CLOSED SURFACES AND CHARACTER VARIETIES ERICCHESEBRO 2 Abstract. The powerful character variety techniques of Culler and Shalen 1 canbeusedtofindessentialsurfacesinknotmanifolds. Weshowthatmodule 0 structuresonthecoordinateringofthecharactervarietycanbeusedtoidentify 2 detected boundary slopes as well as when closed surfaces are detected. This approachalsoyieldsnewnumbertheoreticinvariantsforthecharactervarieties n ofknotmanifolds. a J 0 1 1. Introduction ] T Suppose that N is a compact, irreducible 3-manifold with torus boundary and G that X is an irreducible algebraic component of the SL C-character variety for N. 2 . It is known that the dimension of X is at least one [6]. It is also known that X h t contains a great deal of topological information about N. a m Let X(cid:101) → X be a birational map from a smooth projective curve. This map is defined for all but finitely many points. These points are called ideal points. For [ γ ∈ π1N and χ ∈ X, define Iγ(χ) = χ(γ). This determines a rational function 1 Iγ: X(cid:101) → C. This paper concerns the following landmark result of Culler and v Shalen. 1 3 Theorem1(Culler-Shalen). Ifxˆisanidealpoint,thereisanassociatednon-empty 1 essential surface Σ in N. 2 1. (1) Σ may be chosen to have empty boundary if and only if Iα is regular at xˆ for every peripheral element α∈π N. 0 1 2 (2) Otherwise, there exists a unique slope with the property that if α ∈ π1N 1 represents this slope then I is regular at xˆ. In this case, every component α : of ∂Σ represents the slope corresponding to α. v i X WhenanessentialsurfacearisesfromXandthistheorem,wesaythatthesurface is detected by X. r a Since 1983, case (2) of Theorem 1 has been carefully studied. Many famous papers have provided both applications and insight for this case. Notably, the Culler-Shalennormof[9]andtheA-polynomialof[6]cantellusexactlywhichslopes are detected. These tools are remarkably effective and have been used in proofs of several famous theorems including the Smith Conjecture [22] and the Cyclic Surgery Theorem [9]. In contrast, there is very little in the literature concerning case (1) of Theorem 1. It has also been difficult to interpret global properties of X in terms of the topologyofN. Thispaperintroducesnewfunctionswhichrelatedetectedboundary slopes and essential closed surfaces in the manifold to module structures of the coordinate ring C[X]. In the spirit of [7], these functions also contain number theoretic information about X. 1 2 ERICCHESEBRO ThetraceringT(X)isthesubringofthecoordinateringC[X]whichisgenerated by{Iγ|γ ∈π1N}. DefineTQ(X)tobethesmallestQ-algebrainC[X]whichcontains T(X). Suppose that α ∈ π1N is primitive and peripheral. Then C[X], TQ(X), and T(X) have the structure of a C[I ]-module, a Q[I ]-module, and a Z[I ]-module, α α α C Q Z respectively. Let Rk (α), Rk (α), and Rk (α) denote the ranks of these modules. X X X Note that the functions I are well-defined on slopes. Hence, these rank functions α may be viewed as functions on S, the set of slopes for N. It is clear that C Q Z (1) Rk ≤ Rk ≤ Rk . X X X The following theorem is the main theorem of this paper. Theorem 2. Let X∈{C,Q,Z}. Then (1) The function RkX: S →Z+∪{∞} is constant with value ∞ if and only if X X detects a closed essential surface. X (2) Otherwise, Rk (α)=∞ if and only if X detects the slope α. X We begin by proving the theorem for X = C. A straightforward application of this case yields a proposition of independent interest. Proposition1.1. LetX◦ betheunionoftheirreduciblecomponentsX(cid:48) oftheSL C 2 character variety such that (1) X(cid:48) detects a closed essential surface if and only if X does, and (2) for each slope α, X(cid:48) detects a surface with boundary slope α if and only if X does. Then X◦ is defined over Q. This proposition, together with Theorem 1, also allows us to improve the result to include X=Q. As an application of this result, we prove another proposition. Proposition 1.2. If X contains the character of a non-integral representation which detects a closed essential surface, then X detects a closed essential surface. Together with data compiled by Goodman, Heard, and Hodgson, this proposi- tion gives many examples of hyperbolic knot manifolds for which the hyperbolic componentsoftheircharactervarietiesdetectclosedessentialsurfaces. Wealsouse this proposition in our proof that the main theorem holds in the case X=Z. In addition to our main result, we begin an investigation of the basic properties X of the functions Rk and include computations for several examples. It is easy to X prove the following two propositions. Proposition 1.3. SupposethatN isaknotmanifoldandH (N;Z)∼=Z, generated 1 by α. Let X ⊆X(N) be the curve consisting of all abelian representations of π N. A 1 Then C Z Rk (α) = Rk (α) = 1. XA XA Proposition 1.4. If N is a knot manifold and X is a norm curve component of C X(N) then Rk (α)≥2 for every slope α. X As a corollary to Theorem 2, we have the following. Corollary 1.5. SupposeN istheexteriorofatwo-bridgeknotand(cid:104)µ,β|ωβ =µω(cid:105) is the standard presentation for π N. If X ⊂ X(N) is an irreducible algebraic 1 CLOSED SURFACES AND CHARACTER VARIETIES 3 component defined over Q then X is defined by an irreducible polynomial of the form n−1 (cid:88) In − p (I )Ij . µβ j µ µβ j=0 The set {Iµjβ}0n−1 is a free basis for C[X] as a C[Iµ]-module, TQ(X) as a Q[Iµ]- module, and T(X) as a Z[I ]-module. µ We point out that, with the appropriate definitions, these results also hold for the PSL C-character variety. In what follows, we use X to indicate that we are 2 working in the SL C setting and Y in the PSL C setting. When N ⊂S3 is a knot 2 2 exterior, we reserve λ,µ ∈ π N to be a longitude, meridian pair. We write X 1 A and Y for the algebraic sets of abelian characters. When N is hyperbolic, we use A the notation X and Y to indicate algebraic components of the character varieties 0 0 which contain a discrete faithful character. Below, we list the results of our calculations in five different examples. The last example is of particular interest. It shows that the inequalities (1) need not be equalities and that, although T(X) must always be torsion free as a Z[I ]-module, σ it need not be a free module. (1) Suppose N is the exterior of a trefoil knot. Then X(N) = X ∪X where A 0 X is irreducible. We have 0 Z Rk (µ) = 1. X0 This shows that the converse to Proposition 1.3 does not hold. (2) Suppose N is the exterior of the figure-eight knot. We have Z Z Rk (µ) = Rk (µ) = 2 X0 Y0 C Z Rk (λ) = Rk (λ) = 4 Y0 Y0 RkC (µ2λ) = RkZ (µ2λ) = 5. Y0 Y0 (3) SupposeN =M . Thenπ N =(cid:104)γ,η|γηγ−2ηγη3(cid:105). Defineµ=(η2γηγ)−1 003 1 andλ=(γηγ)−1ηγη. Thenµandλareprimitive,peripheral,andgenerate a peripheral subgroup of N. We have C Z Rk (µ) = Rk (µ) = 4. X0 X0 Also, I ∈T(Y ) and µ 0 C Z Rk (µ) = Rk (µ) = 2. Y0 Y0 Note that, since Y is a norm curve, these ranks achieve their minimum 0 possible value. In contrast to examples (1) and (2), the PSL C-rank is 2 strictly smaller than the SL C-rank (at µ). As with examples (1) and (2), 2 all Z[I ]-modules are free. µ (4) Suppose N is the exterior of the knot 8 . N is hyperbolic, so we consider 20 X . The diagram in Figure 1 gives a Wirtinger presentation for π N which 0 1 reduces to (cid:10)µ,γ|µγ(µγµ)−1γ3(µγµ)−1γµγ−2(cid:11) where γ =ησ. The functions I , I , and I give an embedding of X into C3. I and µ γ γµ 0 γ I both satisfy irreducible integral dependencies in Z[I ][x] of degree 5. γµ µ It follows that B = {IiIj |0 ≤ i,j ≤ 4} is a generating set for C[X ] as γ γµ 0 4 ERICCHESEBRO η σ µ Figure 1. The knot 8 labelled with Wirtinger generators 20 a C[Iµ]-module, TQ(X0) as a Q[Iµ]-module, and T(X0) as a Z[Iµ]-module. Notice that |B|=25. A Groebner basis argument shows that relations amongst the elements of B are plentiful and, in fact, {1,I ,I2,I3,I2I ,I } is a free basis for γ γ γ γ γµ γµ each of the above modules. Hence, C Z Rk (µ) = Rk (µ) = 6 X0 X0 and T(X ) is a free Z[I ]-module. 0 µ (5) SupposeN istheoncepuncturedhyperbolictorusbundlefromSection5of [11]. Then π N = (cid:104)α,β,τ|τατ−1 = (βαβ)−1,τβτ−1 = βα(βαβ)−3(cid:105). The 1 elements τ and λ = [α,β] form a basis for the peripheral subgroup of N and the functions t=I u=I v =I w =I τ ατ βτ αβτ x=I y =I z =I α β αβ give an embedding of X(N) into C7. For (cid:15)∈{0,1}, we have an irreducible algebraic component X ⊂ X(N) which contains a discrete faithful charac- (cid:15) ter. (a) {1,z,z2,z3,y,zy,z2y,z3y} is a free basis for C[X ] as a C[I ]-module. (cid:15) λ C Hence, Rk (λ)=8. X(cid:15) (b) {1,z,z2,z3,y,zy,z2y,z3y,t,zt,z2t,z3t,yt,zyt,z2yt,z3yt} is a free ba- sis for T(X ) as a Q[I ]-module. Hence, RkQ (λ)=16. (cid:15) λ X(cid:15) (c) {1,y,y2,y3,t,u,v,w,x,z,yt,yu,yx,y2x,xt,xu,vy}generatesT(X )as (cid:15) aZ[I ]-module. However, T(X )istorsionfreebutnotfreeasaZ[I ]- λ (cid:15) λ Z module. In particular, Rk (λ)=17. X(cid:15) The projection (t,y,z): X ∪X → C3 is an isomorphism onto its image 0 1 and the relation Q Z Rk (λ)<Rk (λ) X(cid:15) X(cid:15) reflects the fact that the inverse of this isomorphism is not defined over Z. The equality Q C Rk (λ)=2·Rk (λ) X(cid:15) X(cid:15) reflects the fact that X is not defined over Q. (cid:15) 1.1. Acknowledgements. We thank Steve Boyer, Brendan Hasset, and Kelly McKinnie for helpful conversations and suggestions. CLOSED SURFACES AND CHARACTER VARIETIES 5 2. Algebraic Geometry Throughoutthissectionwetakektobeanalgebraicallyclosedfield. Allvarieties aredefinedoverk. WhenXisanirreduciblevarietywewritek[X]andk(X)todenote theringofregularfunctionsonXandthefunctionfieldforX,respectively. Ifp∈X, defineO tobetheringofgermsoffunctionswhichareregularonneighborhoods X,p of p. We have a surjective homomorphism k[X]→k given by f (cid:55)→f(p). Let m be p themaximalidealwhichisthekernelofthismap. ByTheorem3.2part(c)of[13], O is isomorphic to the localization k[X] . We use this isomorphism to identify X,p mp these two rings and think of O as a subring of k(X). X,p Definitions 2.1. Suppose A⊆B are commutative rings and 1 ∈A. B (1) An element b ∈ B is integral over A if there is a number n ∈ Z+ and {α }n−1 ⊂A such that i 0 bn+α bn−1+···+α = 0. n−1 0 This equation is called an integral dependence relation of b over A. (2) B is integral over A if every element of B is integral over A. (3) The integral closure A of A in B is the set of all elements of B which are integral over A. (4) If A=A then A is integrally closed in B. By Corollary 5.3 of [1], the integral closure of A in B is a ring. The following characterization of integral closures is useful. Theorem 3 (Corollary 5.22 of [1]). Suppose A is a subring of a field K. The integralclosureofAinK istheintersectionofallvaluationringsofK thatcontain A. Definitions 2.2. Suppose that X is an affine variety. (1) X is a normal variety if k[X] is integrally closed in k(X). (2) A normalization of X is an irreducible normal variety Xξ and a regular birational map ξ: Xξ →X where k[Xξ] is integral over ξ∗(k[X]). If X is an affine variety and p∈X then the quotient m /m2 has the structure of p p a k-vector space. Definitions 2.3. Suppose that X is an affine variety. (1) A point p∈X is non-singular if dim(cid:0)m /m2(cid:1) equals the dimension of X. p p (2) X is non-singular if every point in X is non-singular. Theorem 1 in Chapter II, Section 5.1 of [21] states that non-singular affine vari- eties are normal. It is clear that if m is principal then dim(cid:0)m /m2(cid:1)=1. Theorem 2 in Chapter p p p II, Section 5.1 of [21] implies that if X is a normal affine algebraic curve and p∈X then m is principal. Thus, we have the following well-known theorem. p Theorem 4. Suppose X is an irreducible affine algebraic curve. X is non-singular if and only if X is normal. The following theorem is a direct consequence of Theorems 4 and 5 in Chapter II, Section 5.2 in [21] (and the first part of the proof of Theorem 4). 6 ERICCHESEBRO Theorem5. IfXisanirreducibleaffinealgebraiccurvethenXhasanormalization ξ: Xξ → X. Moreover, the normalization is unique, affine, and its coordinate ring is the integral closure of ξ∗(k[X]). The projective coordinates on projective space Pn give distinguished open affine subsets{U }n whichcoverPn. SoifX⊂Pn isanirreducibleprojectivevarietythen i 0 we have the distinguished affine open subsets U ∩X which cover X. We say that X i is non-singular if U ∩X is non-singular for every i=0,...,n. i Definition 2.4. Suppose that X is an irreducible affine curve and let ξ: Xξ → X the normalization of X. As in [13, Chapter 6], there is a smooth projective curve X(cid:101) (unique up to isomorphism) so that Xξ is isomorphic to an open set in X(cid:101). The projective variety X(cid:101) is called the smooth projective model for X. Henceforth, we identify Xξ with its image in X(cid:101) and we let ι: X(cid:101) (cid:57)(cid:57)(cid:75) Xξ be the rational map which is the identity on Xξ. Definition 2.5. The ideal points of X are the points in the set I(X)=X(cid:101)−Xξ. It follows from the uniqueness of normalizations and smooth projective models that I(X) is well-defined. We have dominant birational maps X(cid:101) ι (cid:47)(cid:47)Xξ ξ (cid:47)(cid:47)X The induced maps k(cid:0)X(cid:101)(cid:1)(cid:111)(cid:111) ι∗ k(cid:0)Xξ(cid:1)(cid:111)(cid:111) ξ∗ k(X) are isomorphims. Moreover, ξ∗(cid:0)k[X](cid:1)⊆k[Xξ]. Now suppose that Y is an affine variey and ϕ: X → Y is a regular map with ϕ(X)=Y. Then ϕ∗: k(Y)→k(X) is a field monomorphism and ϕ∗(k[Y])⊆k[X]. Definition 2.6. Aholeinϕ: X→Y isapointpˆ∈I(X)suchthat(ϕξι)∗(k[Y])⊆ O . X(cid:101),pˆ Remark 1. If pˆ is a hole in ϕ: X → Y then, since pˆ∈/ Xξ, there is an element f ∈k[X] with (ξι)∗(f)∈/ O . Also, if ϕ is a birational map, then ϕ is surjective X(cid:101),pˆ if and only if ϕ has no hole. Lemma 2.7. If ϕ: X→Y has a hole then k[X] is not integral over ϕ∗(k[Y]). Proof. Take a distinguished open affine set U ⊂ Pn with pˆ∈ U . Set X = U ∩ i i pˆ i X(cid:101). By Theorem 4, Xpˆ is a non-singular affine curve so k[Xpˆ] is integrally closed. Recall that O is isomorphic to the localization k[X ] . Proposition 5.13 of [1] Xpˆ,pˆ pˆmpˆ gives that this localization is integrally closed. The inclusion induced isomorphism k(X(cid:101)) → k(Xpˆ) restricts to an isomorphism OX(cid:101),pˆ→OXpˆ,pˆ. Hence OX(cid:101),pˆ is integrally closed. The point pˆ is a hole in ϕ: X → Y so (ϕξι)∗(k[Y]) ⊆ O . Since O is X(cid:101),pˆ X(cid:101),pˆ integrally closed, every element of k(X(cid:101)) which is integral over (ϕξι)∗(k[Y]) is an element of O . We have f ∈ k[X] with (ξι)∗(f) ∈/ O , hence (ξι)∗(k[X]) is not X(cid:101),pˆ X(cid:101),pˆ integral over (ϕξι)∗(k[Y]) and so k[X] is not integral over ϕ∗(k[Y]). (cid:3) CLOSED SURFACES AND CHARACTER VARIETIES 7 Theorem 6. Suppose X is an irreducible affine algebraic curve and ϕ: X → Y is a regular map with ϕ(X) = Y. Then ϕ has no hole if and only if k[X] is integral over ϕ∗(k[Y]). Proof. ByLemma2.7,weneedonlyshowthatifϕhasnoholethenk[X]isintegral over ϕ∗(k[Y]). Assume, to the contrary, that k[X] is not integral over ϕ∗(k[Y]). Since ξ∗ is injective, this implies that k[Xξ] is not integral over (ϕξ)∗(k[Y]). Taking integral closures in k(Xξ) we have k[Xξ] = k[Xξ] (cid:41) (ϕξ)∗(k[Y]). Take f ∈ k[Xξ]−(ϕξ)∗(k[Y]). By Theorem 3, there is a valuation ring R ⊂ k(Xξ) with f ∈/ R and k ⊂(ϕξ)∗(k[Y])⊆R. Claim: ι∗(R)=O for some pˆ∈X(cid:101). X(cid:101),pˆ Using the claim, ι∗(f) ∈/ O so pˆ ∈ I(X). Moreover, (ϕξ)∗k[Y] ⊆ R so X(cid:101),pˆ (ϕξι)∗(k[Y])⊆ι∗(R)=O . That is, pˆis a hole in ϕ, a contradiction. X(cid:101),pˆ The claim follows from Corollary 6.6 of [13] since it gives an open set XR in X(cid:101) and a point pˆ∈X such that ι∗(R)=O =O . (cid:3) R XR,pˆ X(cid:101),pˆ It is well-known that, in this setting, k[X] is integral over f∗(k[Y]) if and only if k[X] is a finitely generated f∗(k[Y])-module (see for example Proposition 5.1 and Corollary 5.2 of [1]). Hence, we have the following immediate corollary. Corollary 2.8. Suppose X is an irreducible affine algebraic curve and ϕ: X → Y is a regular map with ϕ(X)=Y. Then ϕ has no hole if and only if k[X] is a finitely generated ϕ∗(k[Y])-module. Remarks 1. Suppose that k[X] is a finitely generated ϕ∗(k[Y])-module. (1) Since X is irreducible, k[X] has no zero divisors. Hence k[X] is a torsion free ϕ∗(k[Y])-module. (2) We can be more concrete about a basis for k[X] as a ϕ∗(k[Y])-module. Let {x }m becoordinatefunctionsforX. ByTheorem6, eachx isintegralover i 1 i ϕ∗(k[Y]). Let n be the degree of an integral dependence for x and define i i S = {xα11···xαmm|0 ≤ αi < ni}. Every element of k[X] may be expressed as a ϕ∗(k[X])-linear combination of the elements from the finite set S. (See Proposition 2.16 of [1].) 3. Character varieties, boundary slopes, and closed essential surfaces Definition 3.1. A knot manifold is a connected, compact, irreducible, orientable 3-manifold whose boundary is an incompressible torus. LetN beaknotmanifoldandΓ=π (N). DenotethesetofSL C-representations 1 2 of Γ as R(N) and the set of characters of representations in R(N) as X(N). Let t: R(N) → X(N) be the map which takes representations to their characters. In [10], it is shown that R(N) and X(N) are affine algebraic sets defined over C and the map t is regular. We will refer to R(N) and X(N) as the representation variety and character variety for N. 8 ERICCHESEBRO Culler and Shalen have revealed deep connections between essential surfaces in N andthecharactervarietyX(N). Weoutlinesomeoftheirresultsinwhatfollows. For more background, see [10], the survey article [23], or Chapter 1 of [9]. Definitions 3.2. Suppose that X is a non-empty algebraic subset of X(N). (1) Givenγ ∈Γ, thetrace functionforγ onXistheregularfunctionI ∈C[X] γ defined by I (χ)=χ(γ). γ (2) Let T(X) be the subring (with 1) in C[X] generated by {I |γ ∈Γ}. T(X) is γ called the trace ring for X. The following proposition gives that, as a ring, T(X) is finitely generated. Proposition 3.3 (Proposition 4.4.2 of [23]). Let {γ }n be a generating set for i 1 Γ. Then T(X) is generated, as a ring, by the constant function 1 along with the functions in the set (cid:110) (cid:12) (cid:111) I (cid:12)V =γ ···γ where 1≤k ≤n and 1≤j <···<j ≤n . V (cid:12) j1 jk 1 k The smallest C-algebra containing T(X) is the coordinate ring C[X]. Therefore, a generating set {γ }n for Γ gives an embedding of X into C2n−1 by taking the i 1 functionsI tobecoordinatefunctions. Itisstraightforwardtoseethatunderthis V embedding, X(N) is cut out by polynomials with coefficients in Z. Wehavearegularmap∂: X(N)→X(∂N)fromX(N)tothecharactervarietyfor the peripheral subgroup (well-defined up to conjugation) of Γ given by restricting characters. For a non-empty algebraic subset X of X(N), let ∂X denote the Zariski closure of ∂(X). Definitions 3.4. (1) The unoriented isotopy class of an essential simple closed curve in ∂N is called a slope. We denote the set of slopes on N as S. (2) A boundary slope is a slope which is realized as a boundary component of an essential surface in N. Eachslopecorrespondstoapair{±a}⊂H (∂N;Z). TheinverseoftheHurewicz 1 isomorphism is an isomorphism H (∂N;Z)→π ∂N. This gives a monomorphism 1 1 e: H (∂N;Z) → Γ which is well-defined up to conjugation. Since traces are in- 1 variant under inverses and conjugation, the function {±a} (cid:55)→ I is well-defined. e(a) When α={±a} is a slope we write I =I . α e(a) Assume now that X⊆X(N) is an irreducible affine curve and let X(cid:101) ι (cid:47)(cid:47)Xξ ξ (cid:47)(cid:47)X be the corresponding maps and varieties as defined in Section 2. The following theorem is well known and fundamental, see Theorem 2.2.1 and Proposition 2.3.1 of [10]. It is a translation of Theorem 1 into the language of Sections 2 and 3 of this paper. Theorem 7 (Culler-Shalen). For every ideal point xˆ of X, there is an associated non-empty essential surface Σ in N. (1) Σ may be chosen to have empty boundary if and only if (∂ξι)∗(C[∂X]) ⊆ O . X(cid:101),xˆ CLOSED SURFACES AND CHARACTER VARIETIES 9 (2) Otherwise, there exists a unique slope α such that I ∈O . In this case, α X(cid:101),xˆ every component of ∂Σ represents the slope α. Theorem 7 is broadly applicable. For instance, if N is the exterior of a non- trivial knot in S3 then Kronheimer and Mrowka show that X(N) contains a curve with infinitely many irreducible characters [16]. Also, if the interior of N admits a complete hyperbolic structure with finite volume and χ is the character of a ρ discrete faithful representation then there is a unique algebraic component X ⊆ 0 X(N) which contains χ . Furthermore, X is a curve (see [24] and [23]). We refer ρ 0 to such a curve as a hyperbolic curve. Definitions 3.5. (1) Suppose that X ⊆ X(N) is an irreducible affine curve and xˆ is an ideal point of X. A non-empty essential surface Σ ⊂ N is associated to xˆ if it is contained in a surface given by xˆ and Theorem 7. (2) Suppose that X is an algebraic subset of X(N). A surface Σ is detected by X if there is an ideal point xˆ of an irreducible affine curve in X so that Σ is associated to xˆ. (3) IfanessentialsurfaceΣ⊂N isdetectedbyX(associatedtoxˆ)and∂Σ(cid:54)=∅ theboundarysloperepresentedbyacomponentof∂ΣisdetectedbyX(asso- ciatedtoxˆ). Theslopeisstronglyorweaklydetected(associated)depending on whether xˆ satisfies case (1) or case (2) of Theorem 7, respectively. Remark 2. It is natural to ask if, whenever N contains a closed essential surface, there is a closed essential surface in N detected by X(N). The author is not certain if the answer is known to be no, however there are compelling reasons to believe that the answer is no, see for example [5] or Remark 5.1 in [11]. We conclude this section with some applications of the work in Section 2. Theorem 8. Suppose N is a knot manifold and X is an irreducible algebraic subset of X(N). The following are equivalent. (1) X does not detect a closed essential surface. (2) dim(X)=dim(∂X)=1 and ∂: X→∂X does not have a hole. (3) C[X] is integral over ∂∗(C[∂X]). (4) C[X] is a finitely generated ∂∗(C[∂X])-module. Proof. As mentioned earlier, conditions (3) and (4) are equivalent by Proposition 5.1 and Corollary 5.2 of [1]. Ifdim(X)>dim(∂X)thenC[X]istranscendentalover∂∗(C[∂X])andsocondition (3) cannot hold. The inequality dim(X) > dim(∂X) also implies that X contains a curveC towhichTheorem7applies. Thisyieldsaclosedessentialsurfacedetected by X. Hence, conditions (2) and (3) both imply that dim(X)=dim(∂X). We have established that each of the four conditions imply that dim(X) = dim(∂X). By Proposition 2.4 of [6], dim(X) ≥ 1. Also dim(∂X) ≤ 1, otherwise for any slope α on ∂N, there is a curve in X to which Theorem 7 can be applied to give an essential surface in N with boundary slope α. This contradicts Hatcher’s theorem [15] that N has only finitely many boundary slopes. So, if any condition (1) through (4) holds, then dim(X)=dim(∂X)=1. The theorem now follows from Theorems 6, 7, and Corollary 2.8. (cid:3) 10 ERICCHESEBRO Given a trace function I ∈ C[X], let C[I ] denote the C-subalgebra of C[X] γ γ generated by I . Consider the regular map I : X → C and the induced map γ γ I∗: C[x] → C[X]. If I is non-constant on X then I∗ is injective and C[I ] is γ γ γ γ naturally isomorphic to C[x]. Moreover, Z[x] is naturally isomorphic to the Z- submodule Z[I ] ⊆ T(X) generated by all finite powers of I . For instance, if α is γ γ a slope which is not detected by X, since dim(X)≥1, Theorem 7 implies that I is α non-constant on X. Theorem 9. Suppose N is a knot manifold, X is an irreducible algebraic subset of X(N), and α is a slope. The following are equivalent. (1) X does not detect a closed essential surface and α is not detected by X. (2) C[X] is integral over C[I ]. α (3) As a C[I ]-module, C[X] is a finitely generated and free. α Proof. As with conditions (3) and (4) of the previous theorem, we know that (3) implies(2)andthat(2)impliesthatC[X]isafinitelygeneratedC[I ]-module. C[X] α is an integral domain so, as a C[I ]-module, C[X] is torsion free. C[I ] is a PID so α α C[X] is a free C[I ]-module. α It remains to show that (1) and (2) are equivalent. Assume first that (2) holds. Theorem 8 shows that X does not detect a closed essential surface and dim(X) = 1. Hence I is not constant. We have the regular α mapI : X→CandI (X)=C. SinceC[X]isintegraloverC[I ],Theorem6shows α α α thatI : X→Cdoesnothaveahole. ByTheorem7,αisnotstronglydetectedby α X. Since X does not detect a closed essential surface, α cannot be weakly detected by X. Now suppose that (1) holds. As in the proof of Theorem 8, we have that dim(X) = 1. Consider the regular map I : X → C. Either I (X) = C or I is α α α constant on X. But since dim(X)=1, Theorem 7 implies that if I is constant on α X then α is strongly detected by X or X detects a closed essential surface. So we must have I (X)=C. α By Theorem 6, it suffices to show that the regular map I does not have a hole. α Again, we appeal to Theorem 7 and notice that if I has a hole then either α is α strongly detected or X detects a closed essential surface. (cid:3) Definition 3.6. Let X ⊆ X(N) be an irreducible algebraic subset. The C-rank function is the map RkC: S →Z+∪{∞} X where RkC(α) is the rank of C[X] as a C[I ]-module. X α Remarks 2. Using Theorem 9, we make the following observations. C (1) If X does not detect a closed essential surface, Rk (α) = ∞ if and only if X α is strongly detected by X. C (2) X detects a closed essential surface if and only if Tr (S)={∞}. X (3) ByLemma1.4.4of[9],thereareonlyfinitelymanyboundaryslopesstrongly detected by X. Avoiding this finite set, choose α ∈ S. Then X detects a C closed essential surface if and only if Rk (α)=∞. X For X an irreducible algebraic component of X(N) we define X◦ to be the union of the irreducible components X(cid:48) of X(N) such that