q-terms, SINGULARITIES AND THE EXTENDED BLOCH GROUP

Size: px
Start display at page:

Transcription

1 q-terms, SINGULARITIES AND THE EXTENDED BLOCH GROUP STAVROS GAROUFALIDIS Dedicated to S. Bloch on the occasion of his sixtieth birthday. Abstract. Our paper originated from a generalization of the Volume Conjecture to multisums of q- hypergeometric terms. This generalization was sketched by Kontsevich in a problem list in Aarhus University in 2006; [Ko]. We introduce the notion of a q-hypergeometric term (in short, q-term). The latter is a product of ratios of q-factorials in linear forms in several variables. In the first part of the paper, we show how to construct elements of the Bloch group (and its extended version) given a q-term. Their image under the Bloch-Wigner map or the Rogers dilogarithm is a finite set of periods of weight 2, in the sense of Kontsevich-Zagier. In the second part of the paper we introduce the notion of a special q-term, its corresponding sequence of polynomials, and its generating series. Examples of special q-terms come naturally from Quantum Topology, and in particular from planar projections of knots. The two parts are tied together by a conjecture that relates the singularities of the generating series of a special q-term with the periods of the corresponding elements of the extended Bloch group. In some cases (such as the 4 1 knot), the conjecture is known. Contents 1. Introduction A brief summary of our results What is a q-term? The Bloch group From q-terms to the Bloch group An example Acknowledgement 5 2. Proof of Theorem From q-terms to the extended Bloch group The extended Bloch group The Rogers dilogarithm Two cousins of the extended Bloch group: K3 ind (C) and CH 2 (C, 3) q-terms and potential functions Proof of Theorem Proof of Proposition Proof of Theorem Special q-terms, generating series and singularities What is a special q-term? From special q-terms to generating series Examples of special q-terms from Quantum Topology 16 Date: August 31, The author was supported in part by NSF Mathematics Classification. Primary 57N10. Secondary 57M25. Key words and phrases: Bloch group, extended Bloch group, algebraic K-theory, regulators, dilogarithm, Rogers dilogarithm, Bloch-Wigner dilogarithm, potential, Bethe ansatz, special q-hypergeometric terms, asymptotic expansions, knots, homology spheres, Habiro ring, Volume Conjecture, periods, Quantum Topology, resurgence, Gevrey series. 1

2 2 STAVROS GAROUFALIDIS 5.4. An ansatz for the singularities of L np t (z) For completeness Motivation for the special function Φ and the potential function Laplace s method for a q-term A comparison between the Bloch group and its extended version 18 References Introduction 1.1. A brief summary of our results. Our paper originated from a generalization of the Volume Conjecture to multisums of q-hypergeometric terms. This generalization was sketched by Kontsevich in a problem list in Aarhus University in 2006; [Ko]. Our paper expands Kontsevich s problem, and reveals a close and precise relation between special q-hypergeometric terms (defined below), elements of the extended Bloch group and its conjectural relation to singularities of generating series. Oddly enough, setting q 1 also provides a relation between special hypergeometric terms and elements of the extended additive Bloch group. This is explained in a separate publication; see [Ga1] and [Ga2]. Our paper consists of two, rather disjoint parts, that are tied together by a conjecture. In the first part, we introduce the notion of a q-term, and assign to it elements of the Bloch group and its extended version. The image of these elements of the extended Bloch group under the Rogers dilogarithm is a finite set of periods in the sense of Kontsevich-Zagier. In the second part of the paper we introduce the notion of special q-term, its corresponding sequence of polynomials, and its generating series. Examples of special q-terms come naturally from Quantum Topology, and in particular from planar projections of knots. Finally, we formulate a conjecture that relates the singularities of the generating series of a special q-term with the periods of the corresponding elements of the extended Bloch group. In some cases (such as the 4 1 knot, the conjecture is known, and implies the Volume Conjecture to all orders, with exponentially small terms included What is a q-term? The next definition, taken from [WZ], plays a key role in our paper. Definition 1.1. A q-term t in the r + 1 variables k (k 0, k 1,..., k r ) N r+1 is a list that consists of An integral symmetric quadratic form Q(k) in k, An integral linear form L in k and a vector ǫ (ǫ 0,..., ǫ r ) with ǫ i ±1 for all i, An integral linear form A j in k for j 1,...,J A q-term t gives rise to an expression of the form J (1) t k (q) q Q(k) ǫ L(k) (q) ǫj A Q(q) j(k) valid for k N r+1 such that A j (k) 0 for all j 1,...,J. Here, the standard q-factorial of a natural number n and the q-binomial for 0 m n defined by (see for example [St]): n (2) (q) n (1 q j ), ( ) n m q (q) n (q) m (q) n m q-terms can be easily constructed and may be combinatorially encoded by matrices of the coefficients of the forms A j, L, Q much in the spirit of Neumann-Zagier and Nahm; see [NZ, Ko, Na].

3 q-terms, SINGULARITIES AND THE EXTENDED BLOCH GROUP The Bloch group. Let us recall the symbolic-definition of the Bloch group from [B1]; see also [DS, Ne2, Su1, Su2]. Below F denotes a field of characteristic zero. Definition 1.2. (a) The pre-bloch group P(F) is the quotient of the freeabelian group generated by symbols [z], z F \ {0, 1}, subject to the 5-term relation: [ [ ] [ ] y 1 x 1 1 x (3) [x] [y] + x] 1 y y (b) The Bloch group B(F) is the kernel of the homomorphism (4) ν : P(F) F F, [z] z (1 z) to the second exterior power of the abelian group F defined by mapping a generator [z] to z (1 z). The second exterior power G G of an abelian group G is defined by: (5) G G G Z G/(a b + b a) Recall the Bloch-Wigner function (see for example, [NZ, Eqn.39]) (6) D 2 : C ir, z D 2 (z) : ii(li 2 (z) + log(1 z)log z ), which is continuous on C and analytic on C \ {0, 1}. Since D 2 satisfies the 5-term relation, we can define a map on the pre-bloch group of the complex numbers: (7) R : P(C) ir, z D 2 ([z]) and its restriction to the Bloch group (denoted by the same notation): (8) R : B(C) ir, z D 2 ([z]) 1.4. From q-terms to the Bloch group. In order to state our results, let us introduce some useful notation. Given a linear form A in r + 1 variables k (k 0, k 1,..., k r ) let us define (9) v i (A) a i, where A(k) a i k i. If z (z 0,..., z r ) C and A is an integral linear form A in k (k 0, k 1,...,k r ), let us abbreviate (10) z A r i0 z vi(a) i. If Q is a symmetric quadratic form in k (k 0, k 1,...,k r ), we can write it in the form: Q(k) 1 2 Q ij k i k j + i,j0 i0 QL i k i. Q gives rise to the linear forms Q i (for i 0,...,r) in k defined by Q i (k) i0 Q ij k j. The next definition associates to a q-term t the complex points of an affine scheme defined over Q. j0

4 4 STAVROS GAROUFALIDIS Definition 1.3. Given a q-term t, let X t denote the set of points z (z 0,...,z r ) (C ) r+1 that satisfy the following system of Variational Equations: J (11) z Qi ǫ vi(l) (1 z Aj ) vi(ǫjaj) 1 for i 0,...,r. Here, C C \ {0, 1}. Generically, X t is a zero-dimensional set. In that case, X t (Q ) r+1, where Q is the set of algebraic numbers. The next definition assigns elements of the pre-bloch group to a q-term t. Definition 1.4. Given a q-term t, consider the map: (12) β t : X t P(C) given by: (13) z β t (z) : ǫ j [z Aj ]. The next theorem, communicated to us by Kontsevich, assigns elements of the Bloch group to a q-term t. In [Za2], Zagier attributes the result to Nahm [Na]. Theorem 1. (a) The map β t descends to a map (14) β t : X t B(C) which we denote by the same name. (b) The image of R β t : X t ir is a finite subset of ir P, where P is the set of periods in the sense of Kontsevich-Zagier; [KZ] An example. We illustrate Theorem 1 with the following example which is nontrivial and of interest to Quantum Topology. Consider the special q-term (15) t n (q) q a n(n+1) 2 ǫ n (q) b n where a Z, ǫ ±1 and b N. In other words, Then, r 0, k (n), Q(n) a (16) X t {z C z a (1 z) b ǫ 1} The map (12) is given by: n(n + 1), J b, A j (n) n, j 1,..., b. 2 (17) X t B(C), z β t (z) : a[z]. Equation (16) and the corresponding element of the Bloch group was also studied by Lewin, using his method of ladders. In that sense, the Variational Equations (11) is a generalization of the method of ladders. Theorem 1 is a practical way of constructing elements of the Bloch group B(C) that are typically defined over number fields. Even if X t is not 0-dimensional, its image under R β t always is finite. This finiteness is positive evidence for Bloch s Rigidity Conjecture, which states that B(Q) Q B(C) Q.

5 q-terms, SINGULARITIES AND THE EXTENDED BLOCH GROUP Acknowledgement. An early version of this paper was discussed with M. Kontsevich in the IHES and University of Miami and with S. Bloch in Chicago. The author wishes to thank S. Bloch and M. Kontsevich for generously sharing their ideas with the author and for their hospitality in the IHES in the fall of 2006 and in Chicago in the spring of The paper was finished in the spring of 2007, and updated further in the summer of 2007 to incorporate the final extension of the Rogers dilogarithm due to Goette and Zickert. The author wishes to thank D. Thurston and C. Zickert for many stimulating conversations. 2. Proof of Theorem 1 This section is devoted to a proof of Theorem 1, using elementary symbol-like manipulations. See also [Za2, p.43]. Zagier attributes the following proof to Nahm. Proof. (of Theorem 1) Let us fix z (z 0,...,z r ) that satisfies the Variational Equations (11). We will show that ν(β t (z)) 0 C C, where ν is given by (4). We compute as follows: On the other hand, we have: Thus, β t (z) ǫ j (z Aj (1 z Aj )). z Aj r i0 ǫ j (z Aj (1 z Aj )) ǫ j z vi(aj) i. i0 z vi(aj) i (1 z Aj ) ǫ j v i (A j )(z i (1 z Aj )) i0 i0 ( z i (1 z Aj ) vi(ǫjaj)). Since z satisfies the Variational Equations (11), after we interchange the j and i summation, we obtain that: β t (z) On the other hand, we have: z i z Qi i0 i0 z i i0 z i ((1 z Aj ) vi(ǫjaj)) J (1 z Aj ) vi(ǫjaj) z i (z Qi ǫ vi(l) ) i0 i0 z i z Q z i i0 z i z Qi z L ǫ. i0 z i i0 r z j0 Q z i z j j i,j z vi(l) i ǫ Q z i z j (z i z j )

6 6 STAVROS GAROUFALIDIS Since Q is a symmetric bilinear form and is skew-symmetric, it follows that z i z Qi 0. In addition, we claim that for ǫ ±1, we have: i0 (18) z ǫ 0 C C. Indeed, if ǫ 1, then If ǫ 1, then Since C (C ) 2, the result follows. Thus, z 1 z (1.1) z 1 + z 1. z 2 ( 1) 2(z ( 1)) z (( 1) 2 ) z 1 0. ν(β t (z)) 0 C C. This proves that β t takes values in the Bloch group, and concludes part (a) of Theorem 2. Part (b) follows from a stronger result; see part (c) of Theorem 2 below. The idea is that any analytic function is constant on a connected component of its critical set. In our case, there exists a potential function whose points are the complex points of an affine variety defined over Q by the Variational Equations. The complex points of an affine variety have finitely many connected components. Finiteness of the image of R β t follows. 3. From q-terms to the extended Bloch group In the remainder of the first part of the paper, we will extend our results from the Bloch group to the extended Bloch group. As a guide, given a q-term t, we will assign (a) A potential function V t ; see Definition (b) A set of Logarithmic Variational Equations (see (11)), which typically have a finite set of solutions, given by algebraic numbers. The variational equations may be identified with the critical points X t of the potential V t ; see Proposition (c) A map ˆβ t : X t B(C) where B(C) is the extended Bloch group; see Definition 3.1 and Theorem 2. When ˆβ t is composed with a map ˆR (given in Equation (29)), it essentially coincides with the evaluation of the potential on its critical values; see Theorem 2. (d) The image of the composition e ˆR/(2πi) ˆβ t is a finite set of complex numbers, given by exponentials of periods; see Theorem 2. In the last part of the paper, which was a motivation all along, we mention that quantum invariants of 3-dimensional knotted objects (or general statistical-mechanical sums) give rise to special q-terms, a fact that was initially observed in [GL1] The extended Bloch group. Our aim is to associate elements of the extended Bloch group to every q-term. The extended Bloch group was introduced by Neumann in his investigation of the Cheeger-Chern- Simons classes and hyperbolic 3-manifolds; see [Ne1]. The definition below is called the very-extended Bloch group by Neumann, see also [DZ]. There is a close relation between the Bloch group B(F) and K3 ind (F), (indecomposable K-theory of F); see [B1, B2] and [Su1, Su2], as well as Section 3.3 below. Unfortunately, the relation among B(F) and K3 ind (F) is known modulo torsion, and the torsion does not match. For F C, this is exactly what the extended Bloch group captures. The idea is that: torsion is encoded by the choice of the branch of the logarithms.

7 q-terms, SINGULARITIES AND THE EXTENDED BLOCH GROUP 7 More precisely, consider the doubly punctured plane (19) C : C \ {0, 1} and let Ĉ denote the universal abelian cover of C. We can represent the Riemann surface of C as follows. Let C cut denote C cut open along each of the intervals (, 0) and (1, ) so that each real number r outside [0, 1] occurs twice in C cut. Let us denote the two occurrences of r by r + 0i and r 0i respectively. It is now easy to see that Ĉ is isomorphic to the surface obtained from C cut 2Z 2Z by the following identifications: (x + 0i; 2p, 2q) (x 0i; 2p + 2, 2q) for x (, 0) (x + 0i; 2p, 2q) (x 0i; 2p, 2q + 2) for x (1, ) This means that points in Ĉ are of the form (z, p, q) with z C and p, q even integers. Moreover, Ĉ is the Riemann surface of the function C C 2, z (Log(z), Log(1 z)). where Log denotes the principal branch of the logarithm. Consider the set FT : {(x, y, y x, 1 x 1 1 y 1, 1 x 1 y )} (C ) 5 of 5-tuples involved in the 5-term relation. Also, let FT 0 : {(x 0,...,x 4 ) FT 0 < x 1 < x 0 < 1} and define FT Ĉ5 to be the component of the preimage of FT that contains all points ((x 0 ; 0, 0),...,(x 4 ; 0, 0)) with (x 0,..., x 4 ) FT 0. See also [DZ, Rem.2.1]. We can now define the extended Bloch group, following [GZ, Def.1.5]. Definition 3.1. (a) The extended pre-bloch group P(C) is the abelian group generated by the symbols [z; p, q] with (z; p, q) Ĉ, subject to the relation: 4 (20) ( 1) i [x i ; p i, q i ] 0 for ((x 0 ; p 0, q 0 ),...,(x 4 ; p 4, q 4 )) FT. i0 (b) The extended Bloch group B(C) is the kernel of the homomorphism (21) ˆν : P(C) C C, [z; p, q] (Log(z) + pπi) ( Log(1 z) + qπi) (c) We will call the 2-term complex (21) the extended Bloch-Suslin complex. For a comparison between the extended Bloch-Suslin complex and the Suslin complex, see Section 6.3. Remark 3.2. There are four versions of the extended Bloch group, depending on whether (p, q) Z 2 versus (p, q) (2Z) 2, and whether we include the transfer relation of [Ne1, Eqn.3]. For a comparison of Neumann s version of the extended Bloch group with the above definition, see [GZ, Rem.4.3] The Rogers dilogarithm. The extended Bloch-Suslin complex (21) is very closely related to the functional properties of a normalized form of the dilogarithm. In fact, it would be hard to motivate the extended Bloch-Suslin complex without knowing the Rogers dilogarithm and vice-versa. Following [Ne1, DZ], the Rogers dilogarithm is the following function defined on the open interval (0, 1): (22) L(z) Li 2 (z) + 1 π2 Log(z)Log(1 z) 2 6

8 8 STAVROS GAROUFALIDIS where z Li 2 (z) 0 Log(1 t) dt t is the classical dilogarithm function. In [GZ, Defn.1.5], Goette and Zickert extended L(z) to a multivalued analytic function on Ĉ as follows (see [Ne1, Prop.2.5]): n1 z n n 2 (23) ˆR : Ĉ C/Z(2) (24) (25) ˆR(z; p, q) : Li 2 (z) + 1 π2 (Log(z) + πip)(log(1 z) + πiq) 2 6 L(z) + πi 2 (qlog(z) + plog(1 z)) π2 pq 2. where as usual, for a subgroup K of (C, +) and an integer n Z, we denote K(n) (2πi) n K C. Let us comment a bit on the properties of the Rogers dilogarithm. Remark 3.3. A computation involving the monodromy of the Li 2 function shows that the dilogarithm Li 2 has an analytic extension: (26) Li 2 : Ĉ C/Z(2) See [Oe, Prop.1]. Compare also with [Ne1, Prop.2.5]. In [GZ, Lem.2.2] it is shown that ˆR takes values in C/Z(2) and that the Rogers dilogarithm is defined on the extended pre-bloch group; we will denote the extension by ˆR. In other words, we have: (27) ˆR : P(C) C/Z(2). In [DZ] and [GZ] it was shown that the Rogers dilogarithm coincides with the C/Z(2)-valued Cheeger-Chern- Simons class (denoted Ĉ2 in [DZ]). For a discussion on this matter, see [DZ, Thm.4.1]. Remark 3.4. It is natural to ask why to modify the dilogarithm as in Rogers extension. The motivation is that ˆL (but not the dilogarithm function Li 2 (z) nor the Bloch-Wigner dilogarithm) satisfies the extended 5-term relation FT; see [DZ, Sec.2] and [GZ, Lem.2.2]. Remark 3.5. For the purposes of comparing the Rogers dilogarithm with other special functions (such as the entropy function discussed in [Ga1]) It is easy to see that the derivative of the Rogers dilogarithm is given by: (28) L (z) 1 ( log(1 z) + log(z) ). 2 z 1 z Thus, ˆR descends to a map: (29) ˆR : B(C) C/Z(2) which can be exponentiated to a map: (30) e 1 2πi ˆR : B(C) C. The maps R and ˆR of the Bloch group and its extended version are part of the following commutative diagram:

9 q-terms, SINGULARITIES AND THE EXTENDED BLOCH GROUP 9 ˆR e 2πi B(C) C/Z(2) C R ii e 2πi B(C) ir R + For a proof of the commutativity of the left square, see [DZ, Prop.4.6]. Let us end this section with a remark. Remark 3.6. Even though the regulators R and ˆR are defined on the (extended) pre-bloch groups, the following diagram is not commutative: ˆR B(C) C/(2π 2 Z) B(C) R ir ii 3.3. Two cousins of the extended Bloch group: K3 ind (C) and CH 2 (C, 3). The Bloch group B(F) of a field has two cousins: K3 ind (F) and the higher Chow groups CH 2 (F, 3), also defined by Bloch; see [B2]. A spectral sequence argument (attributed to Bloch and Bloch-Lichtenbaum) and some low-degree computations imply that for every infinite field F we have an isomorphism: (31) K ind 3 (F) CH 2 (F, 3). For a detailed discussion, see [E-V, Prop ]. On the other hand, in [Su1, Thm.5.2] Suslin proves the existence of a short exact sequence: (32) 0 Tor(µ F, µ F ) K ind 3 (F) B(F) 0. where Tor(µ F, µ F ) is the unique nontrivial extension of Tor(µF, µ F ), where µ F is the roots of unity of F. For F C, the above short exact sequence becomes: (33) 0 Q/Z K ind 3 (C) B(C) 0. Moreover, Suslin proves in [Su1] that B(C) is a Q-vector space. On the other hand, in [GZ, Thm.3.12] Goette and Zickert prove that the extended Bloch group fits in a short exact sequence: (34) 0 Q/Z ˆψ B(C) B(C) 0, for an explicit map ˆψ. Equations (31), (33), (34), together with the fact that B(C) is a Q-vector space, imply the following. Proposition 3.7. There exist abstract isomorphisms: (35) B(C) K ind 3 (C) CH 2 (F, 3). In addition there are maps (sometimes also known by the name of cycle maps or Abel-Jacobi maps): R : K3 ind (C) HD(Spec(C), 1 Z(2)) C/Z(2) R : CH 2 (F, 3) HD 1 (Spec(C), Z(2)) C/Z(2) where H D denotes Deligne cohomology. For an explicit formula for R, see [KLM-S, Sec.5.7]. Let us end this section with a problem and a question. Recall that the extended Bloch group can be defined for any subfield F of the complex numbers, as discussed in [Ne1].

10 10 STAVROS GAROUFALIDIS Problem 3.8. Define explicit isomorphisms in (35) that commute with the maps R, R and R. We will come back to this problem in a forthcoming publication; [GZ]. For a careful relation between B(F) and K ind 3 (F) in the case of a number field F, see [Zi] q-terms and potential functions. In this section, we will assign elements of B(C) to a q-term t; see Theorem 2 below. Definition 3.9. Consider the following multivalued function on C : ( ) 1 π 2 (36) Φ : Ĉ C/Z(1), Φ(x) 2πi 6 Li 2(x). Remark 3.3 shows that indeed Φ takes values in C/Z(1). The function x Φ(e 2πix ) appears in work of Kashaev on the Volume Conjecture; see [Ks]. The relevance of the special function Φ is that it describes the growth rate of the q-factorials at complex roots of unity; see Lemma 6.1 in Section 6.1. The next definition associates a potential function to a q-term. Definition Given a q-term t as in (1), let us define its potential function V t by: (37) V t (z) 1 2πi Q(Log(z)) + 1 2πi Logǫ Log(zL ) + ǫ j Φ(z Aj ) where z (z 0, z 1,...,z r ) and Log(z) (Log(z 0 ),..., Log(z r )). Let X t and denote the critical points of the potential V t. Since V t (z) is a multivalued function, let us explain its domain. Given a q-term t as in (1), let denote J r (38) H t {z (C ) r+1 z L (1 z Aj ) (1 z i ) 0}. Let D t denote the universal abelian cover of (C ) r+1 \ H t. Observe that for every i 0,...,r, j 1,...,J there are well-defined analytic maps: i0 (39) π i, π Aj, π L : (C ) r+1 \ H t C given by π i (z) z i, π Aj (z) z Aj and π L (z) z L ; using the notation of (10). Lifting them to the universal abelian cover, gives rise to analytic maps: (40) ˆπ i, ˆπ Aj, ˆπ L : D t Ĉ. We denote the image of z D t under these maps by z i, z Aj and z L respectively. With these conventions we have: Lemma Equation (37) defines an analytic function: (41) V t : D t C/Z(1). The next proposition describes the critical points and the critical values of the potential function. Proposition (a) The critical points z of V t are the solutions to the following system of Logarithmic Variational Equations: (42) ǫ j v i (A j )Log(1 z Aj ) + Q z i (Log(z)) + Logǫ v i (L) 0

11 q-terms, SINGULARITIES AND THE EXTENDED BLOCH GROUP 11 for i 0,...,r. (b) There is a map: (43) Xt X t, z (π(z 1 ),..., π(z r )) where π : Ĉ C is the projection map. For A A j, (j 1,...,J) or A L, let p z,a 2Z (or simply, p A if z is clear) be defined so that we have: (44) Log(z A ) v i (A)Log(z i ) p z,a πi. Given w (x; p 0, q 0 ) Ĉ and even integers p, q 2Z let us denote i0 (45) T p 1 T q 0 (w) : (x; p 0 + p, q 0 + q) Ĉ In other words, T 1 and T 0 are generators for the deck transformations of the Z 2 -cover: (46) π : Ĉ C. We need one more piece of notation: for z D t, consider the corresponding elements z L and z i of Ĉ for i 0,...,r. We denote by z L/2 Ĉ the unique element of Ĉ that solves the equation: (47) Log(z L ) Log(z L/2 ) Definition With the above conventions, consider the map (48) ˆβt : X t P(C) given by: (49) w ˆβ t (w) : [z L/2 ; 0, 2 Logǫ πi ] [z L/2 ; 0, 0] + v i (L)Log(z i ) 0. Our next definition assigns a numerical invariant to a special term t. Definition For a q-term t, let (50) CV t {e Vt(z) z satisfies (42)}. i0 ǫ j [T pz,a j 1 (z Aj )] Our next main theorem links the exponential of the critical values of V t with the values of ˆβ t. Let P denote the set of periods in the sense of Kontsevich-Zagier, [KZ]. P is a countable subset of the complex numbers. Theorem 2. (a) The map ˆβ t descends to a map (51) ˆβt : X t B(C) which we denote by the same name. (b) We have a commutative diagram: X t ˆβ t B(C) e V t C e 1 2πi ˆR (c) CV t is a finite subset of e P. Thus, we get a map: (52) CV : q-terms Finite subsets of e P.

12 12 STAVROS GAROUFALIDIS Example Let us continue with the Example 1.5. When a 1, b 2 and ǫ 1, the Variational Equations (16) have solutions X t {z ±1 z e 2πi/6 }. In that case, the corresponding elements of the extended Bloch group are and their values under the map ˆR are given by: 2[z ±1 ; 0, 0] + [z 1/2 ; 0, 2] [z 1/2 ; 0, 0], 0 ± i whose imaginary part equals to the Volume Vol(4 1 ) of the 4 1 knot; see [Th]. Moreover, CV t {e ± 1 2π Vol(41) } { , }. 4. Proof of Theorem 2 In this section we will give the proofs of Proposition 3.12 and Theorem Proof of Proposition It suffices to show that for every i 0,...,r, 2π 1z i V t z i is given by the left hand side of Equation (42). This follows easily from the definition of V t and the elementary computation: (53) Φ (x) 1 2πi Log(1 x). x 4.2. Proof of Theorem 2. The proof is similar to the proof of Theorem 1, once we keep track of the branches of the logarithms. Fix a q-term t and consider the map given by (49). Without loss of generality, we assume that Q(k) 1 k i k j 2 Let us fix z (z 0,...,z r ) that satisfies the Variational Equations (11). We will show that ˆν(ˆβ t (z)) 0 C C, where ˆν is given by (21). We have where It follows that ˆβ 1,t (z) i,j0 ˆβ t (z) ˆβ 1,t (z) + ˆβ 2,t (z) ǫ j [T pz,a j 1 (z Aj )] ˆβ 2,t (w) [z L/2 ; 0, 2 Logǫ πi ] [z L/2 ; 0, 0]. where ˆν(ˆβ 1,t (z)) ˆν(ˆβ 2,t (z)) ˆν(ˆβ t (z)) ˆν(ˆβ 1,t (z)) + ˆν(ˆβ 2,t (z)) ǫ j (Log(z Aj ) + p z,aj πi) ( Log(1 z Aj )) Log(z L/2 ) 2Logǫ 2Log(z L/2 ) Logǫ. On the other hand, using Equation (44), we have:

13 q-terms, SINGULARITIES AND THE EXTENDED BLOCH GROUP 13 (Log(z A ) + p z,a πi) Log(1 z A ) v i (A)(Log(z i ) Log(1 z A )) i0 Log(z i ) (v i (A)Log(1 z A )). Since z satisfies the Logarithmic Variational Equations (42), after we interchange the j and i summation, we obtain that: ˆν(ˆβ 1,t (z)) i0 i0 i0 Log(z i ) ( ǫ j v i (A j )Log(1 z Aj )) Log(z i ) ( ǫ j v i (A j )Log(1 z Aj )) i0 Log(z i ) ( Q z i (Log(z)) + Logǫ v i (L)). Since Q is an integral symmetric bilinear form and is skew-symmetric, it follows that Moreover, i0 Log(z i ) ( Q z i (Log(z)) 0. Thus, which implies that Log(z i ) (Logǫ v i (L)) i0 v i (L)Log(z i ) Logǫ i0 (Log(z L ) + p z,l πi) Logǫ Log(z L ) Logǫ. ˆν(ˆβ 1,t (z)) Log(z L ) Logǫ ˆν(ˆβ t (z)) (2Log(z L/2 ) + Log(z L )) Logǫ. On the other hand, reducing Equation (47) modulo πiz, and using (44) we have: In other words, we have: 0 Log(z L ) Log(z L/2 ) v i (L)Log(z i ) i0 Log(z L ) Log(z L/2 ) (Log(zL ) + πip z,l ) 1 2 (Log(zL ) + 2Log(z L/2 )) + πip z,l (Log(zL ) + 2Log(z L/2 )). (54) Log(z L ) + 2Log(z L/2 ) πi 2 Z.

14 14 STAVROS GAROUFALIDIS Since Logǫ πiz, it follows that ˆν(ˆβ t (z)) 0 and concludes part (a) of Theorem 2. For part (b), observe first that for every w Ĉ and every even integers p and q we have: ˆR([T p 1 T q 0 (w)]) ˆR([w]) + πi (qlog(z) + plog(1 z)). 2 Moreover, by the definition of Φ and by Equation (44), for any integral linear form A A j for j 1,...,J or A L, we have: Φ(z A ) ( 1 ˆR([T pz,a 1 (z A )]) 1 2πi 2 Log(zA )Log(1 z A ) πi ) 2 p z,alog(1 z A ). Thus, using the definition of the potential function from Equation (37), we obtain that: where T πi 2 V t (z) 1 2πi R(ˆβ t (z)) + T 1 + T 2 ǫ j Log(z Aj )Log(1 z Aj ) πi 2 p z,a j Log(1 z Aj ) j T 2 1 2πi Q(Log(z)) 1 2πi Logǫ Log(zL ) 1 2πi ( ˆR([z L/2 ; 0, 2Logǫ ] [z L/2 ; 0, 0]) πi 1 2πi Q(Log(z)) 1 2πi Logǫ Log(zL ) Logǫ 2πi Log(z L/2 ). Using (44), and interchanging i and j summation, and using the Logarithmic Variational Equations (42), it follows that: ǫ j Log(z Aj )Log(1 z Aj ) j0 Thus, ( ǫ j v i (A j )Log(z i ) p z,aj πi)log(1 z Aj ) j0 i0 i0 Log(z i ) ǫ j v i (A j )Log(1 z Aj ) i0 j0 Log(z i )( Q z i Logǫ v i (L)) ǫ j p z,aj πilog(1 z Aj ) j0 ǫ j p z,aj πilog(1 z Aj ). j0 T 1 1 4πi i0 Log(z i ) Q (Log(z)) + Logǫ z i 4πi Since Q is a symmetric bilinear form, it follows that 1 Log(z i ) Q (Log(z)) Q(Log(z)) 0. 2 z i i0 v i (L)Log(z i ). This, together with Equation (47) implies that T 1 + T 2 0. Exponentiating, it follows that e Vt(z) e 1 2πi ˆR(ˆβ t(z)) C which concludes part (b) of Theorem 2. For part (c), since an analytic function is constant on a connected component of its sets of critical points, and since the set of critical points X t is the set of complex points of an affine variety defined over Q, it follows that X t has finitely many connected components. Thus, CV t is a finite subset of C. Moreover, since i0

15 q-terms, SINGULARITIES AND THE EXTENDED BLOCH GROUP 15 the value of the map (29) at a point [z; p, q] with z Q is a period, in the sense of Kontsevich-Zagier, and since every connected component of X t has a point w so that z e 2πiw (Q) r+1 (due to the Variational Equations (11)), it follows that CV t is a subset of e P. This concludes the proof of Theorem Special q-terms, generating series and singularities The second part of the paper assigns to germs of analytic functions L np t (z) and L p t (z) to a special q- term, and formulates a conjecture regarding their analytic continuation in the complex plane minus a set of singularities related to the image of the composition ˆR ˆβ t What is a special q-term? In this section we introduce the notion of a special q-term. Of course, every special q-term is a q-term. Examples of special q-terms that come naturally from Quantum Topology are discussed in Section 5.3 below. Definition 5.1. A special q-hypergeometric term t (in short, special q-term) in the r + 1 variables k (k 0, k 1,..., k r ) N r is a list that consists of An integral symmetric quadratic form Q(k) in k, An integral linear form L in k and a vector ǫ (ǫ 0,..., ǫ r ) with ǫ i ±1 for all i, Integral linear forms B j, C j, D j, E j in k for j 1,...,J The list t satisfies the following condition. Its Newton polytope P t defined by (55) P t {w R r B j (1, w) C j (1, w) 0, D j (1, w) E j (1, w) 0, j 1,...,J} is a nonempty rational compact polytope of R r. This accurate but rather obscure definition is motivated from the fact that a special q-term t gives rise to an expression of the form J ( ) (56) t k (q) q Q(k) ǫ L(k) Bj (k) (q) Dj(k) Z[q ±1 ] C j (k) q (q) Ej(k) valid for k N r+1 such that B j (k) C j (k) 0 and D j (k) E j (k) 0 for all j 1,...,J From special q-terms to generating series. We now discuss how a special q-term t gives rise to a sequence of Laurent polynomials (a t,n (q)) and to a generating series G t (z). Given a special q-term t in r + 1 variables k (k 0,...,k r ), it will be convenient to single out the first variable k 0, and denote it by n as follows: (57) n k 0, k (k 1,...,k r ) With the above convention, we have k (n, k ). Definition 5.2. A special q-term t gives rise to a sequence of Laurent polynomials (a t,n (q)) as follows: (58) a t,n (q) t n,k (q) Z[q ±1 ] k np t N r where the summation is over the finite set np t N r of lattice points of the translated Newton polytope of t. Definition 5.3. A special q-term t gives rise to a power series L np t (z) defined by: (59) L np t (z) a t,n (e 2πi n )z n. The next lemma proves that L np t (z) is an analytic function at z 0. Lemma 5.4. For every special q-term t, L np t (z) is analytic at z 0. n0

16 16 STAVROS GAROUFALIDIS Proof. Fix a special q-term t as Definition (56). Let f(q) 1 denote the sum of the absolute values of the coefficients of a Laurent polynomial f(q) Q[q ±1 ]. It suffices to show that there exists C > 0 so that (60) a t,n (q) 1 C n for all n. In that case, since e 2πi/n 1, it follows that a t,n (e 2πi n ) C n for all n, thus L np t (z) is analytic for z < 1/C. Now, recall that for natural numbers a, b with a b 0 we have: ( ) ( ) ( ) a a a N[q ±1 ], 1 2 a. b q b q b (see for example, [St]). In addition for natural numbers c, d with c d 0 we have: (q) c c (1 q j ), (q) c 1 2 c d 2 c. (q) d (q) d jd+1 Equation (58) implies that a t,n (q) 1 t n,k (q) 1 k np t N r Since the number of terms in the above sum is bounded by a polynomial function of n, and the summand is bounded by an exponential function of n, Equation (60) follows Examples of special q-terms from Quantum Topology. Quantum Topology gives a plethora of special q-terms t to knotted 3-dimensional objects whose corresponding sequence α t,n (q) depends on the knotted object itself. A concrete example is given in [GL1, Sec.3]. To state it, let J K,n (q) Z[q ±1 ] denote the colored Jones polynomial of a knot K, colored by the n dimensional irreducible representation of sl 2 and normalized to equal to 1 at the unknot; see [Tu]. In [GL1, Lem.3.2] the following is shown. Lemma 5.5. Let β denote a braid whose closure is a knot K. Then, there exists a special q-term t β such that (61) J K,n (q) a tβ,n(q) for all n N. It follows that a tβ,n(e 2πi n ) is the n-th Kashaev invariant of K; [Ks]. Please observe that the special q-term t β depends on the braid β, and that a fixed knot K can always be obtained as the closure of infinitely many braids β. Nonetheless, for all such braids β, the sequence of polynomials (a tβ,n(q)) depend only on the knot K. Remark 5.6. A computer implementation of Lemma 5.5 is available from [B-N]. It uses as input a braid word in the standard generators of the braid group, and outputs the expression (56) of the corresponding special q-term. Quantum Topology constructs many more examples of special q-terms, that depend on a pair (K, g) of a knot K and a simple Lie algebra g, or to a pair (M, g) of a closed 3-manifold M with the integer homology of S 3 and a simple Lie algebra g An ansatz for the singularities of L np t (z). In this section we connect the two different parts of the paper. Namely, a special q-term t gives rise to (a) a finite set CV t C from Definition 3.14, (b) a generating series L np t (z) from Definition 5.3. Conjecture 1. For every special q-term t, the germ L np t (z) has an analytic continuation as a multivalued function in C\(CV t {0}). Moreover, the local monodromy of L np t (z) is quasi-unipotent, i.e., its eigenvalues are complex roots of unity.

17 q-terms, SINGULARITIES AND THE EXTENDED BLOCH GROUP 17 Example 5.7. If the special q-term is given by (15), then Conjecture 1 is known; see [CG]. With the notation of (15), the case of (a, b, ǫ) ( 1, 2, 1) coincides with the Kashaev invariant of the 4 1 knot. In a sequel to this paper [Ga3], we discuss in detail examples of Conjecture 1 for special q-terms that comes from Quantum Topology. 6. For completeness 6.1. Motivation for the special function Φ and the potential function. Recall the special function Φ given by Equation (36). The next lemma is our motivation for introducing Φ. Kashaev informs us that this computation was well-known to Faddeev, and was a starting point in the theory of q-dilogarithm function; see [FK]. Lemma 6.1. For every α (0, 1) we have: (62) [αn] k1 (1 e 2πik N ) e NΦ(e 2πiα )+O( log N N ). Proof. The proof is similar to the proof of [GL2, Prop.8.2], and follows from applying the Euler-MacLaurin summation formula [αn] log( k1 together with the fact that: [αn] (1 e 2πik N )) 1 0 αn k1 1 log(1 e 2πik N ) 0 ( log N log(1 e 2πiαx )dx + O N log(1 e 2πiax )dx 1 ( ) π 2 2πiα 6 Li 2(e 2πiα ). Fix a special q-term t k (q) where k (k 1,..., k r ) and a positive natural number N N. Fix also w (w 0,..., w r ) where w i (0, 1) for i 0,...,r. Let us abbreviate ([w 0 N], [w 1 N],..., [w r N]) by [wn]. Lemma 6.1 implies the following. Lemma 6.2. With the above assumptions, we have: (63) log t [wn] e NVt(e2πiw )+O( log N N ). This motivates our definition of the potential function Laplace s method for a q-term. There is an alternative way to derive the Variational Equations (11) from a q-term t. Since t k is q-hypergeometric, and k (k 1,..., k r ), it follows that for every i 0,...,r we have R i (z 0,..., z r, q) : t n,k 1,...,k i+1,...,k r (q) t n,k1,...,k r (q) Q(z 0,...,z r, q) where z i q ki for i 1,...,r and z 0 q n. It is easy to see that the system of equations: ), R i (z 0, z 1,..., z r ) 1, i 0,...,r. is identical to the system (11) of variational equations. In discrete math, the above system is known as Laplace s method.

18 18 STAVROS GAROUFALIDIS 6.3. A comparison between the Bloch group and its extended version. A comparison between the extended Bloch-Suslin complex and the Suslin complex is summarized in the following diagram with short exact rows and columns. The diagram is taken by combining [GZ, Sec.3] with [Ne1, Thm.7.7], and using the map ˆχ from [GZ, Eqn.3.11] e 0 Q/Z 2πi C C/(Q/Z) 0 ˆψ ˆχ ˆν 0 B(C) P(C) C C K2 (C) 0 ν 0 B(C) P(C) C C K 2 (C) 0 τ exp exp where G G G Z G/(a b + b a) ˆψ(z) ˆχ(e 2πiz ) τ(z) z 2πi ( exp exp)(a b) (e 2πia e 2πib ). In addition, we have the following useful Corollary, from [GZ, 3.14]. Corollary 6.3. For z C we have: The restriction is (2πi) 2 ˆR(ˆχ(e 2πiz )) z ˆR : Ker( B(C) B(C)) C/Z(2) References [AL] M. Abouzahra and L. Lewin, The polylogarithm in algebraic number fields, J. Number Theory 21 (1985) [B-N] D. Bar-Natan, Knot Atlas, [BD] A. Beilinson and P. Deligne, Motivic interpretation of the Zagier conjecture connecting polylogarithms and regulators, in Motives, Proc. Symp. Pure Math. AMS 55 (1984) [B1] S. Bloch, Higher regulators, algebraic K-theory, and zeta functions of elliptic curves, printed version of the Irvine 1978 lectures. CRM Monograph Series, 11 AMS, [B2], Algebraic cycles and higher K-theory, Adv. in Math. 61 (1986) [CG] O. Costin and S. Garoufalidis, Resurgence of 1-dimensional sums of q-factorials, preprint [DS] J.L. Dupont and C.H. Sah, Scissors congruences II, J. Pure Appl. Algebra 25 (1982) [DZ] and C. Zickert, A dilogarithmic formula for the Cheeger-Chern-Simons class, Geom. Topol. 10 (2006) [E-V] P. Elbaz-Vincent, A short introduction to higher Chow groups, in Transcendental aspects of algebraic cycles, London Math. Soc. Lecture Note Ser., 313 (2004) [FK] L.D. Faddeev and R.M. Kashaev, Quantum dilogarithm, Modern Phys. Lett. A 9 (1994) [GL1] S. Garoufalidis and T.T.Q. Le, The colored Jones function is q-holonomic, Geom. and Topology 9 (2005) [GL2] and T.T.Q. Le, Asymptotics of the colored Jones function of a knot, preprint 2005 math.gt/ [GL3] and, Gevrey series in quantum topology, J. Reine Angew. Math., 618 (2008) [Ga1], An extended version of additive K-theory, J. K-Theory 4 (2009) [Ga2], An ansatz for the singularities of hypergeometric multisums, Adv. in Appl. Math. 41 (2008)

19 q-terms, SINGULARITIES AND THE EXTENDED BLOCH GROUP 19 [Ga3], Chern-Simons theory, analytic continuation and arithmetic, Acta Math. Vietnam. 33 (2008) [GZ] S. Goette and C. Zickert, The Extended Bloch Group and the Cheeger-Chern-Simons Class, Geom. Topol. 11 (2007) [Go1] A. Goncharov, Polylogarithms and motivic Galois groups, in Motives, Proc. Symp. Pure Math. AMS 55 (1984) [Go2], Geometry of configurations, polylogarithms and motivic cohomology, Adv. in Math. 114 (1995) [Ks] R. Kashaev, The hyperbolic volume of knots from the quantum dilogarithm, Modern Phys. Lett. A 39 (1997) [KLM-S] M. Kerr, J.D. Lewis and S. Müller-Stach, The Abel-Jacobi map for higher Chow groups, Compos. Math. 142 (2006) [Ko] M. Kontsevich, problem proposed in Aarhus, [KZ] and D. Zagier, Periods, in Mathematics unlimited 2001 and beyond, (2001) [Le] L. Lewin, The inner structure of the dilogarithm in algebraic fields, J. Number Theory 19 (1984) [Na] W. Nahm, Conformal field theory and torsion elements of the Bloch group, in Frontiers in Number Theory, Physics and Geometry II, [NZ] W.D. Neumann and D. Zagier, Volumes of hyperbolic three-manifolds, Topology 24 (1985) [Ne1], Extended Bloch group and the Cheeger-Chern-Simons class, Geom. Topol. 8 (2004) [Ne2], Hilbert s 3rd problem and invariants of 3-manifolds, The Epstein birthday schrift, Geom. Topol. Monogr., 1 (1998) [Oe] J. Oesterlé, Polylogarithmes, Séminaire Bourbaki, Vol. 1992/93. Astérisque No. 216 (1993), Exp. No [St] R.P. Stanley, Enumerative Combinatorics, Volume 1, Cambridge University Press (1997). [Su1] A.A. Suslin, K 3 of a field, and the Bloch group, Translated in Proc. Steklov Inst. Math. 4 (1991) [Su2], Algebraic K-theory of fields, Proceedings of the International Congress of Mathematicians, Vol. 1, 2, Berkeley, (1986) [Su3], On the K-theory of local fields, Journal of Pure and Applied Alg. 34 (1984) [Th] W. Thurston, The geometry and topology of 3-manifolds, 1979 notes, available from MSRI. [Tu] V. G. Turaev, Quantum invariants of knots and 3-manifolds, de Gruyter Studies in Mathematics 18 Walter de Gruyter, [WZ] H. Wilf and D. Zeilberger, An algorithmic proof theory for hypergeometric (ordinary and q) multisum/integral identities, Inventiones Math. 108 (1992) [Za1] D. Zagier, Polylogarithms, Dedekind zeta functiona and the algebraic K-theory of fields, in Arithmetic Algebraic Geometry, Progr. Math. 89 (1991) [Za2], Zagier, Don The dilogarithm function, Frontiers in number theory, physics, and geometry. II, Springer (2007) [Z] D. Zeilberger, A holonomic systems approach to special functions identities, J. Comput. Appl. Math. 32 (1990) [Zi] C. Zickert, The extended Bloch group and algebraic K-theory, preprint 2009 arxiv: School of Mathematics, Georgia Institute of Technology, Atlanta, GA , USA, stavros address: stavros@math.gatech.edu

AN EXTENDED VERSION OF ADDITIVE K-THEORY

AN EXTENDED VERSION OF ADDITIVE K-THEORY STAVROS GAROUFALIDIS Abstract. There are two infinitesimal (i.e., additive) versions of the K-theory of a field F : one introduced by Cathelineau, which is an F

STAVROS GAROUFALIDIS AND THOMAS W. MATTMAN

THE A-POLYNOMIAL OF THE ( 2, 3, 3 + 2n) PRETZEL KNOTS STAVROS GAROUFALIDIS AND THOMAS W. MATTMAN Abstract. We show that the A-polynomial A n of the 1-parameter family of pretzel knots K n = ( 2, 3,3+ 2n)

Advancement to Candidacy. Patterns in the Coefficients of the Colored Jones Polynomial. Katie Walsh Advisor: Justin Roberts

5000 10 000 15 000 3 10 12 2 10 12 1 10 12 Patterns in the Coefficients of the Colored Jones Polynomial 1 10 12 Advisor: Justin Roberts 2 10 12 3 10 12 Introduction Knots and Knot Invariants The Jones

Logarithmic functional and reciprocity laws

Contemporary Mathematics Volume 00, 1997 Logarithmic functional and reciprocity laws Askold Khovanskii Abstract. In this paper, we give a short survey of results related to the reciprocity laws over the

A REAPPEARENCE OF WHEELS

A REAPPEARENCE OF WHEELS STAVROS GAROUFALIDIS Abstract. Recently, a number of authors [KS, Oh2, Ro] have independently shown that the universal finite type invariant of rational homology 3-spheres on the

arxiv: v1 [math.nt] 15 Mar 2012

ON ZAGIER S CONJECTURE FOR L(E, 2): A NUMBER FIELD EXAMPLE arxiv:1203.3429v1 [math.nt] 15 Mar 2012 JEFFREY STOPPLE ABSTRACT. We work out an example, for a CM elliptic curve E defined over a real quadratic

A NOTE ON QUANTUM 3-MANIFOLD INVARIANTS AND HYPERBOLIC VOLUME

A NOTE ON QUANTUM 3-MANIFOLD INVARIANTS AND HYPERBOLIC VOLUME EFSTRATIA KALFAGIANNI Abstract. For a closed, oriented 3-manifold M and an integer r > 0, let τ r(m) denote the SU(2) Reshetikhin-Turaev-Witten

The Extended Bloch Group and the Cheeger Chern Simons Class

Geometr & Topolog (2007) 623 635 623 The Extended Bloch Group and the Cheeger Chern Simons Class SEBASTIAN GOETTE CHRISTIAN K ZICKERT We present a formula for the full Cheeger Chern Simons class of the

On combinatorial zeta functions

On combinatorial zeta functions Christian Kassel Institut de Recherche Mathématique Avancée CNRS - Université de Strasbourg Strasbourg, France Colloquium Potsdam 13 May 2015 Generating functions To a sequence

Some aspects of the multivariable Mahler measure

Some aspects of the multivariable Mahler measure Séminaire de théorie des nombres de Chevaleret, Institut de mathématiques de Jussieu, Paris, France June 19th, 2006 Matilde N. Lalín Institut des Hautes

ON THE CHARACTERISTIC AND DEFORMATION VARIETIES OF A KNOT

ON THE CHARACTERISTIC AND DEFORMATION VARIETIES OF A KNOT STAVROS GAROUFALIDIS Dedicated to A. Casson on the occasion of his 60th birthday Abstract. The colored Jones function of a knot is a sequence of

5.3 The Upper Half Plane

Remark. Combining Schwarz Lemma with the map g α, we can obtain some inequalities of analytic maps f : D D. For example, if z D and w = f(z) D, then the composition h := g w f g z satisfies the condition

Trace fields of knots

JT Lyczak, February 2016 Trace fields of knots These are the knotes from the seminar on knot theory in Leiden in the spring of 2016 The website and all the knotes for this seminar can be found at http://pubmathleidenunivnl/

The Grothendieck-Katz Conjecture for certain locally symmetric varieties

The Grothendieck-Katz Conjecture for certain locally symmetric varieties Benson Farb and Mark Kisin August 20, 2008 Abstract Using Margulis results on lattices in semi-simple Lie groups, we prove the Grothendieck-

Polylogarithms and Hyperbolic volumes Matilde N. Laĺın

Polylogarithms and Hyperbolic volumes Matilde N. Laĺın University of British Columbia and PIMS, Max-Planck-Institut für Mathematik, University of Alberta mlalin@math.ubc.ca http://www.math.ubc.ca/~mlalin

ON POSITIVITY OF KAUFFMAN BRACKET SKEIN ALGEBRAS OF SURFACES

ON POSITIVITY OF KAUFFMAN BRACKET SKEIN ALGEBRAS OF SURFACES THANG T. Q. LÊ Abstract. We show that the Chebyshev polynomials form a basic block of any positive basis of the Kauffman bracket skein algebras

FAMILIES OF ALGEBRAIC CURVES AS SURFACE BUNDLES OF RIEMANN SURFACES

FAMILIES OF ALGEBRAIC CURVES AS SURFACE BUNDLES OF RIEMANN SURFACES MARGARET NICHOLS 1. Introduction In this paper we study the complex structures which can occur on algebraic curves. The ideas discussed

Patterns and Stability in the Coefficients of the Colored Jones Polynomial. Katie Walsh Advisor: Justin Roberts

5000 10 000 15 000 3 10 12 2 10 12 1 10 12 Patterns and Stability in the Coefficients of the Colored Jones Polynomial 1 10 12 2 10 12 Advisor: Justin Roberts 3 10 12 The Middle Coefficients of the Colored

Evgeniy V. Martyushev RESEARCH STATEMENT

Evgeniy V. Martyushev RESEARCH STATEMENT My research interests lie in the fields of topology of manifolds, algebraic topology, representation theory, and geometry. Specifically, my work explores various

SPRING 2006 PRELIMINARY EXAMINATION SOLUTIONS

SPRING 006 PRELIMINARY EXAMINATION SOLUTIONS 1A. Let G be the subgroup of the free abelian group Z 4 consisting of all integer vectors (x, y, z, w) such that x + 3y + 5z + 7w = 0. (a) Determine a linearly

DENSITY SPECTRA FOR KNOTS. In celebration of Józef Przytycki s 60th birthday

DENSITY SPECTRA FOR KNOTS ABHIJIT CHAMPANERKAR, ILYA KOFMAN, AND JESSICA S. PURCELL Abstract. We recently discovered a relationship between the volume density spectrum and the determinant density spectrum

Homological representation formula of some quantum sl 2 invariants

Homological representation formula of some quantum sl 2 invariants Tetsuya Ito (RIMS) 2015 Jul 20 First Encounter to Quantum Topology: School and Workshop Tetsuya Ito (RIMS) Homological representation

THE JACOBIAN OF A RIEMANN SURFACE

THE JACOBIAN OF A RIEMANN SURFACE DONU ARAPURA Fix a compact connected Riemann surface X of genus g. The set of divisors Div(X) forms an abelian group. A divisor is called principal if it equals div(f)

Hilbert s 3rd Problem and Invariants of 3 manifolds

ISSN 1464-8997 (on line) 1464-8989 (printed) 383 Geometry & Topology Monographs Volume 1: The Epstein birthday schrift Pages 383 411 Hilbert s 3rd Problem and Invariants of 3 manifolds Walter D Neumann

TORUS KNOT AND MINIMAL MODEL

TORUS KNOT AND MINIMAL MODEL KAZUHIRO HIKAMI AND ANATOL N. KIRILLOV Abstract. We reveal an intimate connection between the quantum knot invariant for torus knot T s, t and the character of the minimal

On orderability of fibred knot groups

Under consideration for publication in Math. Proc. Camb. Phil. Soc. 1 On orderability of fibred knot groups By BERNARD PERRON Laboratoire de Topologie, Université de Bourgogne, BP 47870 21078 - Dijon Cedex,

CHARACTERISTIC CLASSES

1 CHARACTERISTIC CLASSES Andrew Ranicki Index theory seminar 14th February, 2011 2 The Index Theorem identifies Introduction analytic index = topological index for a differential operator on a compact

FOURIER COEFFICIENTS OF VECTOR-VALUED MODULAR FORMS OF DIMENSION 2

FOURIER COEFFICIENTS OF VECTOR-VALUED MODULAR FORMS OF DIMENSION 2 CAMERON FRANC AND GEOFFREY MASON Abstract. We prove the following Theorem. Suppose that F = (f 1, f 2 ) is a 2-dimensional vector-valued

AN ANSATZ FOR THE ASYMPTOTICS OF HYPERGEOMETRIC MULTISUMS

AN ANSATZ FOR THE ASYMPTOTICS OF HYPERGEOMETRIC MULTISUMS STAVROS GAROUFALIDIS Abstract. Sequences that are defined by multisums of hypergeometric terms with compact support occur frequently in enumeration

FINITELY GENERATED SIMPLE ALGEBRAS: A QUESTION OF B. I. PLOTKIN

FINITELY GENERATED SIMPLE ALGEBRAS: A QUESTION OF B. I. PLOTKIN A. I. LICHTMAN AND D. S. PASSMAN Abstract. In his recent series of lectures, Prof. B. I. Plotkin discussed geometrical properties of the

Kleine AG: Travaux de Shimura

Kleine AG: Travaux de Shimura Sommer 2018 Programmvorschlag: Felix Gora, Andreas Mihatsch Synopsis This Kleine AG grew from the wish to understand some aspects of Deligne s axiomatic definition of Shimura

Lecture 1. Toric Varieties: Basics

Lecture 1. Toric Varieties: Basics Taras Panov Lomonosov Moscow State University Summer School Current Developments in Geometry Novosibirsk, 27 August1 September 2018 Taras Panov (Moscow University) Lecture

Modular forms and the Hilbert class field

Modular forms and the Hilbert class field Vladislav Vladilenov Petkov VIGRE 2009, Department of Mathematics University of Chicago Abstract The current article studies the relation between the j invariant

FRANÇOIS BR U N A U L T (*) Zagier s conjectures on special values of L-functions (**)

R i v. M a t. U n i v. P a r m a ( 7 ) 3 * ( 2 0 0 4 ), 1 6 5-1 7 6 FRANÇOIS BR U N A U L T (*) Zagier s conjectures on special values of L-functions (**) I ll use the following notations through the text.

arxiv: v1 [math.rt] 11 Sep 2009

FACTORING TILTING MODULES FOR ALGEBRAIC GROUPS arxiv:0909.2239v1 [math.rt] 11 Sep 2009 S.R. DOTY Abstract. Let G be a semisimple, simply-connected algebraic group over an algebraically closed field of

Logarithm and Dilogarithm

Logarithm and Dilogarithm Jürg Kramer and Anna-Maria von Pippich 1 The logarithm 1.1. A naive sequence. Following D. Zagier, we begin with the sequence of non-zero complex numbers determined by the requirement

Extensions of motives and higher Chow groups

Extensions of motives and higher Chow groups A. J. Scholl Introduction This note has two purposes: the first is to give a somewhat different description of the higher cycle class map defined by Bloch [3]

Optimistic limits of the colored Jones polynomials

Optimistic limits of the colored Jones polynomials Jinseok Cho and Jun Murakami arxiv:1009.3137v9 [math.gt] 9 Apr 2013 October 31, 2018 Abstract We show that the optimistic limits of the colored Jones

ALGEBRA EXERCISES, PhD EXAMINATION LEVEL

ALGEBRA EXERCISES, PhD EXAMINATION LEVEL 1. Suppose that G is a finite group. (a) Prove that if G is nilpotent, and H is any proper subgroup, then H is a proper subgroup of its normalizer. (b) Use (a)

NEW IDENTITIES INVOLVING SUMS OF THE TAILS RELATED TO REAL QUADRATIC FIELDS KATHRIN BRINGMANN AND BEN KANE

NEW IDENTITIES INVOLVING SUMS OF THE TAILS RELATED TO REAL QUADRATIC FIELDS KATHRIN BRINGMANN AND BEN KANE To George Andrews, who has been a great inspiration, on the occasion of his 70th birthday Abstract.

Math 530 Lecture Notes. Xi Chen

Math 530 Lecture Notes Xi Chen 632 Central Academic Building, University of Alberta, Edmonton, Alberta T6G 2G1, CANADA E-mail address: xichen@math.ualberta.ca 1991 Mathematics Subject Classification. Primary

On Recurrences for Ising Integrals

On Recurrences for Ising Integrals Flavia Stan Research Institute for Symbolic Computation (RISC-Linz) Johannes Kepler University Linz, Austria December 7, 007 Abstract We use WZ-summation methods to compute

Quantum knot invariants

Garoufalidis Res Math Sci (2018) 5:11 https://doi.org/10.1007/s40687-018-0127-3 RESEARCH Quantum knot invariants Stavros Garoufalidis * Correspondence: stavros@math.gatech.edu School of Mathematics, Georgia

Part II. Riemann Surfaces. Year

Part II Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2018 96 Paper 2, Section II 23F State the uniformisation theorem. List without proof the Riemann surfaces which are uniformised

Representation Theory

Part II Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2018 Paper 1, Section II 19I 93 (a) Define the derived subgroup, G, of a finite group G. Show that if χ is a linear character

The kappa function. [ a b. c d

The kappa function Masanobu KANEKO Masaaki YOSHIDA Abstract: The kappa function is introduced as the function κ satisfying Jκτ)) = λτ), where J and λ are the elliptic modular functions. A Fourier expansion

Differential Equations and Associators for Periods

Differential Equations and Associators for Periods Stephan Stieberger, MPP München Workshop on Geometry and Physics in memoriam of Ioannis Bakas November 2-25, 26 Schloß Ringberg, Tegernsee based on: St.St.,

From the elliptic regulator to exotic relations

An. Şt. Univ. Ovidius Constanţa Vol. 16(), 008, 117 16 From the elliptic regulator to exotic relations Nouressadat TOUAFEK Abstract In this paper we prove an identity between the elliptic regulators of

z, w = z 1 w 1 + z 2 w 2 z, w 2 z 2 w 2. d([z], [w]) = 2 φ : P(C 2 ) \ [1 : 0] C ; [z 1 : z 2 ] z 1 z 2 ψ : P(C 2 ) \ [0 : 1] C ; [z 1 : z 2 ] z 2 z 1

3 3 THE RIEMANN SPHERE 31 Models for the Riemann Sphere One dimensional projective complex space P(C ) is the set of all one-dimensional subspaces of C If z = (z 1, z ) C \ 0 then we will denote by [z]

Rigidity, locally symmetric varieties and the Grothendieck-Katz Conjecture

Rigidity, locally symmetric varieties and the Grothendieck-Katz Conjecture Benson Farb and Mark Kisin May 8, 2009 Abstract Using Margulis s results on lattices in semisimple Lie groups, we prove the Grothendieck-

Hyperbolic Knots and the Volume Conjecture II: Khov. II: Khovanov Homology

Hyperbolic Knots and the Volume Conjecture II: Khovanov Homology Mathematics REU at Rutgers University 2013 July 19 Advisor: Professor Feng Luo, Department of Mathematics, Rutgers University Overview 1

Universität Regensburg Mathematik

Universität Regensburg Mathematik Milnor K-theory and motivic cohomology Moritz Kerz Preprint Nr. 11/2007 MILNOR K-THEORY AND MOTIVIC COHOMOLOGY MORITZ KERZ Abstract. These are the notes of a talk given

EXERCISES IN MODULAR FORMS I (MATH 726) (2) Prove that a lattice L is integral if and only if its Gram matrix has integer coefficients.

EXERCISES IN MODULAR FORMS I (MATH 726) EYAL GOREN, MCGILL UNIVERSITY, FALL 2007 (1) We define a (full) lattice L in R n to be a discrete subgroup of R n that contains a basis for R n. Prove that L is

An introduction to arithmetic groups. Lizhen Ji CMS, Zhejiang University Hangzhou , China & Dept of Math, Univ of Michigan Ann Arbor, MI 48109

An introduction to arithmetic groups Lizhen Ji CMS, Zhejiang University Hangzhou 310027, China & Dept of Math, Univ of Michigan Ann Arbor, MI 48109 June 27, 2006 Plan. 1. Examples of arithmetic groups

There s something about Mahler measure

There s something about Mahler measure Junior Number Theory Seminar University of Texas at Austin March 29th, 2005 Matilde N. Lalín Mahler measure Definition For P C[x ±,...,x± n ], the logarithmic Mahler

NUMBER FIELDS WITHOUT SMALL GENERATORS

NUMBER FIELDS WITHOUT SMALL GENERATORS JEFFREY D. VAALER AND MARTIN WIDMER Abstract. Let D > be an integer, and let b = b(d) > be its smallest divisor. We show that there are infinitely many number fields

20 The modular equation

18.783 Elliptic Curves Spring 2015 Lecture #20 04/23/2015 20 The modular equation In the previous lecture we defined modular curves as quotients of the extended upper half plane under the action of a congruence

THE HOMOTOPY TYPE OF THE SPACE OF RATIONAL FUNCTIONS

THE HOMOTOPY TYPE OF THE SPACE OF RATIONAL FUNCTIONS by M.A. Guest, A. Kozlowski, M. Murayama and K. Yamaguchi In this note we determine some homotopy groups of the space of rational functions of degree

Introduction to. Riemann Surfaces. Lecture Notes. Armin Rainer. dim H 0 (X, L D ) dim H 0 (X, L 1 D ) = 1 g deg D

Introduction to Riemann Surfaces Lecture Notes Armin Rainer dim H 0 (X, L D ) dim H 0 (X, L 1 D ) = 1 g deg D June 17, 2018 PREFACE i Preface These are lecture notes for the course Riemann surfaces held

Seifert forms and concordance

ISSN 1364-0380 (on line) 1465-3060 (printed) 403 Geometry & Topology Volume 6 (2002) 403 408 Published: 5 September 2002 G G G G T T T G T T T G T G T GG TT G G G G GG T T T TT Seifert forms and concordance

L 2 BETTI NUMBERS OF HYPERSURFACE COMPLEMENTS

L 2 BETTI NUMBERS OF HYPERSURFACE COMPLEMENTS LAURENTIU MAXIM Abstract. In [DJL07] it was shown that if A is an affine hyperplane arrangement in C n, then at most one of the L 2 Betti numbers i (C n \

A class of trees and its Wiener index.

A class of trees and its Wiener index. Stephan G. Wagner Department of Mathematics Graz University of Technology Steyrergasse 3, A-81 Graz, Austria wagner@finanz.math.tu-graz.ac.at Abstract In this paper,

Power structure over the Grothendieck ring of varieties and generating series of Hilbert schemes of points

arxiv:math/0407204v1 [math.ag] 12 Jul 2004 Power structure over the Grothendieck ring of varieties and generating series of Hilbert schemes of points S.M. Gusein-Zade I. Luengo A. Melle Hernández Abstract

THE QUANTUM CONNECTION

THE QUANTUM CONNECTION MICHAEL VISCARDI Review of quantum cohomology Genus 0 Gromov-Witten invariants Let X be a smooth projective variety over C, and H 2 (X, Z) an effective curve class Let M 0,n (X,

THE S 1 -EQUIVARIANT COHOMOLOGY RINGS OF (n k, k) SPRINGER VARIETIES

Horiguchi, T. Osaka J. Math. 52 (2015), 1051 1062 THE S 1 -EQUIVARIANT COHOMOLOGY RINGS OF (n k, k) SPRINGER VARIETIES TATSUYA HORIGUCHI (Received January 6, 2014, revised July 14, 2014) Abstract The main

The Margulis-Zimmer Conjecture

The Margulis-Zimmer Conjecture Definition. Let K be a global field, O its ring of integers, and let G be an absolutely simple, simply connected algebraic group defined over K. Let V be the set of all inequivalent

Uniformly exponential growth and mapping class groups of surfaces

Uniformly exponential growth and mapping class groups of surfaces James W. Anderson, Javier Aramayona and Kenneth J. Shackleton 27 April 2006 Abstract We show that the mapping class group (as well as closely

Homological Decision Problems for Finitely Generated Groups with Solvable Word Problem

Homological Decision Problems for Finitely Generated Groups with Solvable Word Problem W.A. Bogley Oregon State University J. Harlander Johann Wolfgang Goethe-Universität 24 May, 2000 Abstract We show

Formal Groups. Niki Myrto Mavraki

Formal Groups Niki Myrto Mavraki Contents 1. Introduction 1 2. Some preliminaries 2 3. Formal Groups (1 dimensional) 2 4. Groups associated to formal groups 9 5. The Invariant Differential 11 6. The Formal

20 The modular equation

18.783 Elliptic Curves Lecture #20 Spring 2017 04/26/2017 20 The modular equation In the previous lecture we defined modular curves as quotients of the extended upper half plane under the action of a congruence

Power sums and Homfly skein theory

ISSN 1464-8997 (on line) 1464-8989 (printed) 235 Geometry & Topology Monographs Volume 4: Invariants of knots and 3-manifolds (Kyoto 2001) Pages 235 244 Power sums and Homfly skein theory Hugh R. Morton

Polylogarithms and Double Scissors Congruence Groups

Polylogarithms and Double Scissors Congruence Groups for Gandalf and the Arithmetic Study Group Steven Charlton Abstract Polylogarithms are a class of special functions which have applications throughout

Boundary of Cohen-Macaulay cone and asymptotic behavior of system of ideals

Boundary of Cohen-Macaulay cone and asymptotic behavior of system of ideals Kazuhiko Kurano Meiji University 1 Introduction On a smooth projective variety, we can define the intersection number for a given

Knot Groups with Many Killers

Knot Groups with Many Killers Daniel S. Silver Wilbur Whitten Susan G. Williams September 12, 2009 Abstract The group of any nontrivial torus knot, hyperbolic 2-bridge knot, or hyperbolic knot with unknotting

Research Statement Katherine Walsh October 2013

My research is in the area of topology, specifically in knot theory. The bulk of my research has been on the patterns in the coefficients of the colored Jones polynomial. The colored Jones polynomial,j

Hyperbolic volumes and zeta values An introduction

Hyperbolic volumes and zeta values An introduction Matilde N. Laĺın University of Alberta mlalin@math.ulberta.ca http://www.math.ualberta.ca/~mlalin Annual North/South Dialogue in Mathematics University

COMPLEX VARIETIES AND THE ANALYTIC TOPOLOGY

COMPLEX VARIETIES AND THE ANALYTIC TOPOLOGY BRIAN OSSERMAN Classical algebraic geometers studied algebraic varieties over the complex numbers. In this setting, they didn t have to worry about the Zariski

A density theorem for parameterized differential Galois theory

A density theorem for parameterized differential Galois theory Thomas Dreyfus University Paris 7 The Kolchin Seminar in Differential Algebra, 31/01/2014, New York. In this talk, we are interested in the

Isogeny invariance of the BSD formula

Isogeny invariance of the BSD formula Bryden Cais August 1, 24 1 Introduction In these notes we prove that if f : A B is an isogeny of abelian varieties whose degree is relatively prime to the characteristic

ERGODIC AVERAGES FOR INDEPENDENT POLYNOMIALS AND APPLICATIONS

ERGODIC AVERAGES FOR INDEPENDENT POLYNOMIALS AND APPLICATIONS NIKOS FRANTZIKINAKIS AND BRYNA KRA Abstract. Szemerédi s Theorem states that a set of integers with positive upper density contains arbitrarily

Gorenstein rings through face rings of manifolds.

Gorenstein rings through face rings of manifolds. Isabella Novik Department of Mathematics, Box 354350 University of Washington, Seattle, WA 98195-4350, USA, novik@math.washington.edu Ed Swartz Department

The Affine Grassmannian

1 The Affine Grassmannian Chris Elliott March 7, 2013 1 Introduction The affine Grassmannian is an important object that comes up when one studies moduli spaces of the form Bun G (X), where X is an algebraic

Functions associated to scattering amplitudes. Stefan Weinzierl

Functions associated to scattering amplitudes Stefan Weinzierl Institut für Physik, Universität Mainz I: Periodic functions and periods II: III: Differential equations The two-loop sun-rise diagramm in

Nahm s conjecture about modularity of q-series

(joint work with S. Zwegers) Oberwolfach July, 0 Let r and F A,B,C (q) = q nt An+n T B+C n (Z 0 ) r (q) n... (q) nr, q < where A M r (Q) positive definite, symmetric n B Q n, C Q, (q) n = ( q k ) k= Let

THE HALF-FACTORIAL PROPERTY IN INTEGRAL EXTENSIONS. Jim Coykendall Department of Mathematics North Dakota State University Fargo, ND.

THE HALF-FACTORIAL PROPERTY IN INTEGRAL EXTENSIONS Jim Coykendall Department of Mathematics North Dakota State University Fargo, ND. 58105-5075 ABSTRACT. In this paper, the integral closure of a half-factorial

arxiv: v1 [math.gt] 5 Aug 2015

HEEGAARD FLOER CORRECTION TERMS OF (+1)-SURGERIES ALONG (2, q)-cablings arxiv:1508.01138v1 [math.gt] 5 Aug 2015 KOUKI SATO Abstract. The Heegaard Floer correction term (d-invariant) is an invariant of

On Rankin-Cohen Brackets of Eigenforms

On Rankin-Cohen Brackets of Eigenforms Dominic Lanphier and Ramin Takloo-Bighash July 2, 2003 1 Introduction Let f and g be two modular forms of weights k and l on a congruence subgroup Γ. The n th Rankin-Cohen

Constructing Class invariants

Constructing Class invariants Aristides Kontogeorgis Department of Mathematics University of Athens. Workshop Thales 1-3 July 2015 :Algebraic modeling of topological and computational structures and applications,

Three-dimensional imprimitive representations of PSL 2 (Z) and their associated vector-valued modular forms

Three-dimensional imprimitive representations of PSL 2 (Z) and their associated vector-valued modular forms U-M Automorphic forms workshop, March 2015 1 Definition 2 3 Let Γ = PSL 2 (Z) Write ( 0 1 S =

Strongly Self-Absorbing C -algebras which contain a nontrivial projection

Münster J. of Math. 1 (2008), 99999 99999 Münster Journal of Mathematics c Münster J. of Math. 2008 Strongly Self-Absorbing C -algebras which contain a nontrivial projection Marius Dadarlat and Mikael

Radon Transforms and the Finite General Linear Groups

Claremont Colleges Scholarship @ Claremont All HMC Faculty Publications and Research HMC Faculty Scholarship 1-1-2004 Radon Transforms and the Finite General Linear Groups Michael E. Orrison Harvey Mudd

Bulletin of the Iranian Mathematical Society

ISSN: 1017-060X (Print) ISSN: 1735-8515 (Online) Special Issue of the Bulletin of the Iranian Mathematical Society in Honor of Professor Heydar Radjavi s 80th Birthday Vol 41 (2015), No 7, pp 155 173 Title:

Projects on elliptic curves and modular forms

Projects on elliptic curves and modular forms Math 480, Spring 2010 In the following are 11 projects for this course. Some of the projects are rather ambitious and may very well be the topic of a master

SELF-SIMILARITY OF POISSON STRUCTURES ON TORI

POISSON GEOMETRY BANACH CENTER PUBLICATIONS, VOLUME 51 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 2000 SELF-SIMILARITY OF POISSON STRUCTURES ON TORI KENTARO MIKAMI Department of Computer

Galois Theory of Several Variables

On National Taiwan University August 24, 2009, Nankai Institute Algebraic relations We are interested in understanding transcendental invariants which arise naturally in mathematics. Satisfactory understanding

McGill University Department of Mathematics and Statistics. Ph.D. preliminary examination, PART A. PURE AND APPLIED MATHEMATICS Paper BETA

McGill University Department of Mathematics and Statistics Ph.D. preliminary examination, PART A PURE AND APPLIED MATHEMATICS Paper BETA 17 August, 2018 1:00 p.m. - 5:00 p.m. INSTRUCTIONS: (i) This paper

A topological description of colored Alexander invariant

A topological description of colored Alexander invariant Tetsuya Ito (RIMS) 2015 March 26 Low dimensional topology and number theory VII Tetsuya Ito (RIMS) Colored Alexnader invariant 2015 March 1 / 27

Exercises on chapter 1

Exercises on chapter 1 1. Let G be a group and H and K be subgroups. Let HK = {hk h H, k K}. (i) Prove that HK is a subgroup of G if and only if HK = KH. (ii) If either H or K is a normal subgroup of G