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Skein modules and character varieties Clay, Adam Joseph 2005

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S K E I N M O D U L E S A N D C H A R A C T E R VARIETIES by A D A M JOSEPH C L A Y B.Sc. The University of King's College, 2003 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T OF T H E R E Q U I R E M E N T S FOR T H E D E G R E E OF M A S T E R OF SCIENCE in T H E F A C U L T Y OF G R A D U A T E STUDIES Mathematics T H E U N I V E R S I T Y OF BRITISH C O L U M B I A September 2005 © Adam Joseph Clay, 2005 Abstract We present a survey of the theory of skein modules of manifolds, and an intro-duction to skein algebras of groups. By applying a trick of Doug Bullock, we use 51/(2, C) character varieties to highlight some infinite linearly indepen-dent families of knots in the Kauffman Bracket skein module of a 3-manifold. These families are composed of a knot K, together with all (1, n)-cablings of K. We also exhibit a method of explicit computation based upon the work of Robert Riley, which can identify infinite linearly independent families in the skein algebras of 2-bridge knot groups. ii Table of Contents Abstract ii Table of Contents iii List of Figures ' iv Chapter 1. Introduction 1 1.1 Motivation 1 1.2 Introduction 2 1.2.1 Definition of Knots and Links and Equivalence 2 1.2.2 Diagrams and Local Diagrams 3 Chapter 2. Skein Modules 7 2.1 Skein Modules by Example 7 2.2 The Kauffman Bracket Skein Module 12 2.3 The Relationship Between TTI(M) and 5 ( M ; R,a) 18 2.3.1 The skein module of an Abelian group 25 Chapter 3. Connections with SL (2, C) Character Varieties 29 3.1 The SL(2, C) Character Variety of a Group 29 3.2 The Connection with Skein Algebras 36 3.3 A n Application of Character Varieties 39 3.4 Hyperbolic Knots 46 3.5 Comments 48 Chapter 4. Some Infinite Families of Linearly Independent Knots 49 4.1 Character Varieties of Hyperbolic Knot Complements 49 4.2 Character Varieties of General Knot Complements 50 4.3 A n Explicit Computation for 2-bridge knots 52 4.3.1 Computational Applications to an Infinite Family of 2-Bridge Groups 58 4.4 Questions for Future Research 62 Bibliography 64 iii List of Figures 1.1 A torus with preferred longitude, and the corresponding ribbon 3 1.2 Diagrams of the trefoil and cinqfoil 4 1.3 A tangle diagram 5 1.4 A tangle for which dB n L = four points 5 2.1 The appearance of the links L\, Li in some 3-ball in M 8 2.2 Labelling the arcs inside Bz 9 2.3 The twisted disk with boundary a - c + b - d 9 2.4 The appearance of K\ and K2 in some 3-ball 10 2.5 A n abstract depiction of the surface F 10 2.6 The surface F with broken handles 11 2.7 The appearance of the links L + , LQ and in some 3-ball in M 15 iv Chapter 1 Introduction 1.1 Motivation Skein modules are the foundation of a new algebraic approach to manifold theory, which has knot theory at its core [22], and this new approach is par-ticularly applicable to manifolds of dimension three. Specifically, the skein module of a 3-manifold M is a module whose structure may allow one to dis-tinguish M from many other 3-manifolds. As well, if one succeeds in finding a basis for the skein module of M, then this basis provides an entire family of invariants of links in M. Previous invariants of manifolds, such as the homology or fundamental group of a 3-manifold, have been based upon equivalence relations between submanifolds of M that are relatively weak - namely the relations of being homologous or homotopic. Skein modules involve an importation of classi-cal knot theory into arbitrary 3-manifolds, where one uses the relationship of "similarity of knots" to construct this new algebraic invariant, the skein mod-ule. As most geometric relationships between knots run deeper than (for ex-ample) either homology or homotopy, this new algebraic invariant will likely out-perform classical structures in many ways, as well as providing an arena for the study of knot theory in an arbitrary manifold. At present, one of the main objects of study in the theory of skein modules is the Kauffman bracket skein module, which is a specialization based upon the well-known Jones polynomial of knots. However, as has happened in the past with new structures in algebraic topology, the problem of computing the Kauffman bracket skein module of a given manifold is initially proving to be 1 Chapter 1. Introduction 2 quite difficult. Doug Bullock observed a close relationship between the SL(2, C) charac-ters of the fundamental group of M (characters in the sense of representation theory) and the Kauffman bracket skein module [6]. This has provided a con-nection via which one may be able to translate intractable topological ques-tions about the Kauffman bracket skein module into approachable problems in the world of algebra. New results concerning the structure of this module will have direct im-pact on the theory of links in an arbitrary manifold, as well as the theory of 3-manifolds, both of which are interrelated quite closely. However, some re-sults may have impact beyond even the scope of mathematics, as there is a very explicit relationship between the Kauffman bracket skein module and modern physics, via quantum invariants [1]. 1.2 Introduction 1.2.1 Definition of Knots and Links and Equivalence A knot in a 3-manifold M (typically M = S3) is a piece wise-linearly em-bedded circle: S 1 M. Alternatively we can require that the embedding be smooth, these restrictions can be shown to be equivalent. A link L is a collec-tion of disjoint circles, also piecewise-linearly embedded in M n i=l A knot or link is oriented if each copy of S1 is assigned a preferred orien-tation. A knot or link is called framed if we think of each copy of S1 as a skinny solid torus, together with a preferred longitude on its boundary. This longitude is sometimes referred to as the framing. A better way of thinking of "preferred longitudes" is to think of the knot as an embedded ribbon, instead of a skinny torus with preferred longitude. The two ideas are equivalent, by taking the preferred longitude to be one side of the ribbon: Two knots are considered equivalent if they are ambient isotopic. The for-mal definition of ambient isotopy is as follows: Chapter 1. Introduction 3 Figure 1.1: A torus with preferred longitude, and the corresponding ribbon Definition 1.1. A n isotopy between knots K\ and K<2 in a 3-manifold M is a homotopy H(x,t) between K\ and Ki, such that for each fixed t, the map H(—,t) is a piecewise-linear embedding. A n ambient isotopy between K\ and K2 is an isotopy of M to itself that carries K\ to K2. This means that the ambient space is doing the deforming, and the knots simply "come along for the ride". For framed knots, the isotopy must carry the preferred longitude of one knot to the preferred longitude of the other, or equivalently one ribbon must be carried onto the other. This formal definition is meant to model the cornmonplace idea of untan-gling/knotting a string. In plain English, two knots are equivalent if one can be manipulated (without the string breaking or passing through itself) so that it appears exactly as the other. This notion of equivalence generalizes to links in the obvious manner. 1.2.2 Diagrams and Local Diagrams In the special case where our ambient space is R 3 or 5 3 , there is a less cum-bersome way of thinking of knots and links. Given a knot K we can create a diagram of K, which is essentially a picture of K obtained by projecting into a plane. To the image of the projection, we add information at doubly cov-ered points to create crossings, allowing us to recover the original knot from a diagram. This gives us things as in figure 1.2. Chapter 1. Introduction 4 The trefoil The cinqfoil Figure 1.2: Diagrams of the trefoil and cinqfoil Here, the crossings are the regions appearing as and the 'added infor-mation' is the break in one of the arcs; indicating that the broken arc appears below the unbroken arc in our original knot. The equivalence relation of ambient isotopy can be carried into the world of diagrams. Two diagrams are equivalent if one can be reached from the other via a sequence of planar isotopies and Reidemeister moves, which are local diagram manipulations. There are three such moves: Theorem 1.2. (Reidemeister) Two knots (in S3) are ambiently isotopic if and only if their diagrams are related via a sequence of Reidemeister moves and planar isotopies. A more strict definition of the Reidemeister moves and a proof of this theorem can be found in [26]. If we are not working within 5 3 or M 3 , then global projections are not possible, but we do have "local" diagrams. Suppose that we have a link L in M , and some 3-ball B in M such that B n L / 0. Then we can produce Chapter 1. Introduction 5 a diagram of the part of L that lies inside B by projecting the contents of B onto some plane, and creating crossings as before. The intersection B D L is called a tangle. The part of L which lies outside of B is often referred to as the external wiring of the room B. A tangle diagram appears as: Often we restrict ourselves to tangles with dB n L — four points, as in figure 1.4. Two tangles in B are equivalent if one can be obtained from the other via isotopies of B which fix the points in dB D L. Accordingly, we can rephrase this equivalence in terms of Reidemeister moves on tangle diagrams. A common way of communicating the structure of a link L in some part of a manifold M is to say that L "looks like" a given diagram inside some 3-ball B. This colloquial language is just a way of communicating that there exists a series of ambient isotopies of the tangle LHB, and some plane of projection so that L fl B can be projected to the given diagram. We also sometimes say that a certain equation involving diagrams is true "by tilting your head". This means that the desired equation results from the given equation by rotating all diagrams in the given equation in the same Figure 1.3: A tangle diagram Figure 1.4: A tangle for which dB n L = four points Chapter 1. Introduction 6 direction - usually by 90 degrees. For example, the implication <8>-©-<8>-© follows by tilting your head. We cannot say the same of equations that do not involve pictures. For example, even though 813 = 810 + 3 it is pure nonsense to say that oo oo — = — +w. w 0 Chapter 2 Skein Modules 2.1 Skein Modules by Example We would like to build a setting for the study of knot theory in an arbitrary 3-manifold. Let R be a commutative ring with 1, M an orientable 3-manifold, and £ the set of all links in M considered up to ambient isotopy, including the empty link. If the links in C are oriented we add a subscript "o", and if they are framed we add a subscript "/"• Thus, we have things like C0j; the set of all oriented, framed links in M. Let RC (resp. RC0, etc.) be the free i?-module with basis C (resp C0, etc.). The idea of skein module theory is to start with this structure, and then clev-erly select some family of relations between elements in RC. The skein mod-ule of a 3-manifold M with coefficients in R is then an algebraic invariant of The relations that one chooses can vary greatly. If the relations are too weak, the resulting skein module may be intractible, and of no use. If the relations are too strong, the resulting module may contain no useful information. We illustrate this idea with a very geometric example. We begin with a simple relation that allows us to eliminate all the cross-ings in any given knot. Suppose that L\ and L2 are two oriented knots in a 3-manifold M , and that they are identical everywhere in M, except in a 3-ball where they differ as in figure 2.1. M: relations 7 Chapter 2. Skein Modules 8 Figure 2.1: The appearance of the links L\, L2 in some 3-ball in M In this case, we set L\ = L2 in the skein module, sometimes called 'smooth-ing a crossing'. We write: S2{M) = Remark 2.1. As a consequence of this single relation, we can compute: so that we can smooth crossings of the opposite orientation. With this choice of relations, we find 52(M) = RHi(M; Z ) via the homo-morphism 4>:RC0^ RHi(M;Z), <j>(L) = [L] where we extend this definition linearly to all i?-linear combinations of links. This fact is found in [16], we offer a proof here. Here [L] is the equivalence class of L in the first homology of M. First, to see that <f> descends to an homomorphism $ : 5 2 (M) ->RHi(M;Z), suppose that we have links L\ and I_ i n M that differ in some 3-ball J53, as depicted in figure 2.1. In this case, we break L\ and L2 into arcs (think: 1-chains in singular homology) which lie inside S 3 , and arcs which lie outside B3. Label the arcs inside B3 as in figure 2.2. Outside of B3, L\ and L2 share Chapter 2. Skein Modules 9 Figure 2.2: Labelling the arcs inside Bz the same external wiring, and so denote this common external wiring by e. Then we compute: <j){Ll-L2) = [Ll}-[L2} = [a + b + e]-[c + d + e},_ where addition inside the square brackets corresponds to addition in the first homology group, and the subtraction is taking place inside the group ring RHi(M;-Z). We can see that [a + b + e] = [c + d + e] in Hi(M;Z), as the difference: (a + b + e) — (c + d + e) = a — c + b — d is the boundary of a "twisted" disk, so that [a + b + e] — [c + d + e] = 0 in Figure 2.3: The twisted disk with boundary a — c + b — d RHi ( M ; Z ) . Thus the map $ is well-defined. It is evident that $ is a surjection, because any element of the homology group H\(M\ Z ) can be represented by some link in C0. To see that $ is an monomorphism, we need a lemma: Lemma 2.2. Given any link L, we may choose a representative V of the equivalence class of L in S2 (M) such that L' single copy ofS1. Proof. It suffices to consider the case when L = K\ U K2 has only two com-ponents. We can isotope K\ and K2 so that they are very close to one another Chapter 2. Skein Modules 10 K2 Figure 2.4: The appearance of K\ and K2 in some 3-ball inside some 3-ball in M , and appear as in figure 2.4. Then in S2{M), we find the equality: so that we can apply this equation to figure 2.4 to find V, where L' is the Now suppose that we have two links L\ and L2 such that = &(L2), i.e. $ ( L i - L 2 ) = 0. By the above lemma and by well-definedness of we may assume without loss of generality that L\ — L2 is a knot K, which is mapped to zero under $. This means that K is homologous to zero, so there exists a surface F such that dF = K. By the classification of surfaces, F must ab-stractly appear as in figure 2.5. However, from the calculation in our lemma, connected sum Ki#K2. • K Figure 2.5: A n abstract depiction of the surface F we know that we can break each handle using the relation Chapter 2. Skein Modules 11 smoothing 0 to get a new representative K' of the equivalence class of K, as in figure 2.6. no handles Figure 2.6: The surface F with broken handles Evidently K' is the boundary of a disk, and so it is the unknot. This trivial representative of L\ — Z_ shows that we must have L\ — Z_ = 0 in SiiM), so that $ is an isomorphism $ : S2{M) -> RH\(M\Z) as claimed, thus completing the proof. We will see shortly that by choosing different relations, we can recover a structure based upon the fundamental group of the manifold as well. It is important to maintain the distinction between links and their dia-grams when working through proofs of this nature. The relation (^) ~ Q is a relation between links in M, and not between diagrams of links. At present it is only known how to diagrammatically encode knots and links in S3 or in a handlebody (virtual knots), there is no known way of creating diagrams of links in an arbitrary 3-manifold. For this reason, attempting such proofs in a diagrammatic manner can sometimes lead one astray. Chapter 2. Skein Modules 12 2.2 The Kauffman Bracket Skein Module In this section, we attempt to choose a more useful set of relations. To motivate this choice, be begin with some combinatorics. Suppose that we are working in S3, or M 3 , so that every link under con-sideration admits a diagram. One of the most powerful link invariants in this setting is the Jones polynomial. The Jones polynomial of a link L in S3 can be computed from a diagram of L by using the Kauffman bracket of a link, de-noted (L), which is a polynomial in Z[a ,a - 1 ] . The Kauffman bracket (L) is defined according to.the following recursive (local) diagram manipulations [17]: .«8»=-<©>+^ <©> <"> (L UO) =-(a2+ a-2)(L) (2.2) < O > = 1 (23) We are to interpret each of these equations as rules for mechanically de-composing a diagram into a union of disjoint circles, by eliminating crossings. One finds that since these rules produce only local changes in a knot diagram, crossings can be eliminated in any order, with no effect on the outcome of the calculation. Therefore, eliminating all crossings in a link diagram gives rise to a well-defined polynomial (L), for which there is an explicit combinatorial formula. Next we assign an orientation to the diagram of L to create L0, and com-pute the writhe of L0 (denoted w(L0)). First, assign to each crossing point p in the diagram a value e(p), which is either +1 or - 1 , according to the conven-tion: Then if X is the set of all crossing points in the diagram of L0, we define: w(Lo) = E £(P) We are now in a position to make the following definition: Chapter 2. Skein Modules 13 Definition 2.3. The Jones polynomial of an oriented link L0 in S3 is given by: vLo(t) = - « - M L o ) W i a _ a Example 2.4. We compute the Jones polynomial of the Hopf link by first com-puting the Kauffman Bracket: (GD> - <JDMG£» - +{G5>-'(GD>] * -KQ>-(GD)] Since all the brackets now contain only circles, at this point we may use equation (2.2) to reduce the number of loops inside each bracket to one: = - « V + a" 2)( O ) + 2( O ) - a " V + a~2)( O ) = ( - a 4 - a " 4 ) ( 0 ) Using rule (2.3) we get a final answer of: -(GDH--We are only a step away from finding the Jones polynomial. Depending upon the orientation we choose at this point, we get two possible cases: 1. Hopf+, satisfying w(Hopf+) = 2 2. Hopf^, satisfying w{Hopf-) = —2. This yields the two polynomials VHopf^t) = - a - e ( - a " - a - \ = t _ k = (a-w + a - \ = t _ k = t l + ^ and VHoPfAt) = -a6(-a4-a-i)\ i =(a 1 0 + a 2)| > = n + n Chapter 2. Skein Modules 14 This illustrates a general Fact: If L\ and L2 are identical, except for having opposite orientations, then vL,(t) = vLl(t-1). That this construction defines an oriented link invariant follows from a check that the bracket is invariant under Reidemeister moves II and III, and that the factor a~3w(L°} provides invariance under Reidemeister move I. More details can be found in the original expositions of this idea [18], [19]. This method of calculation using diagrams can be formally justified in the following way: Let V be the set of all knot diagrams, considered up to Reidemeister moves, including the empty diagram. Thus two diagrams in V are considered "the same" if one can be obtained from the other by a sequence of Reidemeister moves. Then if we let R = Zfa^a - 1 ] , the equations (2.1) and (2.2) can be interpreted as equivalences taking place in RV. Let X be the smallest ideal in RV generated by these equivalences, i.e. the smallest ideal containing all expressions of the form: «8»-<©>--<©> <^> and ( L u O ) + ( « 2 + a- 2 )(L}, (2.5) define S = RV/1. Note that in equation 2.4 we have reinterpreted (•), using the brackets to indi-cate that the diagrams in equation 2.4 differ only locally; with the differences appearing as indicated. Within this formal framework, we can properly interpret the calculations in example 2.4 as computing a "nicer" representative of the equivalence class of the diagram (^{^) m S. It is only natural to wish to extend this computational technique to embed-dings S1 M for arbitrary 3-manifolds M. However, this clever machinery of Kauffman is impotent if we adhere to our diagrammatic interpretations, as links in an arbitrary 3-manifold do not admit diagrams. We therefore think in a more general setting. Chapter 2. Skein Modules 15 Definition 2.5. (Przytycki, [22]) Let M be an oriented 3-manifold, R a commu-tative ring with identity, and a 6 flan invertible element. Let B3 denote an arbitrary 3-ball in M. Suppose that L+, LQ and are three links in M that differ from one another only inside B3, where they can be projected to appear as depicted in figure 2.7. Figure 2.7: The appearance of the links L+, L$ and in some 3-ball in M When such a situation exists, the expression L+ — aLo — a~lL00 is called the corresponding skein expression. Let ,!>2,oo be the smallest submodule of RC/ generated by: 1. A l l skein expressions 2. A l l expressions of the form LliQ + (a2 + a~2)L, where L is any link in M and O is the unknot in M. Define the Kauffman bracket skein module1 to be the quotient: S2,oo{M;R,a) = RCf/S2,oo. Hereafter, we suppress the subscript 2, oo when the context is clear. Example 2.6. It is clear that the definition of S = Z[a, a~l]V/l on the previ-ous page and the definition of 5(5 3 , Z[a, a - 1 ] , a) are analagous in some sense, since Reidemeister moves are equivalent to ambient isotopy - the difference to be accounted for is framing. Consider the composition: S(S3, Z[a, a'1}, a)^S A Z[a, a'1] 1The subscript 2, oo in our notation arises from the paper [16], where these subscripts are meant to refer to certain 4-tangles. Chapter 2. Skein Modules 16 where the arrows above have action L h+ diagram ofL - (L) ( Q ) i-> (-a 2 - a~2)(L), each extended linearly to be maps of Z [ a , a _ 1 ] modules. We create our dia-gram via cf> as follows: Suppose that we plan on projecting a link L into some plane P to create our diagram. Before projecting, we isotope L so that the speficified meridian of each framed component is 'parallel' to P, so that an additional twist in the specified meridian contributes ±1 to our writhe: pAsotopy With this convention for creating diagrams, the composition (V>° «/>)(£) H - a 2 - a - 2 ) ( L ) is: 1. Well-defined, because our convention for creating diagrams via <f> en-sures that the different possible diagrams for a framed link all have equal writhe; and because the equivalence relations used in defining S(S3, Z[a , a - 1 ] , a) are precisely the defining relations of the Kauffman bracket. 2. Surjective, as ( ^* ) (p (< i l «" 1 ) -0) = p ( a , a " 1 ) for any polynomial p(a, a~l) 6 Z [ a , a - 1 ] . 3. Injective, as any two links differing by a skein or framing relation will (by definition) have different Kauffman brackets. We conclude that <S(53, Z[a, a - 1 ] , a) ^ Z [a, a x], in fact we can see that <S(5 3 ,Z[a,a - 1 ],a) is free on the basis 0. Chapter 2. Skein Modules 17 For an arbitrary 3-manifold M, one may wonder if S(M, Z[a, a - 1 ] , a) ad-mits such free bases - or at least an amenable set of generators. It is known that not all such modules are free, but it is not known which modules are free and which are not. Computing bases and sets of generators is an extremely difficult problem over which much ink has been spilled. Remark 2.7. Suppose that we can find a free basis for some S(M; R, a), say {[LdJLaULa],...}. Then if we are given any knot K in M, [K] e S(M\ R, a) has a unique representation relative to this basis as a finite sum: n [K] = ri[Li] for some n. i=i In this case the ring elements {r\,r2,rs, • • • } form a set of invariants of K in M, each analagous to the Jones polynomial in the case ofS3. In this definition we have chosen our variable a arbitrarily, but a more deliberate choice can simplify the matter. If we choose our invertible ring el-ement to be —1, then the skein module S(M; R, — 1) enjoys additional struc-ture. Lemma 2.8. S(M; R, -1) is a commutative algebra, with the product of two links given by taking their disjoint union, and identity [0]. Proof. Distributivity of this product follows immediately, since we extend our pair-wise definition of the product to all formal sums of links in precisely the way which conforms to the distributive law. Therefore, at issue is the com-mutativity and associativity of this product. Observe that with a = - 1 , skein relations become: where the second equality follows from tilting one's head by 90 degrees. Thus, for any link L, [L] is independent of crossing changes. In particular this means that [Li] • = [Li U L2] is independent of the relative positioning of L\ and L2, so that [Li] • [L2] = [Lx U L2] = [L2 U L i ] = [L 2 ] • [L{\. Chapter 2. Skein Modules 18 In the same manner we can argue that triple products are independent of our choice of bracketing, so that the product is associative. That the empty set is the identity is immediate from our definition. • There is an easier way of thinking of elements in this specialized skein module. In this skein module, we allow crossing changes. It is a well-known fact (cited in [24]) that considering embedded graphs up to ambient isotopy and crossing changes is equivalent to considering embedded graphs up to homotopy. Therefore in the specialized skein module S(M; R, -1), two links are equivalent if and only if they are homotopic. 2.3 The Relationship Between m (M) and <S(M; R, a) We begin with the definition of a tensor algebra of a module. This is a stan-dard algebraic object, but is not so commonly discussed in algebra classes. Definition 2.9. Let R be a commutative ring, and M an .R-module. Let T°(M) = R Tl{M) = M T2(M) = M <g>M and in general Tr{M) = Af ®---® Af where the tensor above is taken r times. Define the tensor algebra over M to be oo T M = 0 T f c ( M ) k=0 In the langauge of category theory, T is a functor whose action on objects is given by the above equation, and whose action on maps is given by the formula: T(/)(rai <g> • • • <g> mr) = / (mi) <g> • • • <g> f(mr). The object T M arises naturally as the 'free' algebra over a module, in the sense that T is left adjoint to the forgetful functor U mapping from fl-algebras to R-modules. Therefore in a sense, it is not a surprising or artificial structure to r Chapter 2. Skein Modules 19 come across. Multiplication in the algebra T M is given by the tensor product of two elements, and the necessary distributive and associative properties of algebra multiplication follow immediately from the bilinearity and associa-tivity of the tensor product. We use tensor algebras in the following construction, due to Przytycki and Sikora in [24]: Definition 2.10. Let G be a group with identity e, and J? a comutative ring with 1. Denote the group ring over G with R coefficients by RG. Let I be the ideal of TRG generated by the expression e — 2 (here 2 = 1 + 1 € R), together with all expressions of the form: 1. g ®h — h® g 2. g ® h — gh — gh~l where g,h € G. Define the skein algebra of the group G with coefficients in R to be S{G;R) =TRG/I. The elements of a skein algebra are therefore (equivalence classes of) for-mal sums of tensors of elements in G, weighted with coefficients from R. We use square brackets to denote the equivalence class of an element in S(G; R), for example [g\ ® g2 ® 53]. If we fix a ring R, then S(—; R) is a functor from the category of groups to the category of .R-algebras. The action of «S(—; R) on maps is to send a group homomorphism 4> '• G -» G' to the map denoted 4>* • S(G;R) ->• S(G';R), whose action is given completely by: Mb]) = Mg)] for all g G G. The skein algebra of a group satisfies the following properties, whose proofs are largely computational [24]. 1. For any g € G, [g] = [g'1]. Proof. [g <g> e] = [ge + ge'1] = [g + g] = [g] + [g], Chapter 2. Skein Modules 20 whereas we may also compute [g ® e] = [e <g> g] = [eg + eg'1} = [g] + [g'1] so that [g] + [g] = [g] + [g'1], giving [g] = [g'1]. • 2. For any pair g,h E G, we have [gh] — [hg], and consequently [(hg)h-1] = [h-1(hg)] = \g]. Proof. 0 = \g®h]-[h®g] = [gh] + [gh'1) - [hg] - [hg-1] = [9h] - [hg] - [hg'1] + [(hg-1)'1] = [gh] - [hg] where in the last step, the cancellation of [/ag-1] and [(hg-1)-1] follows from property (1). • 3. The Universal Coefficient Theorem If 4> : R <-» R' are rings, and we regard R' as an i?-module with multiplication given by r • r' — (p(r)r' for any r G R and r' G R', then S(G; R') ^ S(G; R) ®R R'. Proof. The proof consists of showing that u : S(G;R)®RR' —> S(G;R'), defined by u([g] ®R r') = [r'g] is an isomorphism. Let X(R) denote the ideal of TRG constructed in definition (2.10), and 1(R!) the analgous ideal in TR'G. Then we have an exact sequence 1(R) TRG -» S(G; R) ->• 0, and because the functor ®R R' is right exact ([11], pp. 378-383), we obtain a second exact sequence X(R) ®R R' -> TRG ®R R' -> S(G- R) ®R Rl ->• 0. Chapter 2. Skein Modules 21 This second exact sequence fits into the commutative diagram I(R) ®R R' • TRG ®R R' > S(G; R) ®R R' > 0 h h « (2.6) l(R') • TR'G > S(G;R') > 0. Here, fa is the isomorphism given by f2({r • gi ® g2 ® • • • ® 9k) ® r') = (r'r) • gx <g> g2 ® • • • ® gk-To see that fa is an isomorphism, note that TRG ®RR' = {R®RG®RG®RRG®---)®RR' = (R ®R R') © (RG ®R R') © {RG ®R RG ®RR')®---since tensors distribute over arbitrary direct sums. Therefore we can consider fa as a map (R ®R R') © (RG ®R R') © (RG ®R RG ®R R') © • • • -> TR'G defined on each component of the direct sum by the same formula as before. We can then see that fa is an isomorphism component-wise on this direct sum, as the restriction of fa is the well known isomorphism (RG © • • • © RG) ®R R' = R'G © • • • © R'G arising from extension ofscalars. We obtain fa by simply restricting fa to the subalgebra T(R) ®R R', and so fa is clearly surjective. These facts allow us to apply the five lemma to diagram 2.6 to conclude that u is an isomorphism. • Remark 2.11. Via an identical proof, we can show that the Universal Coefficient Theorem holds for topologically defined skein modules. Namely, if r : R —> R! is a homomorphism of rings, then S(M; R,a)®R'^ S(M; R',r(a)). (Recall that we have chosen to suppress the subscript 2, oo.) Chapter 2. Skein Modules 22 At last the promised connection with the fundamental group of a manifold emerges, following a proof presented in [24]. Theorem 2.12. If M is a 3-manifold and R is a commutative ring with 1, then S(M;R,-l) = 5(71-1 (M); R). Proof. Define a function on framed links, ip : Cf -> S(iri(M); R), in the fol-lowing manner: Suppose that K is a knot in M , and that -K\(M) is calculated with respect to the base point XQ G M. Then we can connect K to XQ via a path a, yielding a representative element aKoT1 of a conjugacy class in TTI(M) (In doing this we have arbitrarily assigned an orientation to K). Let K denote this conjugacy class. Define ip by the rules ip(K) = —[K] and i/>(0) = —1. First, note that ip is well-defined, as property (1) of group skein modules shows that our choice of orientation for K does not affect ip{K), and property (2) of group skein modules shows that we may connect K to our base point using any path we please, so that our choice of a does not affect ip{K). Suppose a link L in M has components K\, • • • ,Kn. We define iponL G Cf according to the rule iP(L) = V>(#i U • • • LI #„) = {-l)ni>{Kx) ® • • • ® ${Kn) i.e., we extend to all elements in Cf precisely the way that agrees with the algebra multiplication. After extending multiplicatively to all of Cf, we then extend ip linearly to all of RCf. Note that there is no problem regarding the ordering of i/)(Ki), • • • , ip(Kn) in our product, because tensor products have been made abelian. Next we check that ip descends to an algebra homomorphism i> : S{M- R, -1) -> 5(7ri (M) ; R). With our choice of a = — 1, the relations defining <S(M; R, -1) become LUQ + 2L and Chapter 2. Skein Modules 23 In this equation, O is a loop that is contained in some 3-ball, so we can directly compute # U O + 2L) = (-1)^(1) ® V(O) " 2V(£) = [e - 2]V>(£) = 0 by noting that O is the identity in iri(M), and so [O] = [e]. Dealing with the second relation is trickier. First, observe that we can write any skein relation as Li U L+ + Li U L0 + Li U Loo = LX U (L+ + L0 + L M ) where L+ and LQ are knots, and L^  is a two component link. We do this by absorbing into L\ all components of our link, except for the component that intersects our 3-ball of interest. With this choice of L\ we know that one of {LQ, L^} is a two-component link, and the other is a knot. Without loss of generality, we have chosen LQ to be the knot, and L^  to be the two component link, for we can interchange LQ and Loo in the skein relation by tilting our head. By the above considerations, it suffices to show ?/>(L+  L0 + Loo) =0 in the case where L + and LQ are knots, and L^  = K\ U K2 is a two-component link. In this case, choose a base point inside the 3-ball where L+, Lo and Loo differ. Then by connecting K\ and K2 to our base point and carefully choosing orientations, we get a, 8 e TTI (M) such that 1. a ~ K\ and 8 ~ K2 2. a8 ~ L_|  3. ad'1 ~ LQ. Now we may compute (^L+ + L0+Loo) = ^ ( L + H ^ L o J + ^ i - ^ ) = - [ a f l - ^ - ^ - K a ] ® ^ ] =0 and hence ^ descends to i/> : «S(M; i?, -1) -» 5(^1 (Af); fl). To prove that 1/; is an isomorphism, we define an inverse. Let <f>: Ti?7ri(M) ->S(Af; .R,- l) Chapter 2. Skein Modules 24 be the map defined by <t>{l) = where 7 € iri(M) and K1 is a knot representing 7, whose framing we choose arbitrarily. Note that if we are to have <p(r • a) = r • (j)(a), then we must also have 4>(r) — r • 0 for any pure ring element r G TRwi(M). This map is well-defined because: 1. Two homotopic knots will differ from one another by a sequence of am-bient isotopies and crossing changes, and we can change crossings in S(M;R,-1). 2. Choosing a = -1 makes the knots independent of framing, so we may assign any framing to K^. We can convince ourselves of this pictorially by computing using the Kauffman bracket notation, and with a = - 1 : hMH-(l°MK + 2 Having convinced ourselves of well-definedness, we check that </> descends to a homomorphism 4> : 5(TTI(M); R) -»• S(M; R, - 1 ) . 1. Recalling that <p(r) = r • 0, <f>(e - 2) = - O - 2 • 0 = (a2 + <T2) - 0 - 2 - 0 = 0, where the last equality follows when we take a = — 1. 2 . For any loops a, {3, we compute <i>{a®B-B®a) = Ka-Kp-KirKa = 0 since our product in «S(M; R, —1) is commutative. 3. For any loops a, (3: <t>(a®8-aB- aB'1) =Ka-Kp + KaP + Kafi-i = + L+ + LQ = 0. Chapter 2. Skein Modules 25 The constructed homomorphism cf> is the inverse of ip, and so tp is an isomor-phism as claimed. • We construct an explicit connection with S(M; Z[a, a~l],a) via the follow-ing isomorphisms: Corollary 2.13. For any Z-manifold M, we have 5(7n(M),Q = ^ ( M ; C , - 1 ) *L S{M;Z[a,a-\a) ®z[a,a-i]C Proof. The first isomorphism is the result of our theorem. The second arises from applying the Universal Coefficient Theorem to this special case, taking the map r : R —>• R' to be the map Z[a,a _ 1] -> C defined by a i-> — 1. • It is because of this isomorphism that the algebraic object S(G;R) is of in-terest to us. Knowledge of S(G; R) for some finitely generated group G can be translated into information about S(M; Z[a, a - 1 ] , a), whenever M satisfies 7ri (M) = G. In particular, observe that any linear relationship between ele-ments in S(M;Z[a,a~l],a) translates into a linear relationship between the same elements considered as elements of S(iri(M),C), considered as a com-plex vector space. We can therefore state the following: Fact: If a family of elements in <S(7Ti (M), C) are linearly independent, then the same elements considered in S(M; Z[a, a'1], a) are still linearly indepen-dent. 2.3.1 The skein module of an Abelian group We start with some notation. Definition 2.14. For G a group and R a ring, define sym(RG)to be the subal-gebra of RG generated by elements of the form g + g~l for g 6 G. We first need the following fact in order to tackle the skein module of abelian group. Chapter 2. Skein Modules 26 Theorem 2.15. (Przytycki, Sikora, [24]) Let G be an abelian group. Then considered as an R-module, sym(RG) is a free R-module with basis {e} U [g + < ? _ 1 } s e _ , where 1. B = G-{e} if 2 ^OinR 2. B = {geG:g2^e}if2 — 0 in R Proof. First, note that though sym(RG) is generated as an i?-algebra by ele-ments of the form g+g~l, it is also generated as an R-module by such elements, as direct computation reveals: The critical observation is that the right hand side is a sum of elements of the form g + g~x. To show that we indeed have a basis, suppose 0 ^  2 in i?, and that re + n{gi + gT1) + r2(g2 + g^1) + h r„(#„ + c/"1) = 0 in RG. Then since the elements g € G form a basis for RG, we wish to re-bracket this sum as an .R-linear combination of elements in G so as to draw the desired conclusion - that all are zero. The coefficients distribute over the sums gi + g^1 giving a sum of 2n distinct elements in G, provided,-we do not have g^ = gT1 for some gk- In this case the term 2r^gk appears in our sum. Therefore in general, written as a sum of distinct elements in G, our rebracketing of the sum will have coefficients of the form r; and 2rj. Since 2 ^ 0 , we can still use the fact that g € G is a basis of RG to conclude that all the J-J'S are zero. On the other hand, if 2 = 0 in R, then any term of the form g + g"1 — 2g in such a sum is zero, and hence we arrive at the generating set stipulated in In the above proof, we have used the fact that g 6 G is a basis of RG in order to show that a certain generating set of sym(RG) is a basis. We can now use this basis of sym(RG) in a similar manner, to show that a certain generating set of S(G; R) is in fact a basis in the case that G is abelian. Theorem 2.16. (Przytycki, Sikora [24]) Let G be an abelian group, R a commutative ring with 1. Define </>: TRG -> RG by 4>{g) = g + g'1 for all g s G. Then provided (g + g-^ih + h-1 ) = gh + (gh)'1 + gh'1 + (gh-1) (2). • Chapter 2. Skein Modules 27 either 2 ^ 0 in R, or G has no elements of order 2, <f> descends to an isomorphism of modules $ : S(G; R) -> sym(RG). Proof. The map $ is well defined, as we compute: 1. 4>(e - 2) = <f>(e) - 4{2) = e + e~l - 2e = 0 2. (f>(g <g> h — h <g> g) = (f>(g)<f>(h) — <p(h)(f)(g) = 0, since G is abelian (upon expanding and collecting terms). We now construct a generating set of S(G;R), as an A-module. We know that TRG is generated as an JR-module by all finite tensors of elements of G, together with l e i ? . Therefore this set certainly generates S(G; R) as an R-module. However, we can reduce this set. Using the identity g®h = gh+ghr1, we represent any finite tensor as sum of elements in RG, so we can discard all tensors from the generating set. We have now reduced our generating set to G U {1}. However, further using the fact that e = 2 and that g = g~l in S(G; R), we can reduce this generating set to By definition of our map <fr, we have that $(Ti?G) = sym(RG). By the previous theorem, the set 3. Lastly, 4>(g ® h - gh - gh x) <t>{9)4>{h) - 4>{gh) - <t>{gh~l) {g + g-l){h + h-1) -(gh + igh)-1)-^-1 +g~1h) 0, Y = {g + g-1:g€G-{e}}U{e} Chapter 2. Skein Modules 28 is a basis for sym(RG). As a map of jR-modules, $ carries the generating set X bijectively onto the basis Y, and hence $ is an isomorphism of .R-modules, and so is also an isomorphism of algebras. • This gives us a complete description of S(G;R) in the event that G is abelian. Chapter 3 Connections with SX(2, C) Character Varieties 3.1 The SL(2,C) Character Variety of a Group Let G be a finitely generated group. A representation of G in SL(2, C) is a ho-momorphism p : G —> SL(2, C). Therefore we may think of elements p(g) as invertible linear maps C 2 __) C 2 _ There are two mutually exclusive types of representations: irreducible repre-sentations and reducible representations. A representation is called reducible if there exists a proper subspace V of C 2 such that p(g) fixes V for all g G G. That is, (p(g)){v) G V for all g G G and for all v G V. If a representation is not reducible, then it is irreducible. The character of a representation p is the composition X p : G A SL(2,C) *™ce C. Let X(G) denote the set of all characters of the group G. For each g G G, there is a map T9 : X{G) -»• C defined by T9(xp) = Xp{g)- The maps rg satisfy: 1. For any g € G,rg = rg-i. This follows from the identity tr(A) = tr{A~l), which holds in 51,(2, C). 29 Chapter 3. Connections with 5L(2,C) Character Varieties 30 2. Since trace is invariant under conjugation, rg = Th if g and h are conju-gate elements in the group G. Already we have a surprising result, which was proved independently by many people: Theorem 3.1. (Vogt, Fricke, Horowitz, Culler-Shalen [8]) There exists a finite set of elements {9i,-,9n} C G such that every rg is an element of the polynomial ring C [ r 5 l , T 9 J . Proof. Suppose that G has generators {hi,hm}. Let R be the ring generated by all the functions T/iij ...hiT where the ii,...,ir are distinct positive integers < m. The finite set of elements {h^ ...hiT : i i , i r are distinct positive integers < m} will correspond to the finite set of elements {gi,...,gn} stipulated in the state-ment of the theorem. Under this correspondence, C [ T 9 i , r 9 n ] = R, so we prove that rg 6 R for every g € G. The proof will be an induction, which relies heavily upon the following lemma. Lemma 3.2. For any A, B e 5L(2 ,C) , tr(A)tr{B) = tr(AB) + tr{AB~l). Proof. For any A,Be SL(2, C ) , we find that the characteristic polynomial of B is A 2 - tr(B)X + 1 and so by the Cayley-Hamilton Theorem, B satisfies B2 - tr(B)B + / = 0. We rearrange this expression to give B2 + I = tr{B)B Chapter 3. Connections with SL(2, C) Character Varieties 31 and multiply from the right by B 1 A to find BA + B~lA = tr(B)A which gives tr(BA) + tr{B-lA) = tr{B)tr(A) upon taking the trace of both sides, which can equivalently be written as tr(AB) + tr{AB~l) = tr(B)tr{A) by using tr(AB) = tr(BA). • Remark 3.3. From this lemma we get TgTh = Tgh + Tgh-1 for all g,h G G. We now start our first of two inductions. This induction will show that Tg E R whenever g = h^...h^, where i\,...,ir are distinct integers between 1 and m, and k\,...,kr G Z. We will handle the other elements of G with a second induction. Our first induction is on the positive integer fi, defined by m .7=1 where —kj if kj < 0; kj - 1 if kj > 0. Of course there may be many different ways of writing an element g as a product of generators, so that our definition of u.(g) above may not be well-defined. To remedy this, we take fi(g) to be the minimum arising from all possible ways of writing g as a product of generators. As our base case, note that if fi(g) = 0, then all the kj are either 0 or 1, so that Tg G R by definition of R. Chapter 3. Connections with SL(2, C) Character Varieties 32 Claim: Without loss of generality, we can assume k\ / 0,1. Proof of claim: Since n(g) > 0, there is some A;^  that is not one or zero. Choose d to be the smallest integer for which ^ 1,0. In our argument we can re-place the element g with the conjugate g' = hYdl_i...hr1ghil...hid_1, since rg = Tg*, and because /j,(g) = n(g'), as a quick computation shows. The existence of g' proves the claim. We can therefore consider two cases: k\ > 1 and k\ < 0. 1. If k\ < 0, then we can use our remark to write so that we find Note that Tftr1vi'ir11 = T / J H ^ H = Th^g so we get T9 = rh-^Thixg ~ r h \ x g -We show that the right hand side lies in R, completing the induction for the case k± < 0. Since k\ < 0, we compute that ^h^g) = n(g) — 1 and ^(h^g) < n(g) - 1, so that rh2 g, Thi^g G R by the induction hypothesis. By definition, T . - i = € R. So the right hand side lies in R. 2. If k\ > 0, we can proceed in exact analogy with the first case. We write and then note that so that we get Tn = Th T,-l„ — T .-2„. Chapter 3. Connections with SL(2, C) Character Varieties 33 Then both ^(h^g) and ^ (h^g) are strictly less than u-{g), in analogy with before, so that r.-2 0 , T , - I G R by the induction hypothesis. Noting that r/tj G i? (by definition) completes the induction in this second case. Therefore, by induction, T9 € R whenever g = / i * 1 . . , where ii,...,ir are distinct integers between 1 and m, and ki,...,kr G Z . We begin our second induction. Take any element g G G, and we write g = hk* —hk*, where ii,...,ir are no£ necessarily distinct. We induct on r (again taken to be minimal over all ways of writing g as a product of generators) to reach our desired conclusion. The cases r — 1,2 are both covered by our first induction, so that the base case for this second induction holds. If all ii,...,ir are distinct, this is a case we have already dealt with, so assume at least two of i\, ...,ir are equal. Upon replacing g by a conjugate element with equal value r, we may assume without loss of generality that is = %r for some s < r. Then we split g into two pieces X^hk'..hf Y = h!ls^...hf and write Tg = TXY = TXTY - TXY-1-By the induction hypothesis, we clearly have TX,TY G R. Additionally, since is = irr the element XY~l can be written as a "shorter" product of generators than g, namely: XY~l = h'?i...h''s-krh;kr-1...h;ks+1 which has r — 1 terms, so that r ^ y - i G R by the induction hypothesis as well. Therefore rg G R for an arbitrary g G G. • This proof gives a flavour for the techniques involved in this subject, while setting the stage for an even more surprising result. Fix the set of generators of R = C [ r 9 l , T 9 N ] provided by theorem 3 .1 . Then the map t : X(G) -+ C * t(Xp) = (Tgi(Xp),...Tgn(xP)) is an injection. To see this, suppose that t(xp) = t{Xp')> m other words T9i(Xp) = Tgi(Xp') f ° r i = l,—,n. Then given an arbitrary g G G, by theorem Chapter 3. Connections with SL(2, C) Character Varieties 34 3.1 we can write rg as a polynomial in r 9 l , r g n : T9 =P(T9n-,T9n)-Now we check that Xp = Xp' hy verifying that they take the same value on this arbitrary g G G. A quick computation shows so that t is an injection as claimed. Recall that subset V C C 1 is called an algebraic set if V is the set of common zeros of some set S of polynomials contained in C[x\ ,...,xn]. Theorem 3.4. (Culler, Shalen [8]) The set t(X(G)) c C* is the zero set of an ideal in C [ r 9 l , T 9 J , and so is an algebraic set. For different choices of the ele-ments g i , g n , the resulting different parameterizations oft(X(G)) are equivalent via polynomial maps. This theorem has been the foundation for an entire branch of study. The proof is extremely difficult and can be found in [8]. Recall that the coordinate ring of an algebraic set V c C 1 is defined as where 1(V) is the unique largest ideal of polynomials that are identically zero onV: I(V) := {/ G C [ ] • / (ai , . . . ,a„) = 0 for all (au...,an) G V}. There is a more convenient way of thinking of the coordinate ring. The poly-nomials in C[xi,xn] define functions on the algebraic set V simply by re-striction. Two polynomials / , g G C[xi,xn] define the same function on V precisely when / - g = 0 on V, which means that f — g G T(V). Therefore the cosets in C [ V ] can be thought of as polynomial functions restricted to V. xP(g) = Tg(xP) = P(T9i(Xp),-, = P(Tgi(Xp'),-T9n(Xp)) ;T9n (Xp>)) = Tg(Xp') = Xp'(g) Chapter 3. Connections with SL(2, C) Character Varieties 35 If two algebraic sets V, W are equivalent via a polynomial map 4> : V —> W, then (j> : C[W] —> C[V] defined by ^(/) = / o ^ is an isomorphism of C-algebras. From this algebraic fact, we choose not to study the object t(X(G)) of theorem 3.4, but instead the coordinate ring C[t(X(G))] = C [ r g i , T 9 n ] Jl(t(X(G))). We shorten the notation from <C[t(X(G))] to C[X(G)}, as the map t is imma-terial - since t depends on the choice of coordinates, but our final object does not. In this discussion, taking G = TTI(M) for some 3-manifold M can already be used to yield powerful results. In this case, we often shorten X(-K\(M)) to X(M). The following result was first proven by Thurston, then reworded in a much different language by Culler and Shalen. Theorem 3 .5. (Thurston 1, Culler-Shalen [8]) Let M be a compact, orientable 3-manifold. Suppose M has s torus components in its boundary, Tk ^ dM. Let p:7Ti(M) -+SL(2 ,C) be an irreducible representation such that p(ik, (TTI (Tfc))) g {/, -/} for each k. Then any component ofX(M) containing xP has dimension (as a variety) of at least s-3X(M). Corollary 3.6. If M is an n-component link complement, and if TTI(M) admits a representation p as stipulated in theorem 3.5, then X(M) has a component whose dimension is at least n. Proof. Since M is an n-component link complement, we get immediately that dM has n torus components. There is a well known formula for odd-dimensional homology n-manifolds which says that [21], [14]: X(M) = \X{dM), 1 Culler and Shalen attribute this theorem to Thurston, although the source they give appears to never have been published. Chapter 3. Connections with SL(2, C) Character Varieties 36 which we can certainly apply in our case. Therefore, in the particular case that M is a link complement, we know that the component of X(M) containing p has dimension at least 3 s — 3x(dM) = n — -x(torus) = n — 0 = n. • 3.2 The Connection wi th Skein Algebras It is reasonable to believe there could be a connection between SL(2, C ) char-acter varieties and skein algebras, because of the striking similarities between the two defining identities: [<?] ® [h] = [gh] + [gh'1] and tr(A)tr(B) = tr(AB) + triAB"1). This connection was provided by Bullock, in the form of the following theo-rem: Theorem 3.7. (Bullock, [6]) For any group G, there exists a surjective map of alge-bras ip:S(G;C) ^C[X(G)] defined by ip([g]) = rgfor any g G G. Furthermore, ker(tp) = VTj, where Vo denotes the subalgebra of all nilpotent elements in S(G; C ) . Proof. First, note that from its definition, i/> is clearly surjective. Next we show that ip is well-defined, by showing that tp maps the defining ideal of S(G; C ) to zero. Applying the definition of ip to the three types of elements in the ideal, we get: Chapter 3. Connections with SL(2, C) Character Varieties 37 1. ip([e - 2]) = V(N) - M 2 ] ) = re - 2, However, because p(e) = € 5L(2, C) for any representation p, we find that: Te(Xp) = Xp(e) = trace(Id) = 2. Therefore the function r e — 2 is zero when evaluated on any character. 2. tp([g ®h-h®g}) = T 9T,> - 7 7 ^ = 0, where the last equality follows from the commutativity of C[X(G)]. 3. ij}{[g ®h-gh- gh'1}) = TgTh - Tgh - rgh-i - 0, where the last equality is exactly remark 3.3. We can readily observe that y/0 C ker(ip). Given a € \/0 C S(G;C), if the image ip{a) is non-zero, it must be nilpotent in C[X(G)]. However, from the definition of C[X(G)], we know that it cannot contain nilpotents (if the polynomial / is nonzero on some subset A c C, no power fn can be zero on ^4). Therefore ip(a) — 0. That ker(ip) C V0 is much more difficult to prove, and it appears there are only two known proofs. One proof is algebraic, and the other is topologi-cal/ combinatorial, both were created independently of one another. The algebraic proof involves universal representation C-algebras and the Brumfiel-Hilden algebra. The Brumfiel-Hilden algebra is a structure defined in [4] for the purposes of investigating SL(2, C) representations of groups, and is denoted TH<c{G). In [23] it is shown to be isomorphic to S(G; C). The proof of our theorem appears in [4], and is done in the language of Brumfiel-Hilden algebras. The topological and combinatorial proof is extremely lengthly, and deals with resolving trees, Young diagrams and Procesi identities. It is the subject of [6]. Both proofs involve a great deal of tangential material, and so are not presented here. • Also in [6] is a partial proof of the following fact: Chapter 3. Connections with SL(2, C) Character Varieties 38 Proposition 3.8. The skein algebra S(G; C ) is finite dimensional as a complex vector space if and only if dim(X(G)) = 0. Proof. First, we remark that a variety V has dimension zero if and only if the coordinate ring C[V] is finite dimensional as a vector space, the proof of which can be found in Appendix A. Since X(G) is a variety, it is the zero set of some ideal I C C [ r 9 l , r 9 R J . Define a map 4>:C[T9L,...,T9n}^S(G;C) by 4>{TG) = [g]. Since r 9 E C [ r 9 l r 9 J for every g EG, the map <p is surjective. Theorem 10.2 in [6] tells us that yjker(4>) = VI. Then any ideal contains some power of its radical (Chapter 15, Proposition 11 of [11]), so that we get ( V J ) m C ker(<t>) C v7. (3.1) We have the following isomorphisms: <r\Y(nw ~ ^ [ r 9 i ' T9n] / ~ C [ T 9 I , T 9 N ] I C [ X { G ) ] = J i(z(i)) = /VI where the last isomorphism follows from Hilbert's Nullstellensatz. This tells us that C [ T 9 I , . . . , T 9 J / / v t is finite dimensional as a complex vector space, so that C [ T 9 I , . . . , T 9 J / / ( V / r is also finite dimensional. However, recalling the inclusion of equation 3.1, we know that there is a quotient map C [ T 9 I , . . . , T 9 J / C [ T 9 I , . . . , T 9 J / / {Vl)m / ker{<f>) S(G;C) Chapter 3. Connections with SL(2, C) Character Varieties 39 so that S(G; C) is the image of a finite dimensional vector space. Conversely, if S(G; C) is finite dimensional, then the surjection ip from the-orem 3.7 tells us that C[X(G)] is also finite dimensional, and so dim(X(G)) = 0. • Of course, we can restate this theorem as "dim<S(G; C) — oo if and only if dim(X(G)) > 1". Both of these statements are useful to bear in mind. C o r o l l a r y 3.9. Let M be a link complement. Suppose M has boundary components Tk ^¥ dM. Let p :7 r x (M) ->SL(2 ,C) be an irreducible representation such that p(ik.WTk)))£{I,-I} for all k. Then S(M; Z[o, a~l],a) is infinitely generated as a module. Proof. By applying corollary 3.6 and proposition 3.8 we get that S(TTI (M), C) is infinite dimensional, which gives the desired conclusion. • Immediately we have some obvious questions: When dim<S(G; C) < oo, what is the dimension? How does it relate to a manifold M in the event that we take G = -K\(M)1 More obvious would be the question: In any event, what is a basis of S(G; C)? We take some steps towards answering this last question by identifying some linearly independent families of elements in S(G; C). 3 . 3 An Application of Character Varieties We present here an application, due to Doug Bullock in [5]. Fix a representation p of a finitely generated group G. For each fixed p, we get a map evp : C[X{G)} -> C, the evaluation map, defined by eVp{Tg) = Tg(Xp) = Xp{9)-Chapter 3. Connections with SL(2, C) Character Varieties 40 This evaluation "makes sense", as we recall that the elements of the coordi-nate ring C[X (G)] can be thought of as restrictions of polynomial maps to the set of all characters, X(G). Fix an element g e G, and let A; be an arbitrary integer. Recall that ip:S(G;C) ->C[X(G)] is defined by i/)([g]) = rg, and consider the image of [gk] G S(G; C) under the composition evp o tjj. We first calculate that evp o ip{[g0}) = evp o ijj([e}) = eu p(r e) = Te(xP) = XP(e) = 2, and evp o tp([g]) = evp{Tg) = T9(XP) = x P ( o ) . Denote the complex number xp{g) by z. Then using this notation, evp o ip([gk}) = evp o ip{[gk~l] ® [g] - [gk~2]) as g ® h = gh + gh-1 = evpoip([gk-1])z - evpoip({gk-2]). Therefore, we have a recursive formula for evp o ip([gk]) in terms of evp o ip([gk-1}) and evp o tp([gk-2}). Defining polynomials pk {z) by the same recursion, p0(z) = evp o V([<70]) = 2, pi(z) = evpoip(\g]) = z, and in general pk(z) = Zpk-i(z) -Pk-2{z), we have that evp o ,0([9 , f e]) = Pfcl- 2 )- The first few polynomials defined by this recursion are: p0(z) = 2 pi(z) = z p2{z) = z2-2 p3{z)=z3-3z pA{z) = z4 -4z2 + 2 p5(z) = z5 - 5z3 + hz. Chapter 3. Connections with SL(2, C) Character Varieties 41 Remark 3.10. A quick way of computing the kth such polynomial is by making use of the determinant identity Pk(z) = z 1 0 •• • 0 0 0 1 z 1 •• • 0 0 0 0 1 z • • • 0 0 0 0 0 0 •• z 1 0 0 0 0 •• • 1 z 2 0 0 0 •• • 0 1 z where the above matrix is of size k x k. This identity follows from remarking that cofactor expansion along the top row yields the same recursive relationship as the defining relationship of the p^'s. From this identity, or from a quick induction, one can see that deg(pk) = k. Define Mr to be the r x r matrix over C[z±,zr] whose (i, j)-th entry is Pi(zj). Then Lemma 3.11. [6] The determinant \Mr\ is a degree lll±il polynomial in the vari-ables z±, ...,zr. Proof. The proof is by induction. First, the claim is certainly true for r = 1, as Mx = [p!(^)] = [z]. Suppose that up to r — 1, the determinant | M r _ i | is of the stipulated degree. Expanding Mr along the r-th row, we get: r Mr = J2Pr(zi)\Ci\, (3.2) where Cj is Mr with the r-th and i-th rows eliminated. But now the polyno-mials appearing in the columns of C\ still satisfy the defining recursion Po = 2 Pi= z Chapter 3. Connections with SL(2, C) Character Varieties 42 Pk = ZPk-l -Pk-2, but both the r-th row and the column containing polynomials in the vari-able Zi have been eliminated. Therefore, we may apply the inductive hy-pothesis to conduce that \d\ 2 is a degree r^ 2" 1^ polynomial in the variables z i , Z i - i , Z { + i , z r . Therefore, each summand in equation 3.2 has degree: degiprizi^dl) = degiprizi)) + deg(\Ci\) = r + = r±±A. The Zil±L\ degree term of the i-th summand is the only degree r ( r + 1 ) term in equation 3.2 that contains z\, so the degree terms cannot cancel. • Bullock applied this fact in a very clever way to S(G; C). Theorem 3 .12. (Bullock [6]) If there exists rg e C[X(G)] such that the image Tg(X(G)) is open (or whose image contains an open set), then Mb2],brV-} form an infinite linearly independent set in S(G; C), when considered as a C-vector space. Proof. The claim follows by showing that Vr:=span{[g],[g2},...,{gr]}^Cr for every r. The polynomial \Mr \ is non-constant by lemma 3.11, and so can-not be identically zero on the open set Tg(X(G))T C C . Therefore, there must be a point {Tg(Xpi),Tg(Xpi), -Tg{Xpr)) £ Tg(X(G)Y on which \Mr | is non-zero, so that the matrix \Pi(T9(xPi))] = [evPi ° W})}, l<i,j<r is invertible. Define $ :S(G;R) -> C 2d is (r - 1) x (r - 1) Chapter 3. Connections with SL(2, C) Character Varieties 43 by (evpi o V, evP2 o ip,evPr o ^). Then we compute span{H{g]),$([g2)),..., *([/])} span{{evPl o ^ ([g]), ...,evPr o ip{[g})),... (evpi o ip{[gr]),evPr o ip{\gr]))} span{rows of [evPj o i/j([g1})]} Therefore, by invertibility of the matrix [evPj o ip(\g1])], the image of Vr under the map $ is an r-dimensional subspace of V —in other words, the map is surjective. Since Vr is at most r-dimensional, this forces This theorem admits a very nice topological interpretation, by using our isomorphism S(ni(M); C) = S(M; C, -1) to translate the skein module ele-ments {[g], [g2], [g3],...} into knots. Suppose that K is a knot in a 3-manifold M corresponding to an element g e iri(M) that satisfies the hypotheses of theorem 3.12. Then correspond-ing to the elements {[g], [g2], [g3],...} of the skein module S(iri(M), C) are the knots -[Ki],-[K2],-[K3],... G S(M; C, -1), where Ki is an (i, 1)-cabling of the original knot K. It should be noted that the negative signs arising from our isomorphism can be eschewed by using an alternate but isomorphic def-inition of S(G, C) (see [24]), but the correspondence used here introduces a negative sign: [g] — Kg. The knots Ki look like: span{[g},[g2},...,[gr}} = C-as claimed. • Chapter 3. Connections with SL(2, C) Character Varieties 44 inside a tubular neighbourhood of K. Here, the vertical dots are meant to indicate i parallel strands. To see that this corresponds to the element gl in the fundamental group, observe that we may homotope everything inside the indicated box to a single point: which gives us the desired element of the fundamental group. The skein rela-tion [<?'] = [g1-1] ® [g] - ir2} also has a nice topological interpretation. If we resolve the innermost crossing in —Ki using the Kauffman bracket skein relation with a = — 1, we get: which, using our Ki notation, corresponds to Ki_\ U K\ + -£Q-2, upon relaxing and homotoping some of the components into more agreeable positions. This agrees exactly with the right hand side of the skein relation [gl] = [g1'1] ® [g] - [<T2L under the image of our isomorphism, as one would hope. In light of this theorem, we would like to know when there exists a map TG such that Tg(X(G)) contains an open set. Theorem 3.13. (Bullock [6]) If some Zariski component Xo ofX(G) has dimension greater than zero, then there exists T9 that is non-constant on XQ. Consequently, the image T9(X(G)) contains an open set. Chapter 3. Connections with SL(2, C) Character Varieties 45 Proof. Fix some system of coordinates C[T3I , ...,r f l n], and choose a Zariski component XQ of X(G) that has dimension at least 1. Since XQ is of dimen-sion at least one, we can choose two distinct points Xpi a n d Xp2 m -^ o- Since the characters Xpi and Xp2 a r e distinct, we can choose g € G on which they disagree. Then by this choice, Tg(Xpi) = XpAg) + Xpiia) = Tg(xP2) so that TG is non-constant on XQ. In our chosen coordinates, TG is a polyno-mial map, and we have just shown that it is nonconstant on the variety Xo. Irreducible varieties (in general) admit a manifold structure on a dense open subset3 [31], so that we can conclude the polynomial T9 is nonconstant on some open neighbourhood U in Xo- In the chosen coordinates, TG is in fact a polynomial map, and so is holomorphic. Since non-constant holomorphic maps send open sets to open sets, TG sends the open neighbourhood U to an open set. • Corollary 3.14. Let M be a link complement, with boundary components included via the maps ik:dM = Tk^ M. Suppose the fundamental group ofM admits an irreducible representation p-.TTi(M) ->5L(2,Q such that p(u(iri{Tk))) <£ {I, -I}, for each k. Then there is some knot K in M such that the cablings {KUK2,K3,---} are linearly independent in S(M; R, -1). 3The nonsingular points in a variety admit neighbourhoods with manifold structure, and the nonsingular points form a dense open subset [31]. Chapter 3. Connections with SL(2, C ) Character Varieties 46 Proof. By applying corollary 3.6, we are able to conclude that X(M) has a component XQ of dimension at least 1. Hence, by theorem 3.13, there exists some p € 7Ti (Af) such that the image T9(X(M)) contains an open set. Theorem 3.12 then tells us that the infinite family of elements {{9},[92U9%...} is a linearly independent set in S(iri(M), C ) . The isomorphism S (7 r i (M);C) ^S(M;C,-1) gives us the desired conclusion. • We wonder: What knot and link complements admit representations as in corollary 3.14? 3.4 Hyperbolic Knots We begin with a more approachable class of manifolds than a general knot complement, by considering hyperbolic knot complements. We recall some general facts about hyperbolic 3-manifolds [20]. If M is a complete hyperbolic 3-manifold, then there is a universal covering e 3 4 M . From this, Af can be realized as a quotient Af = H 3 y r where T is the subgroup of Isom + (H3) that consists of all isometries 7 satisfy-ing p o 7 = p. Since Af arises as such a quotient, we know Tri(Af) = T '«-»• Isom +(H 3) * PSL(2,C) so that there is a canonical inclusion -n\ (Af) «-»• PSL(2, C ) . Proposition 3.15. (Thurston) Let Af be a hyperbolic manifold. Then the canonical inclusion 0/71-1 (Af) in PSL(2, C ) can be lifted to a representation in SL(2, C ) . Chapter 3. Connections with SL(2, C) Character Varieties 47 For a proof of this fact see [8]. We know immediately that the represen-tation p : 7Ti(M) —¥ 5L(2, C ) arising from this proposition is injective, as the canonical map 7Ti (M) P5L(2, C ) that we lifted is injective. If we further as-sume that M is a hyperbolic knot complement of finite volume, the represen-tation p of 7Ti (M) in 5L(2, C ) provided by Thurston is necessarily irreducible [12], [8]. From this, we immediately get a fact found in much of the literature [9], [10], [2]: Corollary 3.16. If M is a hyperbolic 3-manifold of finite volume, and M is not closed, then some component of X(M) has dimension at least 2,4 and so M con-tains a knot whose (l,i)-cablings form an infinite linearly independent family in 5 ( M ; 1,-1). Proof. We apply corollary 3.6, using the representation provided to us by proposition 3.15. • Remark 3.17. Alarmingly, this already appears to contradict the result of [6] if we choose M to be a small, hyperbolic knot complement of finite volume. In this case we have proven that 5 (M, C , -1) is infinite dimensional, whereas [6] asserts that dimS{M,C, -1) < oo for all small 3-manifolds M. This is because the definition of "small" used in [6] is somewhat nonstandard, in that the incompressible surfaces inside the manifold M are not required to be closed. In this paper we take a small manifold to be a manifold that does not contain any closed, embedded, orientable surfaces that are both incom-pressible and nonboundary parallel. Remark 3.18. Through an entirely different approach, it is shown in proposition 2.4, [7], that in fact any knot complement M satisfies dim(X(M)) > 1. In fact, there can be no components of X(M) having dimension zero! This much more powerful result tells us that S(M; Z[a, a"1], a) is infinitely generated as a module for any knot complement M. 4This bound can be sharpened substantially to X(M) = 1 in the case that M is a small knot complement [7]. Chapter3. Connections with SX(2,Q Character Varieties 48 3 . 5 Comments In light of these facts, we would liket to find elements g such that rg is non-constant on some 1-dimensional Zariski component of X(G), for it is these elements g which will yield infinite linearly independent families M b 2 ] . ! / ] . - } in S(G; C). In particular, if G is the fundamental group of some knot com-plement M , we know that such a linearly independent family must exist, by remark 3.18. In this case, such an element g will correspond to an infinite fam-ily of linearly independent knots {KUK2,K3,---} in<S 2 > 0 0 (M;C,- l ) . Chapter 4 Some Infinite Families of Linearly Independent Knots 4.1 Character Varieties of Hyperbolic Knot Complements We focus on some results of Culler and Shalen, first published in [9]. Let M be a hyperbolic knot complement of finite volume, with torus boundary com-ponent / : T«->- M and let po : TTI(M) SL(2,C) be a lifting of the canonical embedding po : irX{M) ^ PSL(2,C). By comments in [15] and [9], the hypothesis that M has finite volume forces the induced map /* between fundamental groups to be injective. Let Xo be the component of X ( M ) containing the character % P o, which we know (from the work of Thurston) has complex dimension 1. It is shown in [12] that the subgroup Po(MMT))) c PoMM)) consists entirely of parabolic elements. This means that if g G TTI(M) lies in the image of the map /*, then rg(xPo) — xpo (o) — ±2. Therefore, if such maps 49 Chapter 4. Some Infinite Families of Linearly Independent Knots 50 TG are to be constant on Xo, they must take on one of the two values ±2 on the entire variety XQ. Pick an arbitrary g € ir\(M) that lies in the image of /*, and define a subvariety Y C Xo, that can be thought of as the union of the two level sets T~ 1(2) and r~1(—2). Amore strict definition is as follows: for such an element g, define the algebraic set Y' to be the zero locus of the polynomial T 9 ( X ) 2 = 4. Note that Xp0 € Y'. The subvariety Y C Xo that we are seeking is an irre-ducible component of Y' D Xo that contains xp0 • Theorem 4 . 1 . (Culler, Shalen, [9]). The variety Y has complex dimension 0, and so Y = {xPo}, since irreducible zero-dimensional varieties are singleton sets. From this theorem, we reason as follows: Given g € im{f*), the corre-sponding map TG takes on the value +2 or —2 on the character Xp0 G -X"o- How-ever, this is the only character in Xo on which r 5 takes on the value ±2, by our theorem. Therefore the map r s must be nonconstant on the one-dimensional component Xo. By the work of Bullock, this tells us that the infinite family {{gUg2U9%---} is linearly independent in S(G;C), and hence the corresponding knots are linearly independent in S(M;Z[a,a~l],a). As before, the case of hyperbolic knot complements of finite volume is a simpler special case, whose results we can generalize. 4.2 Character Varieties of General Knot Complements Let M be any knot complement, with boundary T included via the map / as before, and with /* injective as before. The authors of [3], [2] have provided us with the following result. For any 1-dimensional component Xo of X ( M ) , one of the following cases holds: 1. For every g G im(f*), the map rg is constant on XQ. Chapter 4. Some Infinite Families ofLinearly Independent Knots 51 2. For every g G im(f*), the map rg is non-constant on Xo. 3. There is exactly one primitive1 element g G im(/*) such that are constant on Xo. A l l other maps rg are nonconstant. Given our angle on this situation, we would like to know the circum-stances under which cases (2) and (3) arise. We have the following partial answer. Theorem 4 .2 . (Boyer, Luft, Zhang, [2]) If M is small, then each one dimensional component X 0 ofX(M) satisfies either case (2) or case (3). From this, we follow a line of reasoning identical to before, and conclude that one of the following two cases hold: 1. Every g G im(f*) gives rise to a linearly independent family of knots in 2. There is exactly one primitive element g G im(/*) whose powers {g,92,g3,---} may not give rise to linearly independent families of knots in <S(M;Z[a,a_1],a). A l l other elements in im(/*) give rise to infinite linearly independent families in 5(M;Z[a,a _ 1 ],a). 1Here, we mean that the element g cannot be written as a power of any other element h E im(ft) Chapter 4. Some Infinite Families of Linearly Independent Knots 52 4.3 An Explicit Computation for 2-bridge knots First, some general facts from the work of Riley in [27]. Define a 2-bridge group of determinant a > 3 (a € Z is odd) to be a group G with presentation (xi,X2 • WXl = X2V)), where and w = x^x^xl3 „eQ-l ±1 for j = l... a - 1. We call these groups 2-bridge groups, because this class of groups subsumes all 2-bridge knot groups [30]. Let C and D be the matrices: C = D = t 1 0 1 t 0 -tu 1 Given a 2-bridge group G, define W 6 GL(Z[t, t 1, u]) by W = CeiD62 •••Dea-1 = wn W12 W21 W22 Define a mapping pt,u : G - > G L ( Z [ M _ 1 , « ] ) by pt,u{x\) — C, and pt,u{x2) — D, where subscript is to indicate that such a map depends on our choice of t and u, which we are to think of as complex numbers. Lemma 4 .3 . If the pair t, u satisfy the polynomial wn + (1 - * ) ^ i 2 = 0, then pt>u defines a representation of G into GL(Z[t, t~l, u]). Proof. The assignment Pt,u(xl) = C ' Pt,u( x2) = D Chapter 4. Some Infinite Families of Linearly Independent Knots 53 defines a homomorphism precisely if pt,u preserves the single relation WX\ = X2W. Under this assignment, the single relation becomes: Wl2 t 1 t 0 " wu Wi2 W21 W22 0 1 —tu 1 W21 W22 or upon multiplying tWn W\i + W\2 tW2\ W2\ + W22 twn -tuWn + W21 tWi2 -tUW\2 + W22 Equating entries and simplifying gives the four polynomial equations: twn = twn (4.1) wn + (1 - t)wu = 0 (4.2) {t - l)w2i + tuwn = 0 (4.3) -u;2i + tuw\2 — 0. (4.4) Equation 4.1 obviously does not concern us, as it is a simple identity. Equation 4.3 can be reduced to an identity if we employ equations 4.2 and 4.4: 0 = (t — l)w2i + tuwn = (t - l)w2i + tu((t - l)wl2) (by 4.2) = {t - \){-tuwi2) + tu{(t - l)wi2 (by 4.4) = 0. To complete the lemma we need only prove that 4.4 is an identity. Define r 0 V = y/—tu 0 -tu Then a quick computation shows that (C€i)T = VD£iV~\ and {Dei)T = VCeiV~l, Chapter 4. Some Infinite Families of Linearly Independent Knots 54 for ei = ± 1 . This allows the clever observation: WT = (CeiDe2 •••D£*-i)T = ( Z ) e « - i ) T ( C e " - 2 ) T - - - ( C £ l ) T = {D£l)T(Ce2)T •••(C€a-1)T since the e/s are palindromic = VCeiV-1VDe2V-1---VD€a-lV~l by our choice of V = VCeiDe2 •••D£«~1V-1 = vwv-\ In other words, W12 W22 Wn —tuW2\ from which we can read off w2\ = —tuw\2- • In light of the importance of this polynomial, we define = wn + (1 - t)wi2. The properties of this polynomial are the subject of [27]. As is standard, let A = Z[t, t - 1 ] . Riley shows that u) always admits a factorization of the form $(t,u) = i f c$i(t,u)$ 2(*,w) • • • $,•(*,«) where k eZ, and each $j e A[u] is irreducible and distinct, and has a leading monic term of the form u n, n > 1. Additionally it is shown that though each factor lies in A[u], no factor can lie in Z[u\. This provides a decomposition of the zero locus of $ into irreducible affine algebraic curves, and each point on such a curve corresponds to a representation pt>u ofG. Also of great importance is a remark in [27], proven by deRham in [25], that 0) is a A-unit multiple of the Alexander polynomial A(t) of the group G. In the case that G is a 2-bridge knot group, there is a nice formula for the Alexander polynomial [13]: A(t) = 1 - f1 + tei+e2 - tei+e2+es + • tEfcTi 1 € < . y/—tU 0 0 1 M i l Wl2 1021 W22 0 /-tu 0 v 7 1 1 ^ Chapter 4. Some Infinite Families of Linearly Independent Knots 55 We will make use of this information shortly. There is also a partial "converse" to this correspondence between repre-sentations and pairs t, u, appearing as lemmas 7 and 8 in [28], and referenced in [27]. Suppose we are given a representation with nonabelian image, <f>: G -»• G L ( 2 , C ) , defined by <j){x\) = M\ and 4>(x2) = M2. Then there exists a matrix U G SL(2,C) and some numbers t, u satisfying wn + (1 — t)wi2 — 0 such that UMxlJ-1 = Vt~ldet(M]) t 1 0 1 , UM2U~l = Vt-ldet{M2) t 0 -tu 1 Furthermore, the pair t, u is unique if M i and M2 have a common eigenvector, otherwise the pair can only be replaced with t~l,u. What this means is that each representation ptyU corresponds to at most two points in the zero locus $(t,u) = 0. We now wish to consider these results as they apply to representations into SL(2, C). Note that if we instead had defined Pt,u(xl) = t 2 C = then the defining relation gets mapped to i i 0 t _ 5 Pt,u i t2 0 1 _ 1 — t^U t 2 WX\ = x2w (H)£ e < + 1 wc = (r^ei+lDW. Since we can cancel the powers of t on both sides, this alternative assignment still defines a representation of G for each pair t, uin the zero locus u) = 0, but whose image now lies in SL(2, C). Therefore, though our original assign-ment of Pt,u{xi) = C, ' pt,u(x2) = D is beneficial for illustrating the correspondence between points satisfying $(t,u) = 0 Chapter 4. Some Infinite Families of Linearly Independent Knots 56 and representations of G, we shall henceforth take pt,u to be the new assign-ment We now consider the characters arising from this new assignment. To sim-plify notation, let For an arbitrary g £ G, we would like to know about the values Tg(xt,u), which is now expressible as a polynomial in t and u. In general, Tg(xt,u) is a n unwieldy mess. However, we have the following lemma when we take u = 0: Proposition 4 .4 . Let g be an arbitrary element of the two-bridge group G, whose generators are x\ and x2. Then if Pt,u{x\) = t 2C, Pt,u{x2) =t 2D. 9 = x h x h x h where ej = ± 1 , we have that Here, s is the exponent sum of g. Proof. We set u = 0 in the matrices C, D, C _ 1 , D _ 1 , and get: C 0 = t 1 0 1 D0 = t 0 0 1 t-1 t-1 0 1 and •o1 t-1 0 0 1 Chapter 4. Some Infinite Families of Linearly Independent Knots 57 By induction on the length of the product, we will prove that an arbitrary product of these four matrices necessarily has the form: t" P(t) 0 1 where a is the exponent sum of the matrix product, and p(t) is some polyno-mial in t. Observe that for a product of length one, the claim holds, as we can see from inspection of C, D, C _ 1 , D - 1 . Assume that the claim holds for some matrix A that is a product of length s of C's and D's with exponent sum k, so that A = tk p(t) 0 1 Then upon multiplying A on the left by Co, C 0 1,DQ, and D0 L, we get: AC0--AC,~L = ADQ = AC0 = tk+l tkp(t) 0 1 tk-l tk~lp{t) 0 1 • tk+l Pit)' 0 1 tk-l Pit)' 0 1 ' so that the claim holds true for an arbitrary product of length s +1. Therefore, with g £ G arbitrary, we can compute a(g) ( Tg(xt,o) = tr(pt,u{g)) =t 2 tri ta^) p(t) 0 1 = t 2 + t 2 . • Suppose that we find two different points (ri,0) and (r2,0) in the zero locus of $(£, if), both of which lie in the same irreducible component of the curve $(t,u) = 0. Chapter 4. Some Infinite Families of Linearly Independent Knots 58 Since $(£,0) = A(t) is the Alexander polynomial, finding ri,r2 amounts to finding roots of the Alexander polynomial, and somehow making an argu-ment that the resulting points lie in the same irreducible component. Then if we find an element g € G such that g-(g) _zisl g ( » ) _ £ ( 9 ) T"! 2 +1"! 2 ^ r 2 2 + J " ! 2 , we will have Tg(Xruo) + Tg(xt,u), so that Tg is a non-constant polynomial on the irreducible component of ii) = 0 that contains the points (ri, 0) and (r2,0). This is sufficient for us to apply Doug Bullock's result, and conclude that the elements are a linearly independent family in S(G, C). 4.3.1 Computational Applications to an Infinite Family of 2-Bridge Groups First, we observe that r ? + r i - n = r « + r - « ^ ^ - ^ = 0 <=» r 2 r V£ + r£ - rfV? - r? = 0 ( « - l ) ( r ? - r J ) = 0 . If we restrict ourselves to real values, the last equation can only be satisfied if 1 ri = — or r± = r 2 . This is not to imply that considering complex roots of the Alexander polyno-mial cannot be fruitful. It is simply more to the point to restrict our attention to real roots for the exposition of these ideas. For p = 0 mod 3 an odd positive integer, we consider the family of 2-bridge knots K v ^ . (For an explanation of this indexing, see [29].) From [13], the corresponding knot group has presentation (xi,X2 • X\W = WX2), Chapter 4. Some Infinite Families of Linearly Independent Knots 59 where and „ , _ „ £ l „ £ 3 . . . ™ e 2 p - 2 UJ — \ 2 1 JP 2p- 1. The e/s obey they pattern: for j = 1 • • • 2p - 2. e i = 1 62 - -1 €3 = -1 64 = 1 e5 = 1 €2p-2 = 1 here, the vertical dots indicate alternating pairs of -1 and +1. This is clearly a 2-bridge group, so we can compute the Alexander polynomial as: A ( i ) = l - t1 + tl~l - t1'1-1 + _ . . . + &u which gives A(i) = 1 - t + 1 - t~l + 1 - t + 1 - r l + 1 + 1 where there are 2p — 1 terms in the sum. Grouping together like terms, we get the simple formula A«) = (^ )( + P « + ( ^ ) r ' . To find the roots of this polynomial, we first multiply the Alexander poly-nomial by a factor of t and then use the quadratic formula on the resulting polynomial. This yields the roots: Chapter 4. Some Infinite Families of Linearly Independent Knots 60 Both roots of the Alexander polynomial are real, and distinct. The two roots are not inverses of one another, because -p + y /2p~=rT _ 2p 2p ~ -p - y/2p=l~ reduces to -Sp2 - 2p - 1 = 0, and considered modulo 3 this equation gives 2 = 0, since p = 0 mod 3. There-fore the roots r\ and r2 satisfy r? + r - » ^ r » + r2-" for all nonzero n. Thus we already have the following fact: If G is a knot group corresponding to one of the knots K_s_, and g £ G has nonzero exponent sum, then TgiXnfi) + Tg(Xr2,o)-It remains to show that the points (ri, 0) and (r2,0) lie ine the same compo-nent of the variety defined by <£(£, u) = 0. Recalling that $(*, u) = t f c $ i ( t , « ) $ 2 ( « , « ) • • • $r ( i ,u) where k e Z, and each $j G A[u] is irreducible and distinct, we see that this must provide a factorization of A(t) over A of the form A ( t ) = 0) = t f c $ ! ( t , 0 ) $ 2 ( t , 0) • • • $ r ( t , 0) = tkCl(t) • • • Cr(t). Here, Ci(t) is the constant term of which lies in A. We consider the possible factorizations of over A. Any such factorization can only have two nonzero roots, correspond-ing to the roots we have already computed. This means in our factorization tkc\ (t) • • • Cr(t), one of two cases may occur: Chapter 4. Some Infinite Families of Linearly Independent Knots 61 1. Two of the Cj's are linear factors, say cn and c m , while all others must be units in A. Then the points ( n , 0) and (r2,0) lie in zero loci of and <3>TO, and so are in separate components of $(£, u) — 0. 2. Only one of the c/s is not a unit, and therefore has the same roots as the Alexander polynomial. Then the points ( n , 0) and (r2,0) lie in the same irreducible component of <&(i, u) = 0. For case (1) to occur, the polynomial P~ l \ . , ^ , (P must factor over the integers into two linear factors. This is not possible, be-cause this polynomial has descriminant 2p — 1 = 2 mod 3, and 2 is not a quadratic residue modulo 3. Therefore ( n , 0) and (r2,0) lie in the same irre-ducible component of u) = 0. This leads us to the following conclusion: Proposition 4 .5 . Suppose that G is the knot group of one of the knots K P . where 2p—X p = 0 mod3 is odd, and that g € G has nonzero exponent sum. Then the elements are linearly independent in the skein module S(G, C). We can use this to make a modest gain in our understanding of the nilrad-ical of skein modules. Proposition 4 .6 . Suppose that G is the knot group of one of the knots K_e_,where 2p— 1 p = OmodS is odd, and that g e G has nonzero exponent sum. Then [g] is not contained in the nilradical ofS(G, C). Proof. This is an immediate consequence of the 'reduction formula' [g] ® [h] = [gh] + [gh'1]. Supposing that [g] ® [g] ® • • • ® [g] = 0, Chapter 4. Some Infinite Families of Linearly Independent Knots 62 we may apply our reduction formula to the left hand side repeatedly until we obtain a sum of elements of the form [gh]. The equation we obtain in this way contradicts the linear independence of the elements • 4 . 4 Questions for Future Research Computationally, skein modules are very difficult to tackle, any new compu-tational techniques would be more than welcome in the field. The approach we have seen for finding infinite linearly independent families could be ex-tended to yield new information by finding additional elements g 6 ir\ ( M ) that yield functions rg which are non-constant on positive dimensional com-ponents of X(M). This is certainly an appealing avenue for future research. Based upon the explicit calculation for the knots K p above, we would 2p— 1 like to come up with better ways of finding pairs of points (ri,0), (r2,0) that lie in the same irreducible component of the curve u) = 0, and use these pairs to create infinite linearly independent families. Alternatively, it seems that this approach of using the roots of the Alexander polynomial is inherently weak in some sense, since it only applies to elements of the group that have nonzero exponent sum. Perhaps there is some alternative approach based upon the calculations of Riley, which will provide more trenchant in-sights into the question When is the map p : S(G; C) C[X(G)] injective? specifically in the case that G is a 2-bridge group. This question was first posed by Przytycki in [23], and the answer is only known in a small num-ber of cases. Recent interest in the A-polynomial has also lead to questions as follows: Suppose that a positive-dimensional component Xo 6 X(M) is defined by some set of polynomials (Pl.P2,P3,-- - ,Pn}, Chapter 4. Some Infinite Families of Linearly Independent Knots 63 and suppose that we find an element g G TT\{M) that is non-constant on the component XQ. It seems reasonable to expect that the linearly independent family of knots arising from this element would bear some connection to the defining polynomials {J>I,P2>P3J • • • ,Pn}- What would this relationship be? This is of particular interest for the following reason. Suppose we have a knot K with complement M. In [7], the authors define the A-polynomial of a knot K as the defining polynomial of a certain one-dimensional algebraic sub-set of X (M). From the work of Doug Bullock, we know that there must exist at least one function rg that is non-constant on this one-dimensional component. The obvious question is: Which elements g G -K\(M) give rise to functions that are non constant on the variety defined by the A-polynomial. Having found these elements, what relationship would they bear to the A-polynomial? A n answer to this question could provide a connection between the skein module of a knot complement and the A-polynomial of the corresponding knot. Bibliography [1] C. Blanchet, N . Harbegger, G. Masbaum, P. Vogel, Topological quantum field theories from the Kauffman bracket, Topology 31 (1992), 685-699. Cited on page(s) 2 [2] Boyer, S., Luff, E., and Zhang, X. On algebraic components of the SL(2, C) character varieties of knot exteriors, Topology 41 (2002), 667-694. Cited on page(s) 47,50,51 [3] Boyer, S., Zhang, X. On Culler-Shalen Seminorms and Dehn Filling, Annals of Mathematics, 2nd Ser., Vol 148, No.3 (1998) 737-801. Cited on page(s) 50 [4] Brumfiel, G.W., Hilden, H .M. 5/(2) representations of finitely presented groups, Contemporary Mathematics 187 (1995). Cited on page(s) 37 [5] Bullock, D. Estimating a Skein Module with SL2 (C) Characters, Proceedings of the American Mathematical Society, Vol. 125, No. 6 (1997), 1835-1839. Cited on page(s) 39 [6] Bullock, D. Rings of SL2(C)-characters and the Kauffman bracket skein module, Comm. Math. Helv. 72 (1997), 521-542. Cited on page(s) 2, 36, 37, 38,41,42,44,47 [7] Cooper, D., Culler, M . , Gillet, H. , Long, D.D., Shalen, P., Plane curves associated to character varieties of 3-manifolds, Invent. Math. 118 (1994) 47-84. Cited on page(s) 47, 63 [8] Culler, M . and Shalen, P. Varieties of Group Representations and Splittings of 3-Manifolds, Annals of Mathematics, 2nd Ser., Vol. 117, No. 1 (1983), 109-146. Cited on page(s) 30, 34,35,47 [9] Culler, M . and Shalen, P., Bounded, imcompressible, separating surfaces in knot manifolds, Invent. Math. 75, (1984), 537-545. Cited on page(s) 47,49,50 64 Bibliography 65 [10] Culler, M . , Gordon, J., Luecke, J., and Shalen, P., Dehn Surgery on Knots, The Annals of Mathematics, 2nd Ser, Vol. 125, No.2 (1987), 237-300. Cited on page(s) 47 [11] Dummit, D., and Foote, R., Abstract Algebra, 2nd edition, John Wiley and Sons, Inc. (1999) Cited on page(s) 20, 38 [12] Fatou, P., Fonctions Automorphes, Vol.2 of Theorie des Fonctions Algebriques, by P. Appell and E. Goursat, Gauthier-Villars, Paris, 1930, pp. 158-160. Cited on page(s) 47,49 [13] Gaebler, R., Alexander Polynomials of Two-Bridge Knots and Links, BSc. Thesis, Harvey Mudd College, 2004. Cited on page(s) 54, 58 [14] Hatcher, A. Algebraic Topology, Cambridge University Press, 2002, pp249. Cited on page(s) 35 [15] Hempel, J. 3-manifolds. Annals of Math. Study, Vol. 86, Princeton University Press (1976). Cited on page(s) 49 [16] Hoste, J., and Przytycki, J., A survey of skein modules of 3-manifolds,in Knots 90, Proceedings of the International Conference on Knot Theory and Related Topics, Osaka Japan), August 15-19,1990, (1992), 363-379. Cited on page(s) 8,15 [17] Kauffman, L., On Knots, Princeton University Press, Princeton. (1987). Cited on page(s) 12 [18] Kauffman, L., State models and the Jones polynomial. Topology 26 (1987), 395-407. Cited on page(s) 14 [19] Kauffman, L., A n invariant of regular isoptopy. Trans. AMS 318 (1990), 417-471. Cited on page(s) 14 [20] Matsuzaki, K. and Taniguchi, M . , Hyperbolic Manifolds and Kleinian Groups, Oxford University Press, New York, 1998, pp. 15-37 Cited on page(s) 46 [21] Maunder, C. Algebraic Topology, Cambridge University Press, Cambridge, 1980. Cited on page(s) 35 Bibliography 66 [22] Przytycki, J. Fundamentals of Kauffman bracket skein modules, George Washington University Mathematics Preprint Series, GWUM-1999-02. Cited on page(s) 1,15 [23] Przytycki, J. and Sikora, A. On skein algebras and 5?2(C)-character varieties, Topology 39 (2000), 115-148. Cited on page(s) 37, 62 [24] Przytycki, J., and Sikora, A. Skein Algebra of a Group, Banach Center Publications, Vol. 42, Knot Theory, (1998), 297-306. Cited on page(s) 18, 19, 22, 26,43 [25] deRham, G. Introduction aux polynomes d'un noeud, Enseignement Math. (2) 13 (1967), 187-194. Cited on page(s) 54 [26] Reidemeister, K. Knotten und Gruppen. Abh. Math. Sem. Univ. Hamburg 5, 7-23,1927. Cited on page(s) 4 [27] Riley, R. Nonabelian representations of 2-bridge knot groups, Quart. J. Math. Oxford (2), 35 (1984), 191-208. Cited on page(s) 52,54,55 [28] Riley, R. Holomorphically parameterized families of subgrups of SL(2, C), Mathematika 32 (1985) no.2 248-264. Cited on page(s) 55 [29] Rolfsen, D. Knots and Links, AMS Chelsea Publishing, Providence, Rhode Island. Cited on page(s) 58 [30] Schubert, H . Knoten mit zwei Brucken, Math. Z., 65 (1956), 133-17. Cited on page(s) 52 [31] Shafarevich. Basic Algebraic Geometry - Varieties in Projective Space, Second Edition. Springer-Verlag, 1994. pp. 92-93,104-106. Cited on page(s) 45 

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