Abstract:
This thesis deals with mathematical objects known as planar algebras. These were
introduced by Vaughan Jones in order to study the so-called ‘standard invariant’
of a II1-subfactor and have provided a powerful pictorial viewpoint with which to
approach various computations in the theory. The first section is devoted to showing that for an arbitrary planar algebra P
satisfying an analogue of the finite depth condition for subfactor algebras, there is a
Pk−1-Pk−1 bimodule isomorphism Pm ⊗Pk−1 Pn → Pm+n−(k−1) for all m, n ≥ k where
k is the ‘depth’ of P. The next section introduces the technical tools that we use to prove the main result. There are three main notions, two of which are those of ‘templates’ and of ‘consequences’. By definition, a template is simply an ordered pair of planar tangles. Consequences of a set of templates roughly correspond to elements of the set obtained by closing the original set under certain planar operations, along with reflexivity and transitivity. Given a planar algebra P together with a subset B of P, there is a notion of a template ‘being satisfied’ in (P,B), and this is the third main notion. It is not hard to see from the definitions that if certain templates are satisfied for (P,B), then so are all their consequences. The main result in this section is a collection of various consequences of a set of templates that we call basic templates.
The third section of this chapter proves the main theorem that subfactor planar algebras of finite depth have a finite skein theory. The approach is to show that the basic templates hold for such a planar algebra (together with the distinguished subset being a basis of Pk where k is the depth) and then use their consequences and the bimodule isomorphism referred to above to deduce the theorem.
The last section gives a very simple proof that finite depth subfactor planar
algebras are actually singly generated and further have a skein theory with this
single generator. This last result is analogous to a finitely presented group having a finitely generated kernel for any surjective homomorphism of a finitely generated free group onto it, and the proof is also an imitation of that proof.
