{centering}
Unified Quantum and Invariants
for Rational Homology 3–Spheres
Irmgard Bühler
Zürich, 2010
Abstract
Inspired by E. Witten’s work, N. Reshetikhin and V. Turaev introduced in 1991 important invariants for 3–manifolds and links in 3–manifolds, the so–called quantum (WRT) invariants. Short after, R. Kirby and P. Melvin defined a modification of these invariants, called the quantum (WRT) invariants. Each of these invariants depends on a root of unity.
In this thesis, we give a unification of these invariants. Given a rational homology 3–sphere and a link inside, we define the unified invariants and , such that the evaluation of these invariants at a root of unity equals the corresponding quantum (WRT) invariant. In the case, we assume the order of the first homology group of the manifold to be odd. Therefore, for rational homology 3–spheres, our invariants dominate the whole set of quantum (WRT) invariants and, for manifolds with the order of the first homology group odd, the whole set of quantum (WRT) invariants. We further show, that the unified invariants have a strong integrality property, i.e. that they lie in modifications of the Habiro ring, which is a cyclotomic completion of the polynomial ring .
We also give a complete computation of the quantum (WRT) and invariants of lens spaces with a colored unknot inside.
Contents

1 Colored Jones Polynomial
 1.1 Links, tangles and bottom tangles
 1.2 The quantized enveloping algebra
 1.3 Universal invariant
 1.4 Definition of the colored Jones Polynomial

1.5 Cyclotomic expansion of the colored Jones polynomial

2 Quantum (WRT) invariant
 2.1 Surgery on links in
 2.2 Gauss sums

2.3 Definition of the quantum (WRT) invariant
 2.3.1 Renormalization of quantum (WRT) invariant

2.3.2 Connection between and invariant

3 Cyclotomic completions of polynomial rings
 3.1 On cyclotomic polynomial
 3.2 Inverse limit
 3.3 Cyclotomic completions of polynomial rings
 3.4 The rings and

3.5 Evaluation

4 Unified invariant
 4.1 Unified invariant of lens spaces
 4.2 Unified invariant of diagonal manifolds

4.3 Definition of the unified invariant: general case

5 Roots in
 5.1 On the module
 5.2 Frobenius maps
 5.3 Existence of th root of in

5.4 Realization of in

6 Unified invariant of lens spaces
 6.1 The quantum invariant of lens spaces with colored unknot inside

6.2 Proof of Lemma 37

7 Laplace transform
 7.1 Definition of Laplace transform

7.2 Proof of Theorem 38
 7.2.1 Part 1: Existence of , odd case
 7.2.2 Part 1: Existence of , even case.
 7.2.3 Part 2: .

7 Laplace transform

6 Unified invariant of lens spaces

5 Roots in

4 Unified invariant

3 Cyclotomic completions of polynomial rings

2 Quantum (WRT) invariant
Introduction
In 1984, V. Jones [16] discovered the famous Jones polynomial, a strong link invariant which led to a rapid development of knot theory. Many new link invariants were defined short after, including the so–called colored Jones polynomial which uses representations of a ribbon Hopf algebra acting as colors attached to each link component. The whole collection of invariants of this spirit are called quantum link invariants.
In the 60’ and 70’ of the last century, Likorish [26], Wallace [35] and Kirby [18] showed, that there is a one–to–one correspondence via surgery between closed oriented 3–manifolds up to homeomorphisms and knots in the –dimensional sphere modulo Kirby–moves. This gives the possibility to study 3–manifolds using knot theory.
In 1989, E. Witten [36] considered quantum field theory defined by the noncommutative Chern–Simons action to define (on a physical level of rigor) certain invariants of closed oriented 3–manifolds and links in 3–manifolds. Inspired by this work, N. Reshetikhin and V. Turaev [33, 34] constructed in 1991 new topological invariants of 3–manifolds and of links in 3–manifolds. The construction goes as follows. Let be a closed, oriented 3–manifold and its corresponding surgery link. The quantum group is a deformation of the Lie algebra and has the structure of a ribbon Hopf algebra. One now takes the sum of the colored Jones polynomial of , normalized in an appropriate way, over all colors, i.e over all finite–dimensional irreducible representations of . Evaluating at a root of unity makes the sum finite and well–defined. These invariants are denoted by . Together they form a sequence of complex numbers parameterized by complex roots of unity and are known either as the Witten–Reshetikhin–Turaev invariants, short WRT invariants, or as the quantum invariants of –manifolds. Since the irreducible representations of the quantum group correspond to the irreducible representations of the Lie group , they are sometimes also called the quantum (WRT) invariants.
R. Kirby and P. Melvin [20] defined the version of the quantum (WRT) invariants by summing only over representations of of odd dimension and evaluating at roots of unity of odd order. These invariants are known as the quantum (WRT) invariants.
They have very nice properties. For example, A. Beliakova and T. Le [5] showed that they are algebraic integers, i.e. for any closed oriented 3–manifold and any root of unity (of odd order). Similar results where also proven for the invariants with some restrictions on either the manifold or the order of the root of unity (see [12], [3], [11], [28]). The full integrality result is conjectured and work in progress.
The integrality results are based on a unification of the quantum (WRT) invariant.
For any integral homology –sphere , K. Habiro [12] constructed a unified invariant whose evaluation at any root of unity coincides with the value of the quantum (WRT) invariant at that root. The unified invariant is an element of a certain cyclotomic completion of a polynomial ring, also known as the Habiro ring. This ring has beautiful properties. For example, we can think of its elements as analytic functions at roots of unity [12]. Therefore, the unified invariant belonging to the Habiro ring means that the collection of the quantum (WRT) invariants is far from a random collection of algebraic integers: together they form a nice function.
In this thesis, we give a similar unification result for rational homology –spheres which includes Habiro’s result for integral homology –spheres. More precisely, for a rational homology –sphere , we define unified invariants and such that the evaluation at a root of unity gives the corresponding quantum (WRT) invariant (up to some renormalization). In the case, we assume the order of the first homology group of the manifold to be odd – the even case turns out to be quite different from the odd case and is part of ongoing research. Further, new rings, similar to the Habiro ring, are constructed which have the unified invariants as their elements. We show that these rings have similar properties to those of the Habiro ring. We also give a complete computation of the quantum (WRT) and invariants for lens spaces with a colored unknot inside at all roots of unity.
Additionally to the techniques developed by Habiro, we use deep results coming from number theory, commutative algebra, quantum group and knot theory. The new techniques developed in Chapters 3 and 5 about cyclotomic completions of polynomial rings could be of separate interest for analytic geometry (compare [27]), quantum topology, and representation theory. Further, even though integrality of the quantum (WRT) invariants does not in general follow directly from the unification of the quantum (WRT) invariants, it does help proving it and a conceptual solution of the integrality problem is of primary importance for any attempt of a categorification of the quantum (WRT) invariants (compare [17]). Our results are also a step towards the unification of quantum (WRT) invariants of any semi–simple Lie algebra (see [34] for a definition of quantum (WRT) invariants). K. Habiro and T. Le announced such unified invariants for integral homology 3–spheres. We expect that the techniques introduced here will help to generalize their results to rational homology 3–spheres.
Plan of the thesis
In Chapter 1, we give the definition of the colored Jones polynomial and state that it has a cyclotomic expansion with integral coefficients. The proof of this integrality result is postponed to the Appendix. This expansion is used for the definition of the unified invariant (Chapter 4).
In Chapter 2, the quantum (WRT) invariants are defined and important facts about (generalized) Gauss sums are stated.
Chapter 3 is devoted to the theory of cyclotomic completions of polynomial rings. For a given , we define the rings and and discuss the evaluation at a root of unity in these rings.
In Chapter 4, the unified invariants and of a rational homology 3–sphere are defined and the main results of this thesis, i.e. the invariance of and and that their evaluation at a root of unity equals the corresponding quantum (WRT) invariant, are proven. Here we use (technical) results from Chapters 6 and 7.
In Chapter 5, we prepare Chapters 6 and 7 by showing that certain roots appearing in the unified invariants exist in the rings and .
In Chapter 6, we compute the quantum (WRT) invariants of lens spaces with a colored unknot inside and define the unified invariants of lens spaces.
In Chapter 7, we define a Laplace transform which we use to prove the main technical result of this thesis, namely that the unified invariant (respectively ) is indeed an element of (respectively of ), where is the order of the first homology group of the rational homology 3–sphere .
The material of Chapters 1 and 2 is partly taken from [12], [26], [20] and [4]. Chapter 3 includes results of Habiro [12, 14]. The case of the results from Chapters 3 to 7 as well as the Appendix appeared in our joint paper with A. Beliakova and T. Le [4]. The case has not yet been published anywhere else.
Acknowledgments
First and foremost, I want to express my deepest gratitude to my supervisor Anna Beliakova. Her encouragement, guidance as well as her way of thinking about mathematics influenced and motivated me throughout my studies.
Further, I would like to thank Thang T. Q. Le for sharing with me his immense mathematical knowledge, his way of explaining, discussing and doing mathematics and for our joint research work.
During my thesis, I spent seven months at the CTQM, University of Aarhus, in Denmark. Further, a part of my PhD was funded by the Forschungskredit of the University of Zurich as well as by the Swiss National Science Foundation.
Chapter 1 Colored Jones Polynomial
In this chapter, we first recall some basic concepts of knot theory and quantum groups. We then define the universal invariant of knots and links which leads us to the definition of the colored Jones Polynomial. In the last section, we state a generalization of Habiro’s Theorem 8.2 of [12] about a cyclotomic expansion of the colored Jones polynomial which we need for the definition of the unified invariant in Chapter 4. The proof of this theorem is postponed to the Appendix.
Throughout this thesis, we will use the following notation. The –dimensional sphere will be denoted by , the –dimensional disc by and the unit interval by . The boundary of a manifold is denoted by . Except otherwise stated, a manifold is always considered to be closed, oriented and 3–dimensional.
1.1 Links, tangles and bottom tangles
A link with components in a manifold is an equivalence class by ambient isotopy of smooth embeddings of disjoint circles into . A one–component link is called a knot. The link is oriented when an orientation of the components is chosen.
A (rational) framing of a link is an assignment of a rational number to each component of the link. It is called when all numbers assigned are integral. A link diagram of a framed link is a generic projection onto the plane as depicted in Figure 1.1, where the framing is denoted by numbers next to each component.
The linking number of two components and of an oriented link is defined as follows. Each crossing in a link diagram of between and counts as or , see Figure 1.2 for the sign. The sum of all these numbers divided by is called the linking number , which is independent of the diagram chosen for .
The linking matrix of a link with components is a matrix with the framings of the ’s on the diagonal and for .
A tangle is an equivalence class by ambient isotopy (fixing ) of smooth embeddings of disjoint –manifolds into the unit cube in with . We define and and call a –tangle if and , where denotes the number of connected components of . Thus a link is a –tangle.
Framing, orientation and diagrams of tangles are defined analogously as for links. See Figure 1.3 for an example of a diagram of an oriented –tangle. Every (oriented) tangle diagram can be factorized into the elementary diagrams shown in Figure 1.4 using composition (when defined) and tensor product as defined in Figure 1.5.
The oriented tangles can therefore be considered as the morphisms of a category with objects , .
In the cube , we define the points for , on the bottom of the cube. An –component bottom tangle is an oriented –tangle consisting of arcs homeomorphic to and the –th arc starts at point and ends at . For an example, a diagram of the Borromean bottom tangle is given in Figure 1.6.
The closure of a bottom tangle is the –tangle obtained by taking the composition of with the element
. See Figure 1.7 for an example.
In [13], Habiro defined a subcategory of the category of framed, oriented tangles . The objects of are the symbols , , where . A morphism of is a –tangle mapping to for some . We can compose such a morphism with –component bottom tangles to get –component bottom tangles. Therefore, acts on the bottom tangles by composition. The category is braided: the monoidal structure is given by taking the tensor product of the tangles, the braiding for the generating object with itself is given by .
1.2 The quantized enveloping algebra
The quantized enveloping algebra is the quantum deformation of the universal enveloping algebra of the Lie algebra . More precisely, it is the –adically complete –algebra generated by the elements and satisfying the relations
where . It has a ribbon Hopf algebra structure with comultiplication (where denotes the –adically complete tensor product), counit and antipode defined by
The universal –matrix and its inverse are given by
where
We will use the Sweedler notation and when we refer to .
As always, the ribbon element and its inverse can be defined via the –matrix and the associated grouplike element satisfies .
By a finitedimensional representation of , we mean a left –module which is free of finite rank as a –module. For each , there exists exactly one irreducible finite–dimensional representation of rank up to isomorphism. It corresponds to the –dimensional irreducible representation of the Lie algebra .
The structure of is as follows. Let denote a highest weight vector of which is characterized by and . Further we define the other basis elements of by for . Then the action of on is given by
where we understand unless . It follows that .
If is a finite–dimensional representation of , then the quantum trace in of an element is given by
where denotes the trace in .
1.3 Universal invariant
For every ribbon Hopf algebra exists a universal invariant of links and tangles from which one can recover the operator invariants such as the colored Jones polynomial. Such universal invariants have been studied by Kauffman, Lawrence, Lee, Ohtsuki, Reshetikhin, Turaev and many others, see [12], [29], [34] and the references therein. Here we need only the case of bottom tangles.
Let be an ordered oriented –component framed bottom tangle. We define the universal invariant as follows. We choose a diagram for which is obtained by composition and tensor product of fundamental tangles (see Figure 1.4). On each fundamental tangle, we put elements of as shown in Figure 1.8. Now we read off the elements on the –th component following its orientation. Writing down these elements from right to left gives . This is the –th tensorand of the universal invariant . Here the sum is taken over all the summands of the –matrices which appear. The result of this construction does not depend on the choice of diagram and defines an isotopy invariant of bottom tangles.
1.4 Definition of the colored Jones Polynomial
Let be an –component framed oriented ordered link with associated positive integers called the colors associated with . Remember that the –dimensional representation of is denoted by . Let further be a bottom tangle with . The colored Jones polynomial of with colors is given by
For every choice of , this is an invariant of framed links (see e.g. [32] and [13, Section 1.2]).
Example 2.
Let us calculate , where denotes the unknot with zero framing. For , we have . We choose for the basis described in Section 1.2. Since , we have
We will need the following two important properties of the colored Jones polynomial.
Lemma 3.
[20, Lemma 3.27]
If is obtained from by increasing the framing of the th component by 1, then
Lemma 4.
[24, Strong integrality Theorem 2.2 and Corollary 2.4]
There exists a number , depending only on the linking matrix of , such that . Further, if all the colors are odd, .
1.5 Cyclotomic expansion of the colored Jones polynomial
Let and have and components. Let us color by fixed and vary the colors of .
For non–negative integers we define
where we use from –calculus the definition
For let
Note that if for some index . Also
The colored Jones polynomial , when is fixed, can be repackaged into the invariant as stated in the following theorem.
Theorem 5.
Suppose is a link in , with having 0 linking matrix. Suppose the components of have fixed odd colors . Then there are invariants
(1.1) 
such that for every
(1.2) 
When , this was proven by K. Habiro, see Theorem 8.2 in [12]. This generalization can be proved similarly as in [12]. For completeness, we give a proof in the Appendix. Note that the existence of as rational functions in satisfying (1.2) is easy to establish. The difficulty here is to show the integrality of (1.1).
Remark 6.
Since unless , in the sum on the right hand side of (1.2) one can assume that runs over the set of all –tuples with non–negative integer components. We will use this fact later.
Chapter 2 Quantum (WRT) invariant
In this chapter, we describe in Section 2.1 a one–to–one correspondence between 3–dimensional manifolds up to orientation preserving homeomorphisms and links up to Fenn–Rourke moves. We then state in Section 2.2 results about generalized Gauss sums and define a variation therefrom. We use this in Section 2.3 where we give the definition of the quantum (WRT) invariants and, for rational homology 3–spheres, a renormalization of these invariants. Finally, we describe the connection between the quantum (WRT) and invariants.
2.1 Surgery on links in
Let be a knot in and its tubular neighbourhood. The knot exterior is defined as the closure of .
A 3–manifold is obtained from by a rational 1–surgery along a framed knot with framing , when is removed from and a copy of is glued back in using a homeomorphism . If , the –surgery is called integral. The homeomorphism is completely determined by the image of any meridian of . To describe this image it is enough to specify a canonical longitude of and an orientation on and . The image will then be a simple closed curve on isotopic to a curve of the form , where and are given by the framing of the knot. The canonical longitude is, up to isotopy, uniquely defined as the curve homologically trivial in and with . For the orientation on and , we choose the standard orientation on which induces an orientation on . The two curves and are then oriented such that the triple is positively oriented. Here is a normal vector to pointing inside , see Figure 2.1.
Theorem 7 (Likorish, Wallace).
Any closed connected orientable 3–manifold can be obtained from by a collection of integral 1–surgeries.
Therefore any ordered framed link gives a description of a collection of 1surgeries and the manifold obtained in this way will be denoted by . Let be an other link in . Surgery along transforms into . We use the same notation to denote the link in and the corresponding one in .
Example 8.
The lens space is obtained by surgery along an unknot with rational framing . Further we have , where denotes the unknot with framing 1.
In [18], R. Kirby proved a one–to–one correspondence between 3–manifolds up to homeomorphisms and framed links up to the two so–called Kirby moves. In [9], R. Fenn and C. Rourke showed that these two moves are equivalent to the one Fenn–Rourke move (see Figure 2.2) and proved the following.
Theorem 9 (Fenn–Rourke).
Two framed links in give, by surgery, the same oriented 3–manifold if and only if they are related by a sequence of Fenn Rourke moves. A Fenn Rourke move means replacing in the link locally by or as shown in Figure 2.2 where the non–negative integer can be chosen arbitrary. The framings and on corresponding components and (before and after a move) are related by .
Proof.
See [9]. ∎
2.2 Gauss sums
We use the following notation. The greatest common divisor of two integers and is denoted by . If does (respectively does not) divide , we write (respectively ).
Further, for odd and positive, the Jacobi symbol, denoted by , is defined as follows. First, . Then for prime, represents the Legendre symbol, which is defined for all integers and all odd primes by
Finally, if , we put
Let where denotes the positive primitive th root of . The generalized Gauss sum is defined as
The values of are well known:
Lemma 10.
For we have
and for
where is defined such that and if and if .
Proof.
See e.g. [21] or any other text book in basic number theory. ∎
Let be a unitary ring and . For each root of unity of order , in quantum topology, the following sum plays an important role:
Here stands for either the Lie group or the Lie group and and . If , is always assumed to be odd.
Let us explain the meaning of the ’s. Roughly speaking, the set corresponds to the set of irreducible representations of where is a root of unity of order .
In fact, the quantum invariants (see next Section 2.3) were originally defined by N. Reshetikhin and V. Turaev in [32, 33] by summing over all irreducible representations of the quantum group where is chosen to be a root of unity of order . The quantum group is defined similarly as (see Section 1.2) where stands for . Similarly as for , there exists exactly one free finite–dimensional irreducible representation of in each dimension. In the case when is chosen to be a primitive th root of unity, only the representations of dimension are irreducible (see e.g. [20, Theorem 2.13]). Since for every irreducible representation of there is a corresponding irreducible representation of the Lie group , the invariant is sometimes also called the quantum (WRT) invariant.
Kirby and Melvin showed in [20, Theorem 8.10], that summing over all irreducible representations of of odd dimension also gives an invariant. Since the Lie group is isomorphic to , where stands for the identity matrix, a representation of dimension of is a representation of if and only if is the identity map. This is true if and only if is odd. Therefore it makes sense to call the invariant introduced by Kirby and Melvin the quantum (WRT) invariant.
As already mentioned above, if is chosen to be a root of unity of order , the irreducible representations of are actually of dimension , and not which we use as upper bound in . But summing up to makes all calculations much simpler and due to the first symmetry principle of Le [24, Theorem 2.5], summing up to or up to does change the invariant only by some constant factor.
For a root of unity, we define the following variation of the Gauss sum:
Notice, that for , is well–defined in since, for odd , .
In the case , the Gauss sum is dependent on a th root of which we denote by .
For an arbitrary primitive th root of unity , we define the Galois transformation
which is a ring isomorphism.
Lemma 11.
Let be a primitive th root of unity and . The following holds.
In particular, is zero for odd, and is nonzero in all other cases.
Proof.
It is enough to prove the claim at the root of unity and then apply the Galois transformation to get the general result.
For , we have odd and
which is nonzero for all and odd .
For , we split the sum into the even and the odd part and get
where and and . Therefore, can only be zero if odd and equal zero, i.e. . Since is a primitive th root of unity, this is true if and only if . ∎
Example 12.
For , we have
(2.1) 
where . Further, for the case, fixing the th root of as , we get