GENERALIZED QUANTUM 6j-SYMBOLS AND THE ...

GENERALIZED QUANTUM 6j-SYMBOLS AND THE COLORED ALEXANDER INVARIANT JUN MURAKAMI WASEDA UNIVERSITY Abstract. We give the Clebsch-Gordan quantum coefficients and the quantum 6j-symbols for the non-integral highest weight representations of the quantized enveloping algebra Uq (sl2 ) at roots of unity. A face model for the colored Alexander invariant, the Conway function, and the Alexander polynomial are constructed by using these 6j-symbols.

Introduction After the discovery of the Jones polynomial in 1984, various invariants of knots, links and 3-maifolds are constructed by using the representations of quantized enveloping algebras. For the simplest quantized enveloping algebra Uq (sl2 ), we still have several variations of invariants. For knots and links, we have the Jones polynomial and the colored Jones invariant. For 3-manifolds, we have the Reshetikhin-Turaev invariant which comes from the linear combination of colored Jones invariants, and the Turaev-Viro invariant which comes from the quantum 6j-symbols. The most popular way of constructing the Jones polynomial and its darivations is to use the skein relation among knot diagrams, and there is another way to use the R-matrix of Uq (sl2 ). The construction using the R-matrix is remarkable since it gives a general method to get knot invariants form any quantum enveloping algebras, and it clarifies the relation between the knot invariants and the solvable lattice models. There are two types of solvable lattice models. One is the vertex model and another is the face model. The R-matrix construction corresponds to the vertex model, and the colored Jones invariant can be expressed as a state sum of a vertex model. There is a general procedure to get a face model from a vertex model and vice versa. For exactly solved models related to Uq (sl2 ), such procedure is actually done in [5]. For knot diagrams, it is done in [11], and the resulting model is written in terms of the the quantum 6j-symbols. By using this model, Turaev [17] generalizes the colored Jones invariant for shadow links. Assuming q to be a root of unity, we have other invariants related to Uq (sl2 ), such as the colored Alexander invariant [1], [13], [7], the logarithmic invariant [14] , and the Hennings invariant [9]. The colored Alexander invariant relates to the central deformation of the N dimensional irreducible representation of Uq (sl2 ), where N is the smallest positive integer with q 2N = 1. This representation can be regarded as a non-integral highest weight representation of Uq (sl2 ). The logarithmic invariant comes from the radical part of a non-semisimple representation of Uq (sl2 ). The Hennings invariant is an invariant of 3-manifolds coming from the right integral of the finite dimensional Hopf algebra which is a quotient of Uq (sl2 ). The aim of this paper is to construct the face model for the colored Alexander invariant, which comes from the non-integral highest weight representations of Uq (sl2 ) at roots of unity. To do this, we first compute the Crebush-Gordan quantum coefficients (CGQC) for the tensor The author was partly supported by JSPS Grand-in-Aid for Scientific Research (No. 19540230). 1

2

JUN MURAKAMI WASEDA UNIVERSITY

representation of two non-integral highest modules, and then construct the generalization of the quantum 6j-symbols. Here, 6j means 6 spins and, usually, spin is represented by a non-negative half integer. But, in this paper, we consider the case that none of such 6 spins is any half integer. We actually compute the CGQC’s and the quantum 6j-symbols for such non-half-integer spins by using the arguments in [10] and [11]. At the last section, the face model for the colored Alexander invariant is constructed by using the generalized quantum 6j-symbols introduced in the previous section. This model is a generalization of those for the Conway function and the Alexander polynomial constructed by O. Viro [19] using the quantum supergroup gl(1|1), and, here we reconstruct them by using Uq (sl2 ). One advantage of the face model is that it can be extended to the invariant of shadow links as in [17]. Another advantage to use the 6j-symbol is its relation to the volume of a hyperbolic tetrahedron. In the case of vertex model, the R-matrix is written in terms of quantum factorials, and each quantum factorial corresponds to the volume of an ideal hyperbolic tetrahedron. On the other hand, it is shown in [16], [15] and [18] that the quantum 6jsymbol somehow corresponds to the volume of any type of hyperbolic tetrahedra, including compact, ideal and truncated tetrahedra, and also corresponds to the volume of spherical tetrahedra. Such relation is applied for discussing the volume conjecture of hyperbolic links in S 2 ×S 1 in [3, 4]. For the colored Alexander invariant, its relation to the hyperbolic volume is argued in [2] and [13]. These arguments suggest some relation between the generalized quantum 6j-symbols constructed here and the volume of certain hyperbolic tetrahedron, which will be discussed in a separate paper. 1. Highest weight representations of Uq (sl2 ) The quantized universal embeloping algebra Uq (sl2 ) is generated by elements K ±1 , X± with the commutation relations K · K −1 = K −1 · K = 1, K X± K −1 = q ±1 X± , (1.1) K 2 − K −2 [X+ , X− ] = . q − q −1 We also give a Hopf algebra structure on Uq (sl2 ) satisfying the following coproduct ∆, antipode S, and counit ε. ∆(X± ) = K ⊗ X± + X± ⊗ K −1 ,

(1.2)

∆(K) = K ⊗ K,

Let (1.3)

[λ]n =

n Y q λ−j+1 − q −(λ−j+1) j=1

q − q −1

,

[λ] = [λ]1 ,

[n]! = [n]n ,

[λ]0 = [0]! = 1.

From (1.1) and (1.2), we have (1.4)

∆(X± p ) =

p X r=0

[p]! X± r K p−r ⊗ X± p−r K −r . [r]! [p − r]! √

From now on, we specialize q to be the 2N -th root of unity exp π N−1 , and q x means √ exp π N−1 x . For any non-half-integer parameter α, we define an N dimensional representation

GENERALIZED QUANTUM 6j-SYMBOLS AND THE COLORED ALEXANDER INVARIANT

3

of Uq (sl2 ) on the vector space Vα with basis eαm (m = 0, 1, · · · , N −1) by the following action. πα (K) eαm =q α−m eαm , p πα (X+ ) eαm = [m] [2α − m + 1] eαm−1 ,

(1.5)

p πα (X− ) eαm = [2α − m] [m + 1] eαm+1 . From (1.5), we have πα (K p ) eαm =q p(α−m) eαm , (1.6)

q πα (X+ p ) eαm = [m]p [2α − m + p]p eαm−p , p

πα (X− ) eαm

q = [2α − m]p [m + p]p eαm+p .

The representation πα is irreducible since α is not a half-integer and is called a highest weight representation with the highest weight α. We define pairing ( , ) : Vα × Vα → R requiring that the vectors eαm be orthomoraml: (eαm , eαn ) = δm,n .

(1.7)

With respect to this pairing, the operators X+ and X− are adjoint: (1.8)

(X± a, b) = (a, X∓ b). 2. Crebsch-Gordan quantum coefficients

Let us consider the tensor product πα ⊗ πβ . By using standart argument about the weight space decomposition, we get the following decomposition of the tensor product. Proposition 1. Let Vα , Vβ be highest weight representations of non-half-integer parameters α, β. If α + β is not a half-integer, then M (2.1) Vα ⊗ Vβ = Vγ . α+β−γ=0,1,··· ,N −1

Let eγm (αβ) be the weight basis of the irreducible component Vγ . The coefficients of the vectors eγm (αβ) in the basis eαm ⊗ eβm is called the Clebsch-Gordan quantum coefficients (CGQC): X · α β γ¸ γ (2.2) em (αβ) = eα ⊗ eβm2 . m1 m2 m q m1 m1 ,m2

The orthogonality of the basis (1.7) assures the following relations: X · α β γ ¸ · α β γ0 ¸ = δmm0 δγγ 0 , (2.3) m1 m2 m q m1 m2 m0 q m1 ,m2

4


and

X· α β γ¸ · α β γ¸ = δm1 m01 δm2 m02 . m1 m2 m q m01 m02 m q

(2.4)

γ,m

We define an inner product in the tensor product Vα ⊗ Vβ : (a ⊗ b, c ⊗ d) = (a, c)(b, d). With respect to this inner product, the operators ∆(X+ ) and ∆(X− ) are adjoint: (∆(X± )(a ⊗ b), c ⊗ d) = (a ⊗ b, ∆(X∓ )(c ⊗ d)). We use the following notations. (2.5)

Bxyz = x + y − z,

Axyz = x + y + z,

Cxyzu = x + y − z − u.

Theorem 1. The Clebsch-Gordan quantum coefficients are gives as follows. (2.6) · ¸ α β γ = δm1 +m2 ,m+Bαβγ q Bαβγ (Bγαβ +1)/2+m2 γ−mβ m1 m2 m q s [2γ − m]![m]![m1 ]![m2 ]![Bαβγ ]![2γ + 1] [2α − m1 ]![2β − m2 ]![Bβγα ]![Aαβγ + 1]![Bγαβ ]! X

min(Bαβγ ,m2 )

(−1)j q j(m−2γ−1)

j=max(Bαβγ −m1 ,0)

[2β − j]![Bγαβ + j]! . [m1 − Bαβγ + j]! [m2 − j]![j]![Bαβγ − j]!

(the generalization of the Majundar formla) (2.7) =δm1 +m2 ,m+Bαβγ ∆αβγ q Bαβγ (Aαβγ +1)/2+αm2 −βm1 p

[m1 ]![2α − m1 ]![m2 ]![2β − m2 ]![m]![2γ − m]![2γ + 1] X

min(Bαβγ ,m2 ) j=max(Bαβγ −m1 ,0)

where ∆αβγ = (2.8)

p

[m2 − j]![Bαβγ

(−1)j q −j(Aαβγ +1) , − j]![Bβγα − m2 + j]![j]![m1 − Bαβγ + j]![2α − m1 − j]!

(the generalization of the van der Waerden formla) [Bβγα ]![Bγαβ ]![Bαβγ ]!/[Aαβγ + 1]!, and we use the convention [x]! = [x]m . [x − m]!

In the above formulas, each quantum factorial of the numerator, say [x]! with a non-integer variable x, can be paired with some quantum factorial [x − y]! of the denominator such that x − y is an integer, and [x]!/[y]! means [x]x−y if x ≥ y and 1/[y]y−x if x < y.


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Proof of Theorem 1. We denote by eγαβ m , m = 0, 1, · · · N − 1 an orthonormal basis in space Vγ ⊂ Vα ⊗ Vβ . The highest weight vector eγαβ in the space Vγ is given by 0 X

³

Bαβγ

eγαβ 0

=

cj

´ eαBαβγ −j ⊗ eβj .

j=0

We know that πα ⊗ πβ (∆(X+ ))

³

eαBαβγ −j

⊗

eβj

´

= πα (K) eαBαβγ −j ⊗ π β (X+ ) eβj + πα (X+ ) eαBαβγ −j ⊗ π β (K −1 ) eβj =q

γ−β+j

p

[j] [2β + 1 −

j] eαBαβγ −j

⊗

eβj−1

+q

−β+j

q [Bαβγ − j] [Bγαβ + 1 + j] eαBαβγ −j−1 ⊗ eβj .

Since π γ (X + ) eγ0 = 0, we get q p q γ−β+1+j [j + 1] [2β − j] cj+1 + q −β+j [Bαβγ − j] [Bγαβ + 1 + j] cj = 0, and so

s cj+1 = − q −(1+γ)

[Bαβγ − j] [Bγαβ + 1 + j] cj . [j + 1] [2β − j]

Consequently

s [Bαβγ ]j [Bγαβ + j]j c0 . [j]j [2β]j

cj = (−1)j q −j(γ+1)

The coefficient c0 is obtained from the condition that the norm of the vector eγαβ is equal 0 to unity: Bαβγ Bαβγ ¯¯ ¯¯ X [Bαβγ ]j [Bγαβ + j]j ¯¯ γαβ ¯¯2 X 2 2 (2.9) ¯¯e0 ¯¯ = (cj ) = (c0 ) q −2j(γ+1) [j]j [2β]j j=0 j=0 µ ¶ −k, Bγαβ + 1 −2 −2 2 = (c0 ) 2 φ1 ; q , q = 1. −2β

Here

µ m φn

a1 , · · · , am ; q, x b1 , · · · , b n

¶ =

X j≥0

(a1 ; q)j · · · (am ; q)j xj (1; q)j (b1 ; q)j · · · (bn ; q)j

is the basic hypergeometric series and (a; q)j =

j Y

(1 − q a+p−1 ).

p=1

For the calculation of the coefficient ck,0 we make use of the formula (1.5.3) in [7], which is µ ¶ (c − b; q)n q bn −n, b . ; q, q = 2 φ1 c (c; q)n

6


Note that this relation holds for any complex numbers b and c. Therefore s s −2 (−2β; q ) [2β]Bαβγ B αβγ (2.10) c0 = q k(Bγαβ +1) = q Bαβγ (α−β+γ+1)/2 , −2 (−Aαβγ − 1; q )Bαβγ [Aαβγ + 1]Bαβγ and

s cj = (−1)j q Bαβγ (Bγαβ +1)/2−j(γ+1)

[Bαβγ ]![2β − j]Bαβγ −j [Bγαβ + j]j [j]![Bαβγ − j]![Aαβγ + 1]Bαβγ

= (−1)j q Bαβγ (Bγαβ +1)/2−j(γ+1) s [Bαβγ ]![2β − j]![2γ + 1]![Bγαβ + j]! . [j]![Bαβγ − j]![Bβγα ]![Aαβγ + 1]![Bγαβ ]!

(2.11)

We set γαβ m eγαβ m := fm ∆(X− ) e0 ,

m = 0, 1, · · · , N − 1.

The coefficients fm are obtained from the orthonormality conditions of the vectors eγαβ m . ³ ´ ¯¯ ¯¯2 γαβ m m ¯¯ = fm 2 eγαβ 1 = ¯¯eγαβ = [m]! [2γ]m fm 2 . 0 , ∆ (X+ X− ) e0 m Therefore

s 1

= fm = p [m]! [2γ]m and

s eγαβ m =

(2.12)

[2γ − m]! , [m]! [2γ]!

[2γ − m]! ∆(X− m ) eγαβ 0 . [m]! [2γ]!

Combining equations (2.11) and (2.12), we get s [2γ − m]![Bαβγ ]![2γ + 1] Bαβγ (Bγαβ +1)/2 eγαβ m = q [m]![Bβγα ]![Aαβγ + 1]![Bγαβ ]! s (2.13) Bαβγ ³ ´ X [2β − j]![Bγαβ + j]! (−1)j q −j(γ+1) ∆(X−m ) eαBαβγ −j ⊗ eβj . [j]![Bαβγ − j]! j=0 From (1.4) and (1.6), ∆(X−m )

³

eαBαβγ −j

⊗

eβj

´ =

m X r=0

=

m X r=0

s

´ ¢³ α ¡ r m−r [m]! β −r m−r eBαβγ −j ⊗ ej ⊗ X− K X− K [r]! [m − r]!

[m]! q (m−r)(γ−β+j)+r(j−β) [r]! [m − r]! ´ [Bγαβ + j]![Bαβγ − j + r]![2β − j]![j + m − r]! ³ α eBαβγ −j+r ⊗ eβj+m−r . [Bγαβ + j − r]![Bαβγ − j]![2β − j − m + r]![j]!


Here we use the convention eαm = 0 if m ≥ N . Substituting it to (2.13), we get s Bαβγ (Bγαβ +1)/2 eγαβ m = q

X

[2γ − m]![Bαβγ ]![2γ + 1] [m]![−Bγαβ ]![Aαβγ + 1]![Bγαβ ]!

Bαβγ

[2β − j]![Bγαβ + j]! X [m]! q (m−r)(γ−β+j)+r(j−β) [j]![Bαβγ − j]! [r]! [m − r]! r=0 s ´ [Bαβγ − j + r]![j + m − r]! ³ α β e ⊗ ej+m−r . [Bγαβ + j − r]![2β − j − m + r]! Bαβγ −j+r m

j

(−1) q

−j(γ+1)

j=0

By putting m1 = Bαβγ − j + r and m2 = j + m − r, s [2γ − m]![Bαβγ ]![2γ + 1] Bαβγ (Bγαβ +1/2) eγαβ m = q [m]![−Bγαβ ]![Aαβγ + 1]![Bγαβ ]! X

Bαβγ

(−1)j q −j(γ+1)

j=0

X

[2β − j]![Bγαβ + j]! [j]![Bαβγ − j]!

[m]! q (m2 −j)(γ−β+j)+(m−m2 +j)(j−β) [m1 − Bαβγ + j]! [m2 − j]!

s

¡ α ¢ [m1 ]![m2 ]! em1 ⊗ eβm2 [2α − m1 ]![2β − m2 ]!

m1 +m2 =Bαβγ +m m1 ≥Bαβγ −j, m2 ≥j

=

X

q Bαβγ (Bγαβ +1)/2+m2 γ−mβ

m1 +m2 =Bαβγ +m

s

[2γ − m]![m]![m1 ]![m2 ]![Bαβγ ]![2γ + 1] [2α − m1 ]![2β − m2 ]![−Bγαβ ]![Aαβγ + 1]![Bγαβ ]!

X

min(k,m2 )

(−1)j q j(m−2γ−1)

j=max(Bαβγ −m1 ,0)

¡ α ¢ [2β − j]![Bγαβ + j]! em1 ⊗ eβm2 . [m1 − Bαβγ + j]![m2 − j]![j]![Bαβγ − j]!

This gives (2.6). Now we obtain (2.7) from (2.6). We use the following relations. Lemma 1. (a) For any non-negative integers a, b such that 0 ≤ a − b ≤ N − 1, (2.14)

X c≥b

q ±((a−b)c−(c−b))

(−1)c = (−1)a δa,b . [a − c]![c − b]!

(b) For any parameters a, b and any non-negative integer c, (2.15)

c X s=0

q ±as [a − b]![b]! q ±bc [a]! = . [s]![b − s]![c − s]![a − b − c + s]! [c]![a − c]!

7

8


(b’) For any parameters a, b and any non-negative integer c, c X (−1)s q ±(a+1)s [b]![a + b + c − s]!

(2.16)

s=0

[a + b]![s]![b − s]![c − s]!

=

q ∓bc [a + c]! . [a]![c]!

(c) For any parameter a, b and a non-negative integer c, we have (2.17)

c X

q ±(a+b−c+2)s

s=0

[a − c]![a + b + 1]! [a − s]![b + s]! = q ±(b+1)c [a]![b]![s]![c − s]! [a]![c]![a + b − c + 1]!

(c’) For any parameter a, b and a non-negative integer c, we have (2.18)

c X

(−1)s q ±(−a+b−c−1)s

s=0

[a]![b + s]! [a]![a − b + c − 1]! = q ±(b+1)c [b]![a + s]![s]![c − s]! [a + c]![c]![a − b − 1]!

Proof. For generic q, the relation (a) is (A.4) of [11], and we can specialize the parameter q to the 2N -th root of unity. For generic q, the relation (b) is true for the case that a and b are non-negative integers by (A.5) in [11]. The both sides of (2.15) are Laurent polynomials with respect to the variable q a , q b , and they are equal for any positive integers a and b. Therefore, these two polynomials are equal and the both sides of (2.15) coinside for any a and b for generic q. Then we can specialize q to the 2N -th root of unity. The proof for the relation (c) is similar to that for the relation (b). The relations (b’) and (c’) comes from the relations (b) and (c) respectively, and the relation [a]! [N − 1 − a − z]! = [a + z]! [N − 1 − a]! for any parameter a and an integer z. ¤ Remark. In the next section, the relations (2.16) and (2.18) are used to rewrite a part of some formulas, and sometimes it is used by the following form. X (−1)s q ±(a+1)s [a + b + c − s]! q ∓bc [a + b]![a + c]! = . [s]![b − s]![c − s]! [a]![b]![c]! s=0

min(b,c)

(2.19)

(2.20)

c X s=0

(−1)s q ±(−a+b−c+1)s

[b]![a − b + c − 1]! [b + s]! = q ±(b+1)c , [a + s]![s]![c − s]! [a + c]![c]![a − b − 1]!

The above formulas are not well-defined since the quantum factorials [a + b + c − s]!, [b − s]!, [a + b]!, [b]!, [b + s]!, [a + s]!, [b]!, [a + c]! are not well-defined. However, (2.19) and (2.20) are applied to a part of a big formula, and combining with the rest of the part we are looking, it is well defined in the sense of (2.8) in the complete formula. Continuation of the proof of Theorem 1. By using (2.15), X q (2β−j)(Bαβγ −u) [2β − m2 ]! q (2β−m2 )(Bαβγ −j) [2β − j]![2β − m2 ]! = , [Bαβγ − j]![m2 − j]![Bβγα ]![2β − m2 ]! u≥j [u − j]![m2 − u]![Bαβγ − u]![Bβγα − m2 + u]!


9

and q −j(2α−m1 ) [Bγαβ + j]![2α − m1 ]! [j]![m1 − k + j]![2α − m1 ]![Bγαβ ]!

=

X v≥0

q −v(Bγαβ +j) [2α − m1 ]! . [v]![j − v]![2α − m1 − v]![m1 − Bαβγ + v]!

Therefore, X j≥0

=

X u≥j≥v≥0

=

X u≥0

[Bαβγ

(−1)j q j(m−2γ−1) [2β − j]![2β − m2 ]![Bγαβ + j]![2α − m1 ]! − j]![m2 − j]![Bβγα ]![2β − m2 ]![j]![m1 − k + j]![Bγαβ ]![2α − m1 ]!

(−1)j q j(u−v)−(j−v))+Bαβγ m2 −2βu−v(Bγαβ +1) [2α − m1 ]![2β − m2 ]! [u − j]![m2 − u]![Bαβγ − u]![Bβγα − m2 + u]![v]![j − v]![m1 − Bαβγ + v]![2α − m1 − v]! (−1)u q Bαβγ m2 −u(Aαβγ +1) [2α − m1 ]![2β − m2 ]! . [m2 − u]![Bαβγ − u]![Bβγα − m2 + u]![u]![m1 − Bαβγ + u]![2α − m1 − u]!

Applying this relation to (2.6), we get (2.7).

¤

3. Quantum 6j-symbols The quantum 6j-symbol is defined by the relation in Figure 1. The left diagram means that the composition of two inclusions Vj → Vj12 ⊗ Vj3 and Vj12 → Vj1 ⊗ Vj2 and the right diagram means that the composition of two inclusions Vj → Vj1 ⊗ Vj23 and Vj23 → Vj2 ⊗ Vj3 . Let ιl : Vj → Vj12 ⊗ Vj3 → (Vj1 ⊗ Vj2 ) ⊗ Vj3 (resp. ιr : Vj → Vj1 ⊗ Vj23 → Vj1 ⊗ (Vj2 ⊗ Vj3 )) be the inclusion corresponding to the left (resp. right) diagram. Then Figure 1 means that X ½j j j ¾ 1 2 12 ι (v) ιl (v) = j3 j j23 q r j23

for v ∈ Vj . The quantum 6j-symbol for non-integral highest weight representations are given as follows. Theorem 2. For a, b, · · · , f satisfying a + b − e, a + f − c, b + d − f, d + e − c ∈ Z, ½ ¾ p a b e (3.1) = (−1)Babe [2e + 1][2f + 1]∆abe ∆acf ∆bdf ∆cde d c f q M X r=m

[r]![Ccdab + r]![Cdeaf

(−1)r [Acde + r + 1]! , + r]![Ccebf + r]![Bbf d − r]![Babe − r]![Baf c − r]!

where m = max(a − d − e + f, 0), M = min(a + b − e, a − c + f ), ∆ is given in (2.7), and Acde , Babe , · · · , Ccdab , · · · are given in (2.5). Remark. Substituting r = z − c − d − e to (3.1), we get the usual formula for the quantum 6j-symbol for integral highest weight representations as in [11]. Proof of Theorem 2. This formula is proved by the same way as in the appendix of [11]. The difference is to use the relations in Lemma 1 which are generalized for non-integer

10


j1

j2

j3

j1

j2

X ½j1 j2 j12 ¾ = j3 j j23 q

j12

j3 j23

j23

j

j Figure 1. The quantum 6j-symbol

parameters. Using the orthogonarity relation (2.3), we have the relation in Figure 2. This gives the following expression of the quantum 6j-symbol. · ¸−1 ½ ¾ a f c a b e = m2 m1 m3 q d c f q

·

X m4 ,m5 ,m6

b d f m5 m4 m1

¸ ·

¸ ·

q

q

a b e m2 m5 m6

e d c m6 m4 m3

¸ . q

Let us put m1 = 0, m3 = 0, m4 = α, then m2 = a+f −c, m5 = b+d−f −α, m6 = e+d−c−α.

a

a

f b

e

½ d

c

=

a b e d c f

f

¾ q

c

Figure 2. Another expression of the quantum 6j-symbol

Using the generalization of Van der Waerden formula (2.7), we have ·

α β γ m1 m2 0

¸ = δm1 +m2 ,Bαβγ (−1)m2 q Bαβγ (Bγαβ +1)/2−m2 (γ+1) q

s

[2α − m1 ]![2β − m2 ]![Bαβγ ]![2γ + 1!] , [m1 ]![m2 ]![Bβγα ]![Aαβγ + 1]![Bγαβ ]!


11

and we compute the quantum 6j-symbol as follows. (3.2) ½ ¾ · ¸−1 X · ¸ a b e a f c b d f = d c f q Baf c 0 0 q Bbdf − α α 0 q α ¸ · ¸ · a b e e d c Baf c Bbdf − α Bdec − α q Bdec − α α 0 q p q (Cacbd +2b+1)Bbdf [Aacf + 1]! [2e + 1][2f + 1]∆abe ∆acf ∆bdf ∆dec = [Bbf d ]![Bdf b ]![Bced ]![Bcde ]! X X q −α(Aacf +2) [Bbf d + α]![2d − α]![Bced + α]! (−1)z q −z(Aade +1) z

[z]![Babe − z]![Cef bc + z]![Bacf − z]!

[α]![Bbdf − α − z]![Cef ad + α + z]!

α

Replacing z by a + b − e − z, (3.2) is rewritten as ½ ¾ X X a b e =D Ez Fz,α , d c f q z

where

p q (Cacbd +2b+1)Bbdf [Aacf + 1]! [2e + 1][2f + 1]∆abe ∆acf ∆bdf ∆dec D= , [Bbf d ]![Bdf b ]![Bced ]![Bcde ]! Ez = (−1)Babe −z q −(Babe −z)(Aade +1)

(3.3)

Fz,α = q −α(Aacf +2) By using (2.14), X

α

Fz,α =

α

P α

[z]![Babe − z]![Baf c − z]![Ccebf + z]!

,

[Bbf d + α]![2d − α]![Bced + α]! . [α]![Cdeaf − α + z]![Bbf d + α − z]!

Fz,α is rewritten as follows.

X (−1)t−s q −α(Aacf +2)−(α−t)(Cbf ce −z)−(α−t)s+s−t [2d − α]![Bbf d + α]![Bced + t]! [Cdeaf − α + z]![α − s]![s − t]![t]![Bbf d + t − z]!

α,s,t

=

1

X

(−1)s q −s(Babe +2f +s−z+1)

X

s

β

where Gs,β =

Gs,β

X

Hs,t ,

(replace α by β + s)

t

q −β(Babe +2f +s−z+2) [2d − s − β]![Bbf d + s + β]! , [Cdeaf − s − β + z]![β]!

(−1)t q t(Cbf ce +s−z−1) [Bced + t]! Hs,t = . [s − t]![t]![Bbf d + t − z]! P P The factor of the sums β Gs,β and t Hs,t are rewritten as follows. X β

Gs,β = q −(Bbf d +s+1)(Cdeaf −s+z)

[Bbf d + s]![−Cdeaf + 2d − z]![Abdf + 1]! [Cdeaf − s + z]![Babe + 2f + s − z + 1]!

.

12


by (2.17), and X

(−1)s q −(Bced +1)s [Bced ]![Ccebf + z]! q −(Bced +1)s [Bced ]![Cbf ce − z + s − 1]! = [s]![Bbf d − z + s]![Cbf ce − z − 1]! [s]![Bbf d − z + s]![Ccebf + z − s]!

Hs,t =

t

by (2.20). Hence, we have X X D Ez Fz,α = D (−1)Babe q −Babe (Aabe +1)−(Bbf d +1)Cdeaf [Bced ]![Abdf + 1]! z

α

X (−1)z q z(Caf de +2d) [−Cdeaf + 2d − z]! [z]![Babe − z]![Baf c − z]! z

(3.4) X s

[s]![Bbf d − z + s]![Ccebf

q −s(Acde +1) [Bbf d + s]! . − s + z]![Cdeaf − s + z]![Babe + 2f + s − z + 1]!

Let us now put s = −u + z, then we have X X X X (3.5) D Ez Fz,α = D I Ju Ku,z , z

α

u

z

where I = (−1)Babe q −Babe (Aabe +1)−(Bbf d +1)Cdeaf [Bced ]![Abdf + 1]! Ju =

(3.6)

Ku,z =

[Bbf d − u]![Ccebf

q u(Acde +1) + u]![Cdeaf + u]![Babe + 2f − u + 1]!

(−1)z q z(−Bcf a −1) [Bbf d − u + z]![−Cdeaf + 2d − z]! . [−u + z]![z]![Bade − z]![Baf c − z]!

By using (2.14), Ku,z is rewritten as follows. X z

Ku,z =

X r

(−1)r q r

X (−1)t q −t(Ccdab +r+1) [−Cdeaf + 2d − t]! [t − r]![Babe − t]![Baf c − t]! t

X (−1)z q −z(Bbf d −r+1) [Bbf d − u + z]! z

[z]![−u + z]![r − z]!

.

By using (2.19) with s = t − r and (2.20) with s = z − u, the sums over t and z are rewritten as follows. X (−1)t q −t(Ccdab +r+1) [−Cdeaf + 2d − t]! (−1)r q r(Cdeaf −2d−1)+Babe Baf c [Bcde ]![Bdf b ]! = , [t − r]![Babe − t]![Baf c − t]! [Ccdab + r]![Babe − r]![Baf c − r]! t X (−1)z q −z(Bbf d −r+1) [Bbf d − u + z]! z

[z]![−u + z]![r − z]!

=

(−1)u q −r(Bbf d −u+1) [r − Bbf d − 1]![Bbf d ]! [r]![r − u]![u − Bbf d − 1]!

(−1)r q −r(Bbf d −u+1) [Bbf d − u]![Bbf d ]! . = [r]![r − u]![Bbf d − r]!


13

Now we compute the summation over u in (3.6) by using (2.15) with s = r − u. (3.7) ½ ¾ X (−1)r q −r(Babe +2f +1) a b e Babe Baf c = DIq [Bcde ]![Bdf b ]! d c f q [r]![Babe − r]![Baf c − r]![Bbf d − r]![Ccdab + r]! r u(Acde +r+1)

X u

[r − u]![Cdeaf

q + u]![Ccebf + u]![Babe + 2f − u + 1]!

q Babe Baf c −Ccebf Cdeaf [Bcde ]![Bdf b ]![Bbf d ]! [Aadf + 1]![Abdf + 1]! X (−1)r [Acde + r + 1]!

= DI

r

[r]![Babe − r]![Baf c − r]![Bbf d − r]![Ccdab + r]![Ccebf + r]![Cdeaf + r]!

= (−1)Babe

p [2e + 1][2f + 1] ∆abe ∆acf ∆bdf ∆dec

M X r=m

[r]![Ccdab + r]![Ccebf

(−1)r [Acde + r + 1]! , + r]![Cdeaf + r]![Babe − r]![Baf c − r]![Bbf d − r]!

where m = max(a − d − e + f, 0), M = min(b + a − e, a + f − c).

¤

4. Colored Alexander invariants In this section, we construct the face model for the colored Alexander invariant [13] by using the method in Section 6 of [11]. Let R0 be the representation of the universal R-matrix ˜ on the tensor of the highest weight representations Vα ⊗ Vβ , i.e. R0 = (πα ⊗ πβ )(R), ˜ R 0 and let R = R P , where P is the transposition defined by P (x ⊗ y) = y ⊗ x. Then R : Vα ⊗ Vβ −→ Vβ ⊗ Vα is given by (4.1) s X [2α − i]![i + n]![j]![2β − j + n]! β β R (eαi ⊗ ej ) = q 2(α−i−n)(β−j+n)+n(n−1)/2 (ej−n ⊗ eαi+n ). [2α − i − n]![i]![j − n]![2β − j]! n This R is represented graphically as follows. i j α@ @ ¡β

ij Rkl :

¡

¡ ¡ ª @ R @

k

R−1

¢ij kl

:

l

i j ¡β α@ ¡ @ ¡ @ ¡ ª R

k

where R±1 eαi ⊗ eβj =

X¡

R±1

¢ij kl

l

eβk ⊗ eαl .

Similarly, the Clebsch-Gordan quantum coefficient given by (2.2) is represented by ·

α β γ m1 m2 m

m1

¸ : q

m2 @ ¡ α@ ¡β @ R¡ ª

γ? m

m γ

?

,

.

α ¡@ β

¡ ¡ ª

m1

@ R @

m2

14


We define an inclusion of the trivial representation in Vα ⊗ VN −1−α by 1 −→

N −1 X

N −1−α q (N −1)(α−i) eαi ⊗ eN −1−i ,

i=0

and a projection from Vα ⊗ VN −1−α to the trivial representation by −1−α eαi ⊗ eN −→ δi,N −1−j q (N −1)(α−i) , j

which are represented by the following diagrams: α ¡@ N − 1 − α ¡ @ , ? ? N −1−i

i

N −1−i

i

? ? . @ ¡ α @¡ N − 1 − α

Proposition 2. The following relations among graphical representations hold: α

α β v ¡ @ ¡ u u sin (2β+1)π sin 2γπ @ ¡ ¡ ¡ ª @ R ¡ ª N = (−1)α+β−γ t , @ R¡ ª@ (2γ+1)π sin 2βπ sin @ @ N

(4.2)

β

?

?

γ α

α

β

v @ u u @ R @ α+β−γ t sin ¡@ ¡ = (−1) Rª sin ¡ ¡ ?

(4.3)

γ

(2α+1)π N (2γ+1)π N

β

¡ @ ¡ ª R¡ @ sin 2γπ @

sin 2απ

?

γ

γ γ

(4.4)

v u (2β+1)π u ¡ sin 2γπ N −1+α+β−γ t sin N ¡ = (−1) R¡ ª ¡@ (2γ+1)π sin N sin 2βπ @ ?

α

@ @ R

β

γ , ? ¡@ ¡ @ ¡ ª R @ α

γ (4.5)

v u (2α+1)π u @ sin 2γπ N −1+α+β−γ t sin N @ = (−1) ¡ @ @ª R sin (2γ+1)π sin 2απ ¡ N ¡ ¡ ª

α

,

?

β

β γ

, ? ¡@ ¡ @ ª ¡ @ R α

β


α

β @ ¡ ¡ ¡ @ @ ¡ @ R¡ ª

(4.6)

α = q tγ −tα −tβ

@ ¡ @ ¡ @ R¡ ª

α

β γ

¡ A ¡ A ½ X ª AU¡ γ tα +tδ −tζ −tξ = q @ β @ R?

ζ

,

γ

γ

(4.7)

β

?

?

α

ξ

α ζ δ ξ

¾ q

β

ξ

?

(4.8)

γ

A ¢ ¢ A ¢ ¢ AU¢® ¢ , @ ¢ @ R¢® ?

δ α

15

δ α

β γ

A ¡ A ½ X U¡ A ª γ −tα −tδ +tζ +tξ q = @ β @ R ? ζ ξ

?

δ

α ζ δ ξ

¾ q

β

γ

A ¢ ¢ A ¢ ¢ AU¢® ¢ . @ ¢ R @ ® ¢ ξ ?

δ

Here tα = α (α + 1 − N ).

Proof. We first prove (4.2). The diagram · α β γ ek = i j

of the right hand side of (4.2) means that ¸ γ eα ⊗ eβj , k q i

and the left hande side means that · ¸ N −1−α γ β γ ek = q (N −1)(α−i) eαi ⊗ eβj , N −1−i k j q where α + β − γ be an integer, 0 ≤ α + β − γ ≤ N − 1, and i + j − k = α + β − γ. By putting j = k = 0 and l = α + β − γ, p · ¸ [2β]![2γ + 1]! α β γ l(2α+1−l)/2 p (4.9) , =q l 0 0 q [2β − l]![l + 2γ + 1]! and (4.10)

q (N −1)(α−l)

p · ¸ q (N −1)(α−l)+(N −1−l)(N −2α+l)/2 [2γ]![2β + 1]! N −1−α γ β p . = N −1−l 0 0 q [−N + 1 + 2γ + l]![N − l + 2β]!

16


So the ratio of (4.10) over (4.9) is

s

q (N −1)(α−l)+(N −1−l)(N −2α+l)/2−l(2α+1−l)/2 s

(4.11) = (−1)α+β−γ

[2β + 1][2β − l]![2γ + l + 1]! [2γ + 1][2β − l + N ]![2γ + l + 1 − N ]! [2β + 1]![2γ + N ]! . [2β + N ]![2γ + 1]!

We also know that sin λπ [λ + N ]! = − N −1 [λ]! 2 sinN Hence (4.11) is equal to

π N

.

v u (2β+1)π u sin 2γπ α+β−γ t sin N (−1) . sin (2γ+1)π sin 2βπ N

The relation ¸ of the Clebsh-Gordan quantum · (4.3) ¸is proved similarly· by using the ratio γ N −1−β α α β γ . and q (N −1)(β−l) coefficients 0 N −1−l 0 q 0 l 0 q The relation (4.4) comes from (4.2) and the following graphical relation. γ

γ

¡ ¡ ¡ ª ¡ ª@ @ = ? R¡ @ ª@ @ ¡@ @ ? ? ¡ @ ª ¡ @ R

α

β

α

γ v u (2γ+1)π u ¡ sin 2βπ N −1−α−β+γ t sin N ¡ . = (−1) @ R ¡ ª ¡ sin (2β+1)π sin 2γπ @ N

β

?

α

@ R @

β

The relation (4.5) is proved by using the reflection of this graphical relation. The R-matrix (4.1) is essentially equivalent to that used in [13], and a positive twist of a string with color λ corresponds to a scalar q 2tλ . Hence the left hand side of (4.6) corresponds to a square root of q 2tγ −2tα −2tβ , which is ±q tγ −tα −tβ . The signature is determined by comparing with the action of the permutation on the tensor product of irreducible representations of the Lie algebra sl2 . The relation (4.7) is proved by using the deformation of the diagram given in Figure 3, the graphical meaning of the 6j-symbol given in Figure 1, and (4.6). The relation (4.8) is proved similarly. ¤

Figure 3. The sequence of deformations


17

R1 R5 R0

−→

R3 R4

R2

R6

K1

K2 L TL

Figure 4. A tangle diagram TL related to the link L and its regions Now, we define the face model for the colored Alexander invariant [13]. Let L = K1 ∪ K2 ∪ · · · ∪ Kr be a link with connected components K1 , K2 , · · · , Kr and let TL be a (1, 1)-tangle diagram obtained from L by cutting at the component K1 as in Figure 4. We associate nonhalf-integer complex numbers λ1 , λ2 , · · · , λr to K1 , K2 , · · · , Kr respectively. These numbers are called the colors of the components. The strings of the tangle diagram TL divide the plane into several regions, which we denote R0 , R1 , · · · , Rd . Especially, R0 , R1 denote the leftmost and rightmost regions respectively. Let α0 be a non-half-integer complex number, α1 be a complex number such that α0 + λ1 − α1 is an integer and 0 ≤ α0 + λ1 − α1 ≤ N − 1. We define a state of TL as a mapping ϕ : {R1 , R2 , · · · , Rd } −→ C which satisfies the following conditions. (i) ϕ(R0 ) = α0 , ϕ(R1 ) = α1 , (ii) If Ki and Kj is adjacent along Kk , Ri is on the right of Kk and Rj is on the left of Kk , then ϕ(Ri ) + λk = ϕ(Rj ) + l, where l is an integer of 0 ≤ l ≤ N − 1. (iii) ϕ(Ri ) is not a half-integer for any i. Note that the third condition is a condition for α0 . Let Zα0 (TL ) be the following state sum: Y X Y Y Wcrossing (c) Wmax (p) Wmin (p), (4.12) Zα0 ,α1 (TL ) = ϕ c : crossing

p : maximal point

q : minimal point

where Wcrossing (c), Wmax (p), Wmin (q) are given as follows. p Ã # −→

λ α

?

β

p Ã # −→

λ α

β

?

v u (2β+1)π u sin 2απ N −1−λ+β−α t sin N , Wmax (p) = (−1) (2α+1)π sin N sin 2βπ v u (2β+1)π u sin 2απ λ+β−α t sin N , Wmax (p) = (−1) (2α+1)π sin N sin 2βπ

18


λ1

λ1

λ2

λ1 α6 α3

56

α2 λ2

α0

α3

α5

α0

α5

α0

λ2

α1

α1

α1

α4

α2

α4

αi = ϕ(Ri ) Figure 5. Vertex-IRF correspondence in [11] 6

α λ

β

−→

"!

q β

α λ

v u (2β+1)π u sin 2απ N −1−λ+α−β t sin N , Wmin (q) = (−1) (2α+1)π sin N sin 2βπ

6

−→

"!

q λ

µ

½

@ @δ

¡ α ¡¡β ¡ γ @ ª ¡ @ R

λ

v u (2β+1)π u sin 2απ λ+α−β t sin N Wmin (q) = (−1) , (2α+1)π sin N sin 2βπ

−→

Wcrossing (c) = q tα +tβ −tγ −tδ

µ

@ δ ¡ ¡ @ α @β ¡γ @ ¡ ª R @

µ α γ λ β δ

½ −→

Wcrossing (c) = q

−tα −tβ +tγ +tδ

¾

µ α γ λ β δ

, q

¾ , q

and Wcrossing for crossings with other orientations are given as follows. ¡ µ ¡ ¡ µ ¡ ¡ ¡@ ¡ ¡ ¡ ¡ , = @ ¡ ¡ @¡ ¡ @ ª ¡ ¡ @ R ¡ @ @

··· .

Theorem 3. Let Zeα0 ,α (TL ) =

(q −

[2λ1 + 1]! −1 q )N −1 [2λ1

+ N ]!

Zα0 ,α1 (TL ).

Then Zeα0 ,α1 (TL ) is equal to the colored Alexander invariant in [13]. Proof. The colored Alexander invariant is defined from the R-matrix in (4.1). To define the colored Alexander invariant, we assign a scalar OTNL (λ1 , · · · , λr ) ∈ Vλ1 from the tangle TL .


19

[2λ1 + 1]! OTNL (λ1 , · · · , λr ) Then is a link invariant. Now we follow the argument of Section (q − q −1 )N −1 [2λ1 + N ]! 6 in [11] giving the correspondence of vertex models and IRF models, which is represented graphically as in Figure 5. The left diagram in Figure 5 correspond to the scalar operator OTNL (λ1 , · · · , λr ) : Vλ1 → Vλ1 . On the other hand, the right diagram in Figure 5 represents Zα0 ,α1 (TL ). Hence Zα0 ,α1 (TL ) = OTNL (λ1 , · · · , λr ) and so Zeα0 ,α1 (TL ) is equal to the colored Alexander invariant. ¤ Remark. The scalar Zα0 ,α1 (TL ) does not depend on the choice of non-integer numbers α0 and α1 .

5. The Conway function and the Alexander polynomial By specializing N to be 2, the colored Alexander invariant becomes the multivariable Alexander invariant ∆, which is also known as Conway’s potential function [8]. Therefore, we can construct the face model for ∆. The colored Alexander invariant is an invariant of framed links. To get an invariant of un-framed knots and links, we modify the weights corresponding to the crossing points by multiplying −q ±(−2λµ+λ+µ) , where the sign ± is + for positibe crossings and − for negative crossings. Then the resulting face model is given as follows. Let ε be 0 or 1. '

λ α

$

−→

α+λ−ε

sin απ , sin(α + λ − ε)π

(−1)1−ε

?

$

'

λ α

α−λ+ε

α

6α − λ + ε &

sin απ , sin(α − λ + ε)π

(−1)ε

?

s −→

sin απ , sin(α − λ + ε)π

(−1)1−ε

%

α+λ−ε6

α

s

−→

λ

λ &

λ

s

s −→

(−1)ε

%

sin απ , sin(α + λ − ε)π

µ

@ α+λ ¡ @ ¡ ¡ α α+λ+µ ¡ @ ¡ ¡ ª α+µ@ R

−→

−q λ+µ ,

20


λ

µ

@ α+λ ¡ @ ¡ ¡ α α + λ + µ − 1 −→ ¡ @ ¡ ¡ ª α+µ @ R

λ

s [2λ][2µ] , [2α + 2λ][2α + 2µ]

q −2α−λ−µ

µ

@ α+λ ¡ @ ¡ ¡α + λ + µ − 1 α ¡ @ ¡ @ R ¡ ª

s −→

q −λ+µ

[2α][2α + 2λ + 2µ] , [2α + 2λ][2α + 2µ]

α+µ−1

λ

α+λ−1

µ

¡ ¡ ¡α + λ + µ − 1 α ¡ @ ¡ R ¡ ª α+µ @ @ @

λ

α+λ−1

s −→

q λ−µ

[2α][2α + 2λ + 2µ] , [2α + 2λ][2α + 2µ]

µ

@ @

¡ ¡ ¡α + λ + µ − 1 α ¡ @ ¡ ¡ ª @ R

s −→

−q 2α+λ+µ

−→

q −λ−µ .

[2λ][2µ] , [2α + 2λ][2α + 2µ]

α+µ−1

λ

α+λ−1

@ @

α

¡ ¡ ¡ ª

µ

¡ ¡ ¡α + λ + µ − 2 @ @ R

α+µ−1

For negative crossings, the weight is given by replacing q of the weight for the positive crossing with the same state by q −1 . Theorem 4. Let Zα20 ,α1 (TL ) be the state sum defined by (4.12) from the above weights, and let 1 Zeα20 ,α (TL ) = − 2λ1 Zα20 ,α1 (TL ). −2λ 1 q −q Then Zeα20 ,α1 (TL ) = ∆(t1 , t1 , . . . , tk ), where ∆ is the Conway potential function in [8], and ti = q −2λi . Proof. It is known that ∆ is defined by the local relations (RI), (RII), . . ., (RVII) in [12]. It is a simple computation to see that Zeα20 ,α1 (TL ) satisfies these relations. We omit the actual computation. ¤


21

Corollary 1. Let L be a link. Then the Alexander polynomial ∆L is given by ∆L (t) = Zα20 ,α1 (TL ), where all the components of TL is colored by λ and t = q −4λ . Remark. Face models for the Conway function and the Alexander polynomial are constructed in [19] by using the representation theory of quantum supergroup gl(1|1). Acknowlagement. The author thanks Professor F. Costantino for discussion. References [1] Y. Akutsu, T. Deguchi and T. Ohtsuki, Invariants of colored links. J. Knot Theory Ramifications 1 (1992), 161–184. [2] J. Cho and J. Murakami, Some limits of the colored Alexander invariant of the figure-eight knot and the volume of hyperbolic orbifolds. To appear in J. Knot Theory Ramifications. [3] F. Costantino, Coloured Jones invariants of links and the volume conjecture. J. Lond. Math. Soc. (2) 76 (2007), 1–15. [4] F. Costantino, 6j-symbols, hyperbolic structures and the volume conjecture. Geom. Topol. 11 (2007), 1831–1854. [5] E. Date, M. Jimbo, T. Miwa and M. Okado, Fusion of the eight vertex SOS model. Lett. Math. Phys. 12 (1986), 209–215. [6] G. Gasper and M. Rahman, Basic hypergeometric series, second edition. Encyclopedia of Mathematics and its Applications, 96, Cambridge University Press, Cambridge 2004. [7] N. Geer and N. Reshetikhin, On invariants of graphs related to quantum sl2 at root of unity. preprint, arXiv:0904.0409. [8] R. Hartley, The Conway potential function for links. Comment. Math. Helv. 58 (1983), 365–378. [9] M. Hennings, Invariants of links and 3-manifolds obtained from Hopf algebras. J. London Math. Soc. (2) 54 (1996), 594–624. [10] A. N. Kirillov, Clebsch-Gordan quantum coefficients. J. Soviet Math. 53 (1991), 264–276. [11] A. N. Kirillov and N. Yu. Reshetikhin, Representations of the algebra Uq (sl2 ), q-orthogonal polynomials and invariants of links. In Infinite-dimensional Lie algebras and groups (Luminy-Marseille, 1988), Adv. Ser. Math. Phys. 7, World Scientific, Teaneck 1989, 285–339. [12] J. Murakami, On local relations to determine the multi-variable Alexander polynomial of colored links. In Knots 90 (Osaka, 1990), de Gruyter, Berlin 1992, 455–464. [13] J. Murakami, Colored Alexander invariants and cone-manifolds. Osaka J. Math. 45 (2008), 541–564. [14] J. Murakami and K. Nagatomo, Logarithmic knot invariants arising from restricted quantum groups. Internat. J. Math. 19 (2008), 1203–1213. [15] J. Murakami and A. Ushijima, A volume formula for hyperbolic tetrahedra in terms of edge lengths. J. Geom. 83 (2005), 153–163. [16] J. Murakami and M. Yano, On the volume of a hyperbolic and spherical tetrahedron. Comm. Anal. Geom. 13 (2005), 379–400. [17] V. G. Turaev, Shadow links and face models of statistical mechanics. J. Differential Geom. 36 (1992), 35–74.

22


[18] A. Ushijima, A volume formula for generalised hyperbolic tetrahedra. In Non-Euclidean geometries, Math. Appl. (N. Y.) 581, Springer, New York 2006, 249–265. [19] O. Viro, Quantum relatives of the Alexander polynomial. St. Petersburg Math. J. 18 (2007), 391-457.

Jun Murakami Department of Mathematics Faculty of Science and Engineering Waseda University 3-4-1 Ohkubo, Shinjuku-ku Tokyo, 169-8555, JAPAN e-mail: [email protected]