Joshua Chen's 2014 honours thesis in mathematics, on "The Temperley-Lieb categories and skein modules", is available here.

### Outline

In this thesis we develop the theory of diagrammatic Temperley-Lieb categories in order to construct examples of spherical fusion categories, which we then use to define Turaev-Viro skein modules for $n$-holed disks.

The basic Temperley-Lieb category contains as endomorphism spaces the Temperley-Lieb algebras, which have many surprising links to statistical mechanics, knot theory and representation theory. From this basic definition one constructs the Temperley-Lieb-Jones categories, which have very nice structure: at roots of unity $q$ they are equivalent as braided spherical tensor categories to the semisimplified category $\operatorname{Rep}U_q(\mathfrak{sl}_2)$ of representations of the quantum algebra $U_q(\mathfrak{sl}_2)$. (We do not address this any further in this thesis, the interested reader may see for example [ST08] or Chapter XII of [Tur94]). Further, the Temperley-Lieb-Jones categories are examples of spherical fusion categories, that is, semisimple spherical linear categories with additional nice properties, and one of the reasons these categories are interesting is that they allow us to construct skein modules.

A skein module is a module associated to a 2-manifold, and is the first step towards constructing a $(2+1)$-dimensional topological quantum field theory (TQFT). Briefly speaking, a $(n + 1)$-dimensional TQFT is a functor from $(n + 1)-\mathbf{Cob}$ to $\mathbf{FdVect}$, assigning to every $n$-manifold a finite-dimensional vector space and to every $(n + 1)$-cobordism between $n$-manifolds a linear transformation between the corresponding vector spaces, in a manner that respects composition and the rigid symmetric monoidal structure of the categories. These were first used to construct topological invariants by Witten in a seminal paper in 1989 [Wit89], and were axiomatized by Atiyah [Ati88] around the same time. (For a general introduction to topological quantum field theory from the algebraic point of view see for instance [Ati88] or the first section of [BD95].)

Following Witten's work, one of the next TQFTs to be discovered was the Turaev-Viro TQFT [TV92], which takes as input a spherical fusion category in order to construct vector spaces (free modules) for 2-surfaces and linear maps for 3-cobordisms. In this thesis we deal only with the 2-dimensional aspect of the theory and use Temperley-Lieb-Jones at roots of unity to construct skein module invariants associated to a specific class of surfaces, namely $n$-holed disks.

The outline of this thesis is as follows. Chapter 1 begins with some preliminary definitions and results. In Chapter 2 we define generic Temperley-Lieb, introduce the all-important Jones-Wenzl idempotents and use them to construct the generic Temperley-Lieb-Jones categories $TLJ$. In Chapter 3 we take a necessary detour into some Temperley-Lieb skein theory, proving the results we will need in order to show that generic $TLJ$ is semisimple, which we do in Chapter 4. In Chapter 5 we consider what happens for $TLJ$ evaluated at a root of unity, and show that after taking the quotient by the negligible ideal we obtain a semisimple category with finitely many simple objects, which is also spherical fusion. Finally, in Chapter 6 we present an alternative approach to constructing the Turaev-Viro skein modules for $n$-holed disks.