands connecting two copies of
__\( \psi \)__. The closed loop can be deleted, the cap makes the diagram
zero, and the third case can be written as a linear combination of
diagrams that have fewer copies of __\( \psi \)__. We can repeat this
process until we simplify down to a scalar multiple of the empty
diagram.

#### The jellyfish algorithm

With his construction of the __\( E_6 \)__ and __\( E_8 \)__ planar algebras, Jones
laid out the template for what is sometimes called the *skein
theoretic* approach to defining a subfactor. The same approach was
used in
[e7]
to construct, and thoroughly analyze, the __\( D_{2n} \)__
planar algebra. The first new subfactor constructed in this way
was the extended Haagerup subfactor
[e8].

As in the __\( E_8 \)__ case, the __\( D_{2n} \)__ planar algebra is defined by a
single uncappable generator and a list of relations, including a
braiding relation of the same form as Figure 1.
The relations are quite simple and powerful, so
[e7]
give a
direct proof that the planar algebra is not trivial, without use
the graph planar algebra.

A key observation in [e7] is that you can use the braiding relation to bring any pair of generators to be adjacent. Then there is another relation that lets you simplify the adjacent pair of generators. In this way, they avoid the need for an Euler characteristic argument.

In retrospect, a similar approach would have been possible in the
__\( E_8 \)__ case. It is more difficult in that two adjacent generators
need to be connected by two strands in order to be simplified.
However, if all of the generators are moved to the top of the
diagram, then it is not hard to show there must be either a generator
connected to itself by a “cup”, or a pair of generators connected
by more than two strands. This would eliminate the need for the
Euler characteristic argument, which is not necessarily an improvement
over Jones’ algorithm, but does provide motivation for the extended
Haagerup planar algebra.

As usual,
[e8]
defines a planar algebra with one generator
and a list of relations, and prove it is nontrivial by embedding
it in the graph planar algebra of the extended Haagerup graph. This
graph planar algebra is too large to analyze as carefully as Jones
does in the __\( E_6 \)__ and __\( E_8 \)__ cases. Instead, a computer search is
used to find an element that satisfies the defining relations.

Analogous to the braiding relation, the extended Haagerup planar
algebra has two *braiding substitute* relations. The simpler
of the two is of the form shown in Figure 2.
Again, the generator is a circle, and we have omitted the distinguished
regions, the shading, and the coefficients in the linear combination.
We have also cheated with the ellipsis, which hides some terms that
are diagrams with no copies of the generator.

Note that one of the terms on the right of the braiding substitute relation has two copies of the generator. However all generators on the right are all closer to the top than the generator on the left. Thus, if we are willing to increase the number of generators in a diagram, we can move them all to the top.

Once we have all of the generators at the top of a closed diagram,
we can then start to decrease the number of generators. If a
generator is connected to itself by a “cup”, then the diagram is
zero. If not, it is not hard to show there must be a pair of
generators that are joined by at least half of their strands. Such
a pair can be simplified by the *quadratic relation*. We can
repeat this process until there are no generators.

The above evaluation algorithm is called the *jellyfish algorithm*,
since the first stage is reminiscent of jellyfish floating to the
top of a tank. The same algorithm has been used to construct other
subfactors, for example in
[e10].
Conversely, it has been used
to place restrictions on the type of graphs that can be principal
graphs of subfactors
[e9].