# Dimension and Trace of the Kauffman Bracket Skein Algebra

## Abstract

Let be a finite type surface and a complex root of unity. The Kauffman bracket skein algebra is an important object in both classical and quantum topology as it has relations to the character variety, the Teichmüller space, the Jones polynomial, and the Witten-Reshetikhin-Turaev Topological Quantum Field Theories. We compute the rank and trace of over its center, and we extend a theorem of the first and second authors in [Math. Z. 289 (2018), pp. 889–920] which says the skein algebra has a splitting coming from two pants decompositions of .

## 1. Introduction

Let be a finite type surface and a complex root of unity. The Kauffman bracket skein algebra is an important object in both classical and quantum topology as it has relations to the character variety, the Teichmüller space, the Jones polynomial, and the Witten-Reshetikhin-Turaev Topological Quantum Field Theories. We recall the definition of in Section 3.

The linear representations of play an important role in hyperbolic Topological Quantum Field Theories. In Reference 13 we prove the Unicity Conjecture of Bonahon and Wong Reference 6 which among other things states that generically all irreducible representations of have the same dimension equal to the square root of the dimension of over its center Here if . is an algebra whose center is a domain then the dimension of over denoted by , is defined to be the dimension of the vector space , over the field of fractions of The calculation of the dimension . is one of the main result of this paper.

We show that where , is the field of fractions of is a division algebra having finite dimension over its center , Thus every element . lies in a finite field extension of and hence has a reduced trace We recall the definition of the reduced trace in Section .2.

The second goal of the paper is to compute the reduced trace of elements of To state the theorem, denote by . the set of all isotopy classes of simple diagrams on surface where a simple diagram is the union of disjoint, non-trivial simple closed curves on , For each . one can define an element such that the set , is a of -basis and is central if and only if is in a certain subset of See Section .3.3 for details. The definition of involves Bonahon and Wong’s threading map Reference 5. As the space -vector has basis hence it is enough to compute the trace of each , .

Along the way we develop tools for determining when a collection of skeins forms a basis for .

The last goal of the paper is to prove that there exists a splitting of over its center coming from pairs of pants decompositions of the surface.

This theorem has an application in defining invariants of links in 3-manifold which will be investigated in a future work.

The paper is organized as follows. In Section 2 we survey results about division algebras that have finite rank over their center, and facts about trace and filtrations of algebras, with the goal of applying these to the Kauffman bracket skein algebra. We follow by introducing the Kauffman bracket skein algebra in Section 3. Its basis is given in terms of simple diagrams, so we describe ways of parametrizing simple diagrams on a surface. We also introduce a residue group. In section 4 we show that after enough twisting the Dehn Thurston coordinates of a simple diagram on a closed surface stabilize to become an affine function of the number of twists. This allows us to define stable Dehn-Thurston coordinates. In Section 5 we introduce a degree map and use it to formulate a criterion for independence of a collection of skeins over its center. Section 6 computes the dimension of the Kauffman bracket skein algebra over its center, proving Theorem 1. In section 7 we find bases for commutative subalgebras of generated by the curves in a primitive non-peripheral diagram on with coefficients in In Section .8 we find a formula for computing the trace, proving Theorem 2. The paper concludes in Section 9 which proves the splitting theorem (Theorem 3).

## 2. Division algebras, trace, filtrations

In this section we survey some well-known facts about division algebras, trace, and filtrations of algebras that will be used in the paper.

### 2.1. Notations and conventions

Throughout the paper , , , , denote respectively the set of natural numbers, integers, rational numbers, real numbers, and complex numbers. Note that Let . be the field with 2 elements.

A complex number is a *root of 1* if there is a positive integer such that and the smallest such positive integer is called the ,*order * of denoted by , .

All rings are assumed to be associative with unit, and ring homomorphisms preserve 1. A *domain* is a ring not necessarily commutative, such that if , with then , or For a ring . denote by the set of all non-zero elements in For example, . is the set of all non-zero complex number.

### 2.2. Algebras finitely generated over their centers

Recall that a *division algebra * is an associative algebra with unit such that every nonzero element has a multiplicative inverse. Note that a commutative division algebra is a field. The following is well-known, and we present a simple proof for completeness.

The above proposition reduces many problems concerning domains which are finitely generated as modules over their centers to the case of division algebras finitely generated over their centers.

### 2.3. Trace

Suppose is a field and is a which is finite-dimensional as a -algebra space. For -vector the left multiplication by , is a operator acting on -linear and its trace is denoted by , The .*reduced trace* is defined by

Again, the following is well-known.

### 2.4. Maximal commutative subalgebras

Suppose is a division algebra with center If . is a maximal commutative subalgebra, then is a field and

This follows directly from Theorem 15.8 in Reference 15. Moreover splits over i.e., , is isomorphic to the algebra of matrices with entries in for some .

### 2.5. Dimension

Suppose a -algebra has center which is a commutative domain. Let be the field of fractions of The .*dimension * is defined to be the dimension of the space -vector .

A *filtration compatible with the product* of is a sequence of of -subspaces such that , and , For any subset . let .

### 2.6. Lattice points in a polytope

Suppose is the standard Euclidean space. A -dimensional*lattice* is any abelian subgroup of maximal rank A .*convex polyhedron* is the convex hull of a finite number of points in and its volume is denoted by -dimensional Let .

## 3. Kauffman bracket skein algebra

The Kauffman bracket skein module of a 3-manifold was introduced independently by Przytycki Reference 26 and Turaev Reference 32Reference 33. In this section we recall the definition of the Kauffman bracket skein algebra of a finite type surface and present some results concerning its center. We also explain how to coordinatize the set of curves on , and use coordinates to define a residue group associated to and a root of 1.

### 3.1. Finite type surface

An oriented surface of the form where , is an oriented closed connected surface and is finite (possibly empty), is called a *finite type surface*. A point in is called a puncture. The genus and the puncture number totally determine the diffeomorphism class of and for this reason we denote , The Euler characteristic of . is which is non-negative only in 4 cases: ,

Since the analysis of these four surfaces is simple and requires other techniques, very often we consider these cases separately.

Throughout this section we fix a finite type surface .

In this paper a *loop* on is a unoriented submanifold diffeomorphic to the standard circle. A loop is *trivial* if it bounds a disk in it is ;*peripheral* if it bounds a disk in which contains exactly one puncture. A *simple diagram* is the union of several disjoint non-trivial loops. A simple diagram is *peripheral* if all its components are peripheral.

If is a smooth map such that and embeds into then the image of is called an *ideal arc*. Isotopies of ideal arcs are always considered in the class of ideal arcs.

Suppose is either an ideal arc or a simple diagram and is a simple diagram. The *geometric intersection number* is the minimum of with all possible , isotopic to and isotopic to We say that . is * -taut*, or and are *taut*, if they are transverse and .

A simple diagram is *even* if is even for every loop It is easy to see that . is even if and only if represents the zero element in the homology group .

Very often we identify a simple diagram with its isotopy class. Denote by the set of all isotopy classes of simple diagrams on Let . be the subset of all classes of even simple diagrams, and be the subset of all peripheral ones. For convenience, we make the convention that the empty set is a peripheral simple diagram. Thus

### 3.2. Kauffman bracket skein algebra

A *framed link* in is an embedding of a disjoint union of oriented annuli in By convention the empty set is considered as a framed link with 0 components and is isotopic only to itself. .

For a non-zero complex number the ,*Kauffman bracket skein module* of at denoted by , is the , space freely spanned by all isotopy classes of framed links in -vector subject to the following *skein relations*

Here the framed links in each expression are identical outside the balls pictured in the diagrams, and the arcs in the pictures are supposed to have blackboard framing. If the two arcs in the crossing belong to the same component then it is assumed that the same side of the annulus is up.

For two framed links and in their product, , is defined by first isotoping , into and into and then taking the union of the two. This product gives the structure of a which is in most cases non-commutative. Let -algebra, be the center of .

The localized skein algebra of is defined by

where is the field of fractions of the center From Proposition .2.1, we have the following corollary.

### 3.3. Chebyshev basis and center

The *Chebyshev polynomials of the first kind* are defined recursively by

They satisfy the product to sum formula,

Suppose is a simple diagram. Some components of may be isotopic to each other. Let …, , be a maximal collection of components of such that no two of them are isotopic. Then there are positive integers such that is the union of parallel copies of with …, , In other words, . Let

If then in general However, if . where