TitleColored Alexander invariants and co. Citation Osaka Journal of Mathematics. 45(2)

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1 TitleColored Alexander invariants and co Author(s) Murakami, Jun Citation Osaka Journal of Mathematics 45(2) Issue Date Text Version publisher URL DOI Rights Osaka University

2 Murakami, J Osaka J Math 45 (2008), COLORED ALEXADER IVARIATS AD COE-MAIFOLDS Dedicated to Professor oriaki Kawanaka on his sixtieth birthday JU MURAKAMI (Received December 5, 2006, revised June 15, 2007) Abstract In this paper, we reconstruct the link invariant of framed links introduced in [1] by the universal R-matrix of U (sl 2 ) and name it the colored Alexander invariant We check that the optimistic limit o-lim of this invariant is determined by the volume of the knot and link cone-manifold for figure eight knot, Whitehead link and Borromean rings We also propose the A-polynomials of these examples obtained from the colored Alexander invariant 1 Introduction ew link invariants are introduced in [1] for colored links They are defined for each positive integer and considered as a generalization of the multivariable Alexander polynomial [12], which corresponds to the case 2 Here we redefine these invariants by using the universal R-matrix of U (sl 2 ) Let Ô exp( 1) be a 2-th root of unity Let U (sl 2 ) be the uantum enveloping algebra corresponding to the Lie algebra sl 2 defined by the following generators and relations: U (sl 2 ) K, K 1, E, F K K 1 K 1 K 1, K E K 1 2 E, K F K 1 2 F, [E, F] K K 1 1 U (sl 2 ) also has a hopf algebra structure with the following coproduct ½ ½(K ) K Å K, ½(E) E Å K + 1 Å E, ½(F) F Å 1 + K 1 Å F The -dimensional irreducible representation of U (sl 2 ) at exp( Ô 1) admits central deformation parametrized by the scalar «correponding to the central element K Mathematics Subject Classification Primary 57M27; Secondary 20G42

3 542 J MURAKAMI Instead of «, we use parameter satisfying «Let (, V ) be the corresponding representation : U (sl 2 ) End(V ) Then is isomorphic to the irreducible representation of the highest weight if is not an integer The R-matrix G in [1] is compatible with the action of U (sl 2 ) on V Å V and V Å V, ie ½(x)G G½(x): V Å V V Å V We modify the above R-matrix G so that it coincides with the representation of the universal R-matrix when is an integer By using this modified R-matrix, we construct a framed link invariant which we call the colored Alexander invariant, and investigate its relation to the hyperbolic volume of the knot and link cone-manifolds Let M 1, 2,, k be the cone manifold obtained from the link L with the cone angle i along the i-th component of L Then we expect the following aive expectation (Volume conjecture for the colored Alexander invariant) If is a hyperbolic cone manifold, M 1, 2,, k If M 1, 2,, k Vol(M 1, 2,, k ) lim ½ is a spherical cone manifold, L 1, 2, 2 2, k 2 2 logl ( 1(2), 2 (2),, k (2)) exp Ô 1 Vol(M 1, 2,, k ) ½ 2 It is known that the above does not hold if one of i s is rational with respect to But if we use the notion of optimistic limit o-lim introduced in [9], we would declare the following conjecture Conjecture If M 1, 2,, k is a hyperbolic cone manifold, If M 1, 2,, k Vol(M 1, 2,, k ) o-lim ½ log 2 is a spherical cone manifold, Vol(M 1, 2,, k ) o-lim ½ Im(log 2 L ( 1(2), 2 (2),, k (2)) L ( 1(2), 2 (2),, k (2))) The optimistic limit o-lim means to apply the saddle point method formally as in [5], [9] and [10] This conjecture would be proved by applying Yokota s theory developed in [13] to the colored Alexander invariant However, in this paper, we show

4 COLORED ALEXADER IVARIATS AD COE-MAIFOLDS 543 some examples instead of proving the conjecture Such relation is first observed in [11] by considering the deformation of the parameter Here, instead of, we deform the highest weights corresponding to the components of a link, and this deformation can be accomplished independently for each component while the perturbation of gives a simultaneous deformation for the all components As an application, a system of A-polynomials for a multi-component link is proposed at the end of the paper 2 R-matrix 21 Representation of U (sl 2 ) We first describe the highest weight representation (, V ) of U (sl 2 ) Let Ú 0, Ú 1,, Ú 1 be the basis of V, and the actions of K, E, F are given by the following: EÚi i + 1 Úi 1 1 EÚ 0 0, i + 1 FÚi Úi+1 1 FÚ 1 0, K Ú i 2i Ú i, É where a a a n 1 We also use the notations x; n i0 x i for positive integer n and x; Universal R-matrix Let R u be the universal R-matrix R u of U (sl 2 ) given as follows: R u 1 2 HÅH ½ n0 1 2n n; n n(n 1)2 (E n Å F n ), where H is an element such that H K H is not an element of U (sl 2 ), but we define the action of H tov so that H K, ie Similarly, H Ú i ( 2i)Ú i 1 2 HÅH Ú i Å Ú j 1 2 ( 2i)( 2 j) Ú i Å Ú j Let R be the R-matrix corresponding to the braid generator where P is the permutation R R u P, P(x Å y) y Å x

5 544 J MURAKAMI Fig 1 R-matrices at positive and negative crossings Since R u is the universal R-matrix, R satisfies the braid relation (1) R 12 R 23 R 12 R 23 R 12 R Representation of the R-matrix The R-matrix R defined in the previous section gives a mapping V Å V V Å V as follows (2) R (Ú i Å Ú j ) n 1 2 R 1 (Ú i Å Ú j ) n ( 2i 2n)( 2 j+2n)+n(n 1)2 i + n; n j + n; n (Ú j n n; n Å Ú i+n ), ( 1) n 1 2 ( 2i)( 2 j) n(n 1)2 j + n; n i + n; n (Ú j+n n; n Å Ú i n ) In the following, R i j kl means the matrix element of R, ie R (Ú i Å Ú j ) k,l R i j kl (Ú k Å Ú l ), and R, R 1 correspond to the crossings as in Fig 1 24 R-matrix in [1] We introduce some symbols for -analogues m n m n (z; n) n 1 j0 (1 z j ), (; m) for m n 0, (; m n) (; n) 0 for m n 0, z, (z; m) (z; n) for m, n 0

6 COLORED ALEXADER IVARIATS AD COE-MAIFOLDS 545 Let G ab cd («, ; ) be the R-matrix given in [1], a c G ab cd («, ; +) d 2bd «sb+tc sd ta d b 2, 2 f («,, 2 F(«, a)f(, b) ) F(«, d)f(, c) u(a+d b c)+ú(ab cd), G ab cd («, ; ) b c 1 2 d a f («,, 2 ) 1,1 2 c 2ac sb+tc «sd ta 1 F(«, b)f(, a) F(«, c)f(, d) u(a+d b c)+ú(ab cd), where f, F are arbitrary functions, s, t, u, Ú are arbitrary numbers, and «, are the parameters for the under and the over paths respectively ote that i + n in i + n; n, i n; n j j n 2 2, 2 ( 1) n jn (+1)n 1 2 n(n 1) j 1 ; n jn (+1)n 1 2 n(n 1) j + n; n Theorem 2 The above R-matrix is eual to G lk ji («, ; ) as follows: R i j kl G lk ji ( 2, 2 ; +), (R 1 ) i j kl Glk ji ( 2, 2 ; ), where the arbitrary parameters and functions are fixed as follows s 1 2, t 0, u 0, Ú 1, F(«, i) 1, f ( 2, 2, 2 ) (12) Proof Compare R i j kl in (2) and G lk ji ( 2, 2 ; +) by putting n l i j k G lk ji ( 2, 2 ; +) 1 2 +in+ jn (+1)n (12)n(n 1) 2i+2i( j n) ( j n)+i+(i+n)( j n) i j i + n; n j + n; n n; n 1 2 ( 2i 2n)( 2 j+2n)+n(n 1)2 i + n; n j + n; n n; n R i j kl Proof for (R i j kl ) 1 is similar

7 546 J MURAKAMI REMARK 3 The action of U (sl 2 ) on V and V +2 are the same one However, our R-matrices concerning to and + 2 are different This is the reason why we use the parameter instead of «3 Colored Alexander invariants 31 Modified invariant Let T be a (1, 1) tangle with parametrized components Let 1 be the parameter corresponding to the open component and 2,, k be the parameters for other components These parameters are called the colors of the components Let O T ( 1,, k ) be the operator in End(V 1 ) constructed by the state sum as in [12] obtained by assigning the R-matrices for crossings of T and the following scalars for maximal and minimal points of T ote that Let (K ( 1) )Ú i (1 ) 2i Ú i, (K ( 1) )Ú i (1 )+2i Ú i T ( 1, 2,, k ) 1 + ; 1 1 O T ( 1, 2,, k ), where 1 is the color of the open component of T Theorem 4 T ( 1, 2,, k ) is an invariant of a colored framed link L with colors 1, 2,, k obtained by closing the tangle T DEFIITIO 5 We write L ( 1, 2,, k ) instead of T ( 1, 2,, k ) and call it the colored Alexander invariant of colored links Proof of Theorem 4 Let L ( 1, 2,, k ) be the function defined as above from f 1 R instead of R, where f is the function in Theorem 2 Since the product of the f terms in L only depends on the framing and linking numbers of L, if is a link invariant, then is also a framed link invariant By the way, «( 1)2 («; 1) 1 2 ( 1) 1 ( 1)( 2)2 1 + ; 1 1 for «2 1, where («; k) É k 1 j0 (1 «j ) Therefore L ( 1, 2,, k ) ( 1) 1 ( 1)( 2)2 ˆL ( 1, 2,, k ),

8 COLORED ALEXADER IVARIATS AD COE-MAIFOLDS 547 Positive twist of type I, type II, negative twist of type I, type II Fig 2 Positive twists and negative twists where ˆ is the isotopy link invariant in Definition 54 of [1] In the definition of ˆ, the function f is normalized as f («, «; 2 ) «( 1)2 so that formulas (311) and (312) in [1] hold In the definition of, we add the factor «( 1)2 1( 1) to the contribution of minimal and maximal point instead of the R-matrix so that the similar formulas hold Therefore, is a framed link invariant 32 Framing contribution The only difference of our and ˆ in [1] is that depends on the framing of the link We check for the four types of twist given in Fig 2 We assume that the string is colored by For the positive twist of type I, the corresponding scalar is obtained by computing the action to Ú 0 (1 ) (+2 2)2 For the positive twist of type II, the corresponding scalar is obtained by computing the action to Ú 1 (1 )+2( 1) ( 2+2)(2 +1) ( )2 (+2 2)2 For the negative twist of type I, the corresponding scalar is obtained by computing the action to Ú 0 (1 ) (12)2 (+2 2)2 For the negative twist of type II, the corresponding scalar is obtained by computing the action to Ú 1 Therefore, we have the following (1 ) 2( 1) (+2)(2+1) (+2 2)2 Proposition 6 Let K f and K 0 be ambient isotopic knots with framing f and 0, we have (3) K (+2 2) f () 2 f K0 ()

9 548 J MURAKAMI 4 Examples Fig 3 Tangle for the Hopf link 41 Hopf link For the positive Hopf link H +, we compute the colored Alexander invariant by using the tangle in Fig 3 1 H + (, ) + ; 1 1 i0 + ; i0 (1 ) 2i (2 i) (2 i) (+1) 2(+1)i + ; 1 1 2(+1) (+1) 1 1 2(+1) + ; 1 1 (+1)(+1) (+) +1 (+1) + ; 1 (+1 )(+1 ) 2 2 Ô 1 sin Ô 1 2 sin (+1 )(+1 ) 2 Ô 1 sin Ô 1 1 (+1 )(+1 ) Here we use the relation ([1], 1392, 1) (4) + ; 2 Ô 1 sin i1 ( + i) 2 Ô 1 sin( + 1) 2 Ô 1 sin Therefore, we get (5) H + (, ) Ô 1 1 (+1 )(+1 ) Similarly, for negative Hopf link H, we have (6) H (, ) Ô 1 +1 (+1 )(+1 )

10 COLORED ALEXADER IVARIATS AD COE-MAIFOLDS 549 Fig 4 Tangle for the figure eight knot 42 Figure eight knot For the figure eight knot 4 1, 4 1 () is given as follows 4 1 () + ; 1 1 i, j + ; 1 1 i, j (1 ) 2i (1 )+2 j R i,0 0,i R i, j i+ j,0 R0,i+ j i, j R j,0 0, j 2( j i) (2 i) + ; 1 1 i, j + ; 1 1 ( 1) i 1 2 ( 2i)( 2 j) i(i 1)2 i + j; i; i i; i 1 2 ( 2 j)( 2i)+ j( j 1)2 i; j ( 1) i ( j i)(2+i+ j+3)2 i + j; i; i + j i; i 0 jk 1 By using Lemma A in Appendix, the above is eual to + ; 1 1 k + ; 1 1 k 1 2 Ô 1 sin ( 1) k 3k2 k2 2 k k (2 j) ( 1) k j 3k2 k2 2 k+ j(3+k+2) k; i; k i; i k j1 (1 4+2k+2 2 j ); k + k + 1; k; k + ; 1 k + k + 1; 2k + 1 (by (4)) (k i + j) + k + 1; 2k + 1

11 550 J MURAKAMI Fig 5 Tangle of Whitehead link Therefore (7) 4 1 () 1 2 Ô 1 sin 1 k0 + k + 1; 2k Whitehead link Let K W be the Whitehead link of framings 0 for both components The colored Alexander invariant of K W is given by K W (, ) ((+1 )2 (1 ) 2 )2 + ; 1 1 0ki j 1 (1 ) 2k (1 )+2( j k) (2 k) 1 2 ( 2 j+2k) (( 2i)( 2 j+2i)+i(i 1))2 j + i; i ( 1) k (( 2k)( 2 j+2k)+k(k 1))2 j; k; k k; k 1 2 ( 2 j+2k)( 2k)+(i k)(i k 1)2 j k; i k k; i k i k; i k + ; 1 1 (i+1)(i 2 j)+i+( i 1) ; i j; i j + i; i 0i j 1 0ki ( 1) k k( i 2 3) 1 i k; i kk; k

12 COLORED ALEXADER IVARIATS AD COE-MAIFOLDS 551 By using Lemma A in Appendix, the above is eual to 0i j 1 0i j 1 0i j 1 (i+1)(i 2 j)+i+( i 1) ;i j;i j +i;i i;i+ ; 1 (i+1)(i 2 j)+i+( i 1) ;i j;i j +i;i i;i+ ; 1 3i ji i 5i 2+2 j + i k1 ( k ) i k1 ;i j;i j +i;i+1+i;i i;i+ ; 1 ow, replacing j by i + l and using Lemma B in Appendix, we have 0i 1 0i 1 i 2 2 i 2i i 2 + ( 1) 4 sin sin i 2 2 i i i + 1; 2i + 1 i; i + ; + i + 1; 2i + 1 i; i + ; 0l 1 i (i+1)( i) i; i 2 + i + 1; 2i ; 2i + 1; 2i + 1 [ 2]i 1 Therefore (8) K ( 1) W (, ) 4sin sin i 2 2 3i 2 [ 2]i 1 1 k ( +1+k (+1+k) ) 2l(i+1) l + i; i l; i i; i i; 2i i + 1; 2i + 1 2i + 1; 2i + 1 i 2 2 3i 2 i;i+1+i;2i +1+i +1;2i +1 2i +1;2i Borromean rings Let K B be the Borromean rings Then K B (,, ) + ; 1 1 i+ jli,i+ jk j (1 ) 2i (1 )+2 j R i,0 0,i R i, j k,i+ j k R0,k i+ j l,k+l i j k+l i j,i+ j k R l,0 R i+ j l,l i, j R j,0 0, j

13 552 J MURAKAMI + ; 1 1 i+ jli,i+ jk j + ; 1 1 i+ jli,i+ jk j ( 1)( ) 2i+2 j (2 i) (2 j) ( 1) k j 1 2 ( 2i)( 2 j) (k j)(k j 1)2 k; k j i j + k; k j k j; k j 1 2 (+2i+2 j 2k 2l)( 2i 2 j+2l)+(k+l i j)(k+l i j 1)2 i j + l; k + l i j ( 1) k+l i j 1 2 ( 2i 2 j+2k)(+2i+2 j 2k 2l) (k+l i j)(k+l i j 1)2 l; k + l i j; k + l i j k + l i j; k + l i j 1 2 ( 2i)( 2 j)+(l i)(l i 1)2 j; l i i; l i l i; l i ( 1) l i ( 1)( ) i(i 3)2 j( j 3)2+k(3k+1)2 l(3l+1)2 2ik+il jk+2 jl+( i j+k+l)+( i+ j k+l)+(i+ j k l) k; j i j+k; k+l i j i j+l; k+l i jl; i; k+l i j k+l i j; k+l i ji+ j k; i+ j ki+ j l; i+ j l By putting i l + r s and j k r, we get + ; 1 1 ls,ks + ; 1 1 k,l,s + ; 1 1 ls,ks ( 1) s ( )+( +2k 2l)(s+1)+s+s(3+s)2 k; s l + s; s k + s; sl; s; s s; s 2 ( 1) r r(2+s+3) s; r r; r sr ( 1) s ( )+( +2k 2l)(s+1)+s+s(3+s)2 k; s l + s; s k + s; sl; s; s s; s 2 (1 2 4 )(1 2 6 ) (1 2 2s 2 ) ( 1) s ( 1)( )+( +2k 2l)(s+1)+s+s(s+3)2 s(2+s+3)2 k; s l + s; s k + s; sl; s; s s; s 2 + s + 1; s

14 COLORED ALEXADER IVARIATS AD COE-MAIFOLDS 553 ls,ks ( 1) s ( )+( +2k 2l)(s+1) Fig 6 Borromean rings k; s l + s; s k + s; sl; s + s + 1; 2s ; s; s 2 ( 1) s ( )+( )(s+1) + s + 1; 2s ; s; s 2 s 2k(s+1) k; s k + s; s 2l(s+1) l; s l + s; s ks ls ( ) s ( 1) s ( )(s+1) + s + 1; 2s ; s; s 2 k 2k(s+1) k + s; s k; s l 2l(s+1) l + s; s l; s By using Lemma B in Appendix, we have k 2k(s+1) (s+1)( s) s; k + s; s k; s s 2 + s + 1; 2s ; 2s + 1; 2s + 1 and 2l(s+1) l + s; s l; s l (s+1)( s)+ s; s 2 + s + 1; 2s ; 2s + 1; 2s + 1

15 554 J MURAKAMI Hence K B (,, ) s ( 1) s s; s2 + s + 1; 2s s + 1; 2s s + 1; 2s ; + ; + ; 2s + 1; 2s Ô sin sin sin [ 2]s 1 Therefore (9) K B (,, ) Ô 1 8 sin sin sin [ 2]s 1 ( 1) s s; s2 + s + 1; 2s s + 1; 2s s + 1; 2s + 1 2s + 1; 2s ( 1) s+1 s; s2 + s + 1; 2s s + 1; 2s s + 1; 2s + 1 2s + 1; 2s Colored Jones invariant For integer,,, the colored Jones invariants V L (, ) of 4 1, K W, K B are given in [3] V 41 () V KW () V K B () i0 + i + 1; 2i + 1, 1 min(,) i 2 2 3i 2 i0 min(,,) i0 + i + 1; 2i i + 1; 2i + 1, 12i + 1; i i + 1; 2i i + 1; 2i i + 1; 2i i + 1; i Comparing with the colored Alexander invariants, the terms in the summations are similar, but the range of the summations are uite different, especially for the link case 5 Volume of cone manifolds 51 Figure eight knot Let 4 1 be the figure eight knot and let (a; x) n É n 1 j0 (1 ax j ) Then, from (7), we have 4 1 () Ô 1 2 sin k (2k+1)(+1) ( 2 2k+2 ; 2 ) 2k+1

16 COLORED ALEXADER IVARIATS AD COE-MAIFOLDS 555 Let Li 2 (z) be the analytic continuation of the following function: Li 2 (x) x 0 log(1 x) x dx ½ k1 x k k 2 Then Li 2 (z) is a multivalued function and its branches are given by Li 2 (z) + 2Ô 1p log z (p, ¾ Z) In the rest of this paper, Li 2 (z) means an appropriate branch of it Following Kashaev s way in [7] (see also [10] for more examples) to consider the relation to the hyperbolic volume, let V (a, z) be the following function Ô 1 (10) V (a, z) 1 log a + log a log z + Li2 az Li 2 z a Here a, z correspond to 2, 2k respectively, and the term Ô 1 log a is added so that the value of V (a, z 0 ) in the following coincide with the volume not only for hyperbolic case but also for spherical case Adding Ô 1 log a to V corresponds to multiplying to The optimistic limit introduced in [9] of the colored Alexander invariants of 4 1 with fixed a is given by V (a, z 0 ) where z 0 is the solution of the following euation V z 0, ie z2 (a 1 + a 1 )z The solution of this euation is given as follows Let a exp Ô 1«and K «figure eight knot cone-manifold with the cone angle «along the knot be the Hyperbolic case (0 «23) z 0 1, cos(arg z 0 ) cos «1 2, Euclidean case («23) z 0 1, Spherical case (23 «43) z 0 real number Comparing with the results of [8], we have and we get the following Ô V (a, z) 1 arccosh(1 + cos «cos 2«) Vol(K «), ««

17 556 J MURAKAMI Theorem 7 For hyperbolic case, ie 0 «(23), Ô 1V (a, z 0 ) Vol(K «) For spherical case, V (a, z 0 ) Vol(K «) 52 Whitehead link Let K W be the Whitehead link Then, from (8), we have K W (, ) Ô 1 4 sin sin 1 (++1) i ( 1) i+1 ( 2 2+i 3)i (2 ; 2 ) ( 2 ; 2 ) i ( 2 2i+2 ; 2 ) 2i+1 ( 2 2i+2 ; 2 ) 2i+1 ( 2 ; 2 ) 2i+1 ow we compute the optimistic limit of K W (, ) Let V (a, b, z) be the following function V a(a, b, z) Ô 1(log a + log b + log z) Li2 (z) Li 2 (az) Li 2 (bz) + Li 2 (z 2 ) log z, + Li 2 a z b + Li 2 z log a + log b log z 2 where a, b, z correspond to 2, 2, 2i respectively, and the term Ô 1(log a + logb) is added so that the value of V (a, b, z 0 ) in the following coincide with the volume not only for hyperbolic case but also for spherical case By using we have (11) Li 2 (z 1 ) Li 2 (z) 1 2 (log Ô 2 z)2 + 1 log z + 3, Ô z V (a, b, z) 1(log a + log b + log z) Li2 (z) + Li 2 (z 2 ) Li 2 (az) Li 2 a z 1 Li 2 (bz) Li 2 b 2 (log 1 a)2 2 (log 1 b)2 2 (log z) Here 2 Ô 1 log z is absorbed by Li 2 (z) as a branch of it The optimistic limit of the colored Alexander invariants of K B with fixed a, b and c is given by V (a, b, c, z 0 ) where z 0 is a solution of the following euation V (a, b, z) z 0

18 COLORED ALEXADER IVARIATS AD COE-MAIFOLDS 557 By putting z (x 1)(x + 1), this euation becomes to (12) x (A2 B 2 + A 2 + B 2 1)x 2 A 2 B 2 x + A 2 B 2 0, where A a + 1 Ô b + 1 Ô 1, B 1 a 1 b 1 Let a exp Ô 1«and b exp Ô 1, then A cot(«2), B cot( 2) and we have (13) Ô V (a, b, z) 1 2 arctan A «x, Ô V (a, b, z) 1 2 arctan B x Comparing with the results of [8], (13) shows that the partial derivatives of V coinside with the lengths of the singular geodesics of W «,, where W «, is the Whitehead link cone-manifold with cone angles «and Hence we get the following Theorem 8 If W «, is hyperbolic, then the volume of W «, is given by Vol(W «, ) V (a, b, z 1) V (a, b, z 2 ) 2 Ô, 1 where z 1 z, z 2 z, z (x 1)(x + 1), Im(x) 0 and x is a root of the cubic euation (12) If W «, is spherical, then the volume of W «, is given by Vol(W «, ) V (a, b, z 1) V (a, b, z 2 ), 2 where z 1 (x 1 1)(x 1 + 1), z 2 (x 2 1)(x 2 + 1), and x 1, x 2 are nonnegative roots of the cubic euation (12) REMARK 9 (1) The above gives a new formula for the volume of W «, without using integral expression (2) For hyperbolic case, Im V (a, b, z 1 ) and Im V (a, b, z 2 ) are eual to the volume of W «, since ā a 1, b b 1 and V (a, b, z 2 ) V (a, b, z 1 ) (3) For spherical case, V (a, b, z 1 ) and V (a, b, z 2 ) are real numbers

19 558 J MURAKAMI 53 Borromean rings Let K B be the Borromean rings Then, from (9), we have,, (K B ) Ô sin sin sin ( 1) s+1 s ( s)s ((2 ; 2 ) s ) 2 ( 2 2s+2 ; 2 ) 2s+1 ( 2 2s+2 ; 2 ) 2s+1 ( 2 2s+2 ; 2 ) 2s+1 (( 2 ; 2 ) 2s+1 ) 2 ow we compute the optimistic limit of K B (,, ) Let V (a, b, c, z) be the following function (14) V (a, b, c, z) Ô 1(log a + log b + log c + log z) 2Li2 (z) + 2Li 2 (z 2 ) Li 2 (az) Li 2 (bz) Li 2 (cz) + Li 2 a z b c + Li 2 + Li 2 z z log2 z (log a + log b + log c) log z, where a, b, c, z correspond to 2, 2, 2, 2s respectively, and the term Ô 1(log a + log b + log c) is added so that the value of V (a, b, c, z 0 ) in the following coincide with the volume not only for hyperbolic case but also for spherical case By using we have Li 2 (z 2 ) 2Li 2 (z) + 2Li 2 ( z), Li 2 (z 1 ) Li 2 (z) 1 2 (log Ô 2 z)2 + 1 log z + 3, V (a, b, c, z) Li 2 (az) Li 2 (bz) Li 2 (cz) + Li 2 a z Li 2 (z 1 ) + 2Li 2 ( z) 2Li 2 ( z 1 ) (log a + log b + log c)(log z Ô 1) b c + Li 2 + Li 2 + Li 2 (z) z z The optimistic limit of the colored Alexander invariants of K B with fixed a, b and c is given by V (a, b, c, z 0 ) where z 0 is a solution of the following euation (15) dv dz 0

20 COLORED ALEXADER IVARIATS AD COE-MAIFOLDS 559 Let a exp Ô 1«, b exp Ô 1, c exp Ô 1, z exp Ô 1, A tan(«2), B tan( 2), C tan( 2) and T tan( 2) Then (15) is transformed to (16) T 4 (A 2 + B 2 + C 2 + 1)T 2 A 2 B 2 C 2 0 Ê x 0 For real parameters «and, let ½(«, ) («+ ) («), where (x) log2 sin t dt is the Lobachevski function Then «2½ 2, a Im Li 2 (az) Li 2 + log a log z 2 z Therefore, for real parameters «,,, and, Im V (a, b, c, z) «2 ½ 2, 2 + ½ 2, + ½ 2 2, 2 2½ 2, 2 ½ 0, 2 ow, consider the case that the parameter z is a real number In this case, we use parameter T satisfying T (z 1)(z + 1) Then the euation (15) is transformed to (17) T 4 + (A 2 + B 2 + C 2 + 1)T 2 A 2 B 2 C 2 0 For 2, let Æ(, ) be the function in [8], By putting t tan, Æ(, ) Æ(, ) 2 tan ow replace t by (z 1)(z + 1), we get «Æ 2, 1 2 ½ d log(1 cos 2 cos 2) cos 2 2(t 2 + tan 2 ) dt log (1 + t 2 )(1 + tan 2 ) t 2 1 (1+tan )(1 tan ) 1 (1 az)(1 az 1 ) dz log (1 + z 2 ) z Let T 0 be the negative root of (17) and z 0, 0 be the real numbers satisfying z 0 (1 + T 0 )(1 T 0 ) and tan 0 T 0 Then 2 Æ «2, 0 + Æ 2, 0 + Æ 2, 0 2Æ 2, 0 Æ(0, 0 ) Re V (a, b, c, z 0 ) Comparing with the results of [6] and [8], we get the following

21 560 J MURAKAMI Theorem 10 For 0 «,, 2, let M «,, be the cone-manifold with singlar set K B whose cone angles of three components of K B are «,, If M «,, is a hyperbolic cone-manifold with 0 «,,, then Vol(M «,, ) Im(V (a, b, c, z 1 )) Im(V (a, b, c, z 2 )) 1 2 Ô 1 (V (a, b, c, z 1) V (a, b, c, z 2 )), where z 1, z 2 correspond to the real solutions of the euation (16) If M «,, is a spherical cone-manifold with «,, 2, then Vol(M «,, ) Re V (a, b, c, z 3 ) Re V (a, b, c, z 4 ) 1 2 (V (a, b, c, z 3) V (a, b, c, z 4 )), where z 3, z 4 correspond to the real solutions of the euation (17) 6 A-polynomials In this section, we introduce a polynomial or a system of polynomials of the above three examples by using the method in [13] For figure eight knot, it coinsides with the A-polynomial, and for link cases, these are generalization of the A-polynomial for links 61 Figure eight knot The A-polynomial of 4 1 is obtained by eliminating the parameter x from the following system of euations V (a, x) x V (a, x) 0, a log L a Here V (a, x) is given by two parameters a and L correspond to the meridian and longitude The resulting polynomial is (18) a 2 L + al + 2a 2 L + a 3 L a 4 L + a 2 L 2 0 In fact, this polynomial coincides with A J(2, 2) (L, M) in Theorem 7 of [4] with a M 2 62 Whitehead link After the above construction of the A-polynomial of 4 1, the system of A-polynomials of the link K W may be obtained by eliminating the parameter z from the following system of euations V (a, b, z) z V (a, b, z) V (a, b, z) 0, a log L a, b log L b, a b

22 COLORED ALEXADER IVARIATS AD COE-MAIFOLDS 561 where V (a, b, z) is the function given by (11) The parameters a, b correspond to the meridians of the first and second components, and L a, L b correspond to the longitudes of them respectively The result is (19) a 2 bl a 3 + (a 2 b 2 ab 2 + a 2 2ab a + b)l a 2 + (a 2 b ab 2 2ab + b 2 a + 1)L a + b 0, ab 2 L b 3 + (a 2 b 2 a 2 b 2ab + b 2 + a b)l b 2 + (ab 2 a 2 b + a 2 2ab b + 1)L b + a 0 63 Borromean rings The system of A-polynomials of the link K W may be obtained by eliminating the parameter z from the following system of euations V z 0, a V a log L a, b V b log L b, c V c log L c, where V V (a, b, c, z) is the function given by (14) The parameters a, b, c correspond to the meridians of the first, second and third components, and L a, L b, L c correspond to the longitudes of them respectively By eliminating z from the first two euations, we get the following Let a exp Ô 1«, b exp Ô 1, c exp Ô 1, A cot(«2), B cot( 2), C cot( 2), D A 2 (B 2 C 2 + B 2 + C 2 + 1), and E A 2 + B 2 C 2, then (20) DL a 4 4E L a 3 2(D 4A 2 + 4B 2 C 2 )L a 2 4E L a + D 0 For L b and c and L c, we get the similar euation corresponding to the symmetry of a, b Appendix Here we show some formulas we used in the computation Lemma A The following formulas hold «i0 ( 1) i i «i «j1 (1 +«+1 2 j ) This formula comes from the following uantized Pascal relation and an induction n n 1 n 1 (A1) s + (n s) s s s 1

23 562 J MURAKAMI Lemma B For any a, b, c such that b, a c are nonnegative integers and a is not an integer, we have the following identity: 1 b k0 k(a+b c+2) a k; a cb + k; b ((b+1)c a) ; b; ba + b + 1; a + b c + 1 a + ; a + b c + 1; b + 1 Proof Downword induction on b STEP 1 If b 1, 1 b k0 On the other hand, k(a+b c+2) a k; a cb + k; b a; a c 1; 1 ((b+1)c a) ; b; ba + b + 1; a + b c + 1 a + ; a + b c + 1; b + 1 a c ; 1; 1a + ; a c + ( 1) a + ; a c + ; 1; 1a; a c Hence the euality holds for b 1 STEP 2 Assume that Lemma holds for b and prove for b 1 First, we use the relation n n and n; n 1; n Apply this to b + k 1; b + k 1, we have b k0 b k0 k(a+b c+1) a k; a cb + k 1; b 1 k(a+b c+1) a k; a c 1; b + k 1 k; k 1; b 1 b k0 ow we use Pascal retation (A1) 1; b 1 b k0 k(a+b c+1) a k; a c b; k k; k k(a+b c+1) a k; a c b; k k; k

24 1; b 1 1 b 1 b b 1 k0 b 1 k0 COLORED ALEXADER IVARIATS AD COE-MAIFOLDS 563 b 1 k0 b k0 k(a+b c+1) a k; a c k b 1; k + k; k k(a+b c+2) a k; a c 1; b + k k; k b b 1 k0 ( b k) b 1; k 1 k 1; k 1 (k+1)(a+b c+2) a k; a c 1; b + k 1 k 1; k 1 k(a+b c+2) a k; a cb + k; b + k k; k (a+2b c+2) k(a+b c+2) (a k; a cb + k; b b k(a+b c+2) a k 1; a cb + k; b + k k; k (a+2b c+2) a k 1; a cb + k; b) 1 ((b+1)c a) ; b; ba + b + 1; a + b c + 1 b a + ; a + b c + 1; b + 1 (a+2b c+2) ((b+1)(c 1) (a 1)) ; b; ba + b; a + b c + 1 a + 1; a + b c + 1; b + 1 a) ((b+1)c ; b; ba + b; a + b c b a + ; a + b c + 1; b + 1 (a + b + 1 (a+b c+1) c) a) ((b+1)c ((b+1)c a) ; b; ba + b; a + b c b a + ; a + b c + 1; b + 1 c a + b c + 1 (bc a) ; b 1; b 1a + b; a + b c a + ; a + b c; b Hence the formula also holds for b 1 References [1] Y Akutsu, T Deguchi and T Ohtsuki: Invariants of colored links, J Knot Theory Ramifications 1 (1992), [2] IS Gradshteyn and IM Ryzhik: Table of Integrals, Series, and Products, Fifth edition, Academic Press, San Diego, CA, 1996

25 564 J MURAKAMI [3] K Habiro: On the colored Jones polynomials of some simple links; in Recent Progress Towards the Volume Conjecture (Japanese) (Kyoto, 2000), Sūrikaisekikenkyūsho Kōkyūroku 1172, 2000, [4] J Hoste and PD Shanahan: A formula for the A-polynomial of twist knots, J Knot Theory Ramifications 13 (2004), [5] RM Kashaev: The hyperbolic volume of knots from the uantum dilogarithm, Lett Math Phys 39 (1997), [6] R Kellerhals: On the volume of hyperbolic polyhedra, Math Ann 285 (1989), [7] A Kirillov: Clebsh-Gordan uantum coefficients, (Russian), Zap auchn Sem Leningrad Otdel Mat Inst Steklov (LOMI) 168 (1988), Anal Teor Chisel i Teor Funktsii 9, 67 84, 188; translation in J Soviet Math 53 (1991), [8] AD Mednykh: On hyperbolic and spherical volumes for knot and link cone-manifolds; in Kleinian Groups and Hyperbolic 3-Manifolds (Warwick, 2001), London Math Soc Lecture ote Ser 299, Cambridge Univ Press, Cambridge, 2003, [9] H Murakami: Some limits of the colored Jones polynomials of the figure-eight knot, Kyungpook Math J 44 (2004), [10] H Murakami, J Murakami, M Okamoto, T Takata and Y Yokota: Kashaev s conjecture and the Chern-Simons invariants of knots and links, Experiment Math 11 (2002), [11] H Murakami and Y Yokota: The colored Jones polynomials of the figure-eight knot and its Dehn surgery spaces, J Reine Angew Math 607 (2007), [12] J Murakami: A state model for the multivariable Alexander polynomial, Pacific J Math 157 (1993), [13] Y Yokota, On the volume conjecture for hyperbolic knots, arxiv:mathqa/ (2000) [14] Y Yokota: From the Jones polynomial to the A-polynomial of hyperbolic knots; Proceedings of the Winter Workshop of Topology/Workshop of Topology and Computer (Sendai, 2002/ara, 2001), Interdiscip Inform Sci 9, 2003, Department of Mathematics Faculty of Science and Engineering Waseda University Ohkubo 3 4 1, Shinjyuku-ku Tokyo Japan murakami@wasedajp