Kiby-Melvin s τ and Ohtsuki s τ fo Lens Saces Bang-He Li & Tian-Jun Li axiv:math/9807155v1 [mathqa] 27 Jul 1998 Abstact Exlicit fomulae fo τ (L(,q)) and τ(l(,q)) ae obtained fo all L(,q) Thee ae thee systems of invaiants of Witten-tye fo closed oiented 3- manifolds: 1 {τ (M), 2;τ (M), odd 3}, whee τ was defined by Reshetikhin and Tuaev [1], and τ was defined by Kiby-Melvin [2] 2{Θ (M,A), 1,wheeA is a 2-th imitive oot of unity}definedbyblanchet, Habegge, Masbamm and Vogel [3] 3 {ξ (M,A), 1,,k + 2, wheea is an -th imitive oot of unity} defined by the fist autho [] And it was oved in [] that they ae equivalent Exlicit fomulae fo {τ (M), 2} and {ξ (M,e ), odd 3} have been obtained fo lens saces in [5], whee e a = ex(2π 1/a) Ohtsuki [6] defined his invaiant τ(m) = λ n (t 1) n Q[[t 1]] n=0 The fist autho is suoted atially by the Tianyuan Foundation of P R China, the second autho is suoted atially by NSF gant DMS 930580 1
fo ational homology 3-shee M, and obtained that τ(l(,q)) = t 3s(q,)t 1 2 t 1 2 t 1 2 t 1 2 fo lens saces L(,q) with odd, whee s(q,) is the Dedekind sum To obtain τ(l(,q)) with odd, Ohtsuki used the fomula fo τ (L(,q)) with an odd ime, odd and not divisible by, found by Kiby-Melvin [7], and Gaoufalidis [8] To obtain τ(l(,q)) with even, exlicit fomulae fo τ in this case ae needed The aim of this note is to fist deive exlicit fomulas of τ fo all L(,q) and all, and then give fomulas of τ(l(,q)) Notice that in [6], the oientation of L(, q) comes fom the continued faction exantion of /q, while in [5], the oientation comes fom that of /q So to calculate τ (L(,q)) with oientation as in [6], we should stat fom the conjugate of the fomula in Theoem 1 in [5], that is ξ (L(,q)) = ξ (L(,q),e ) Let a,b and be integes, denote by (b,) the geatest common diviso of b and If (b,) = 1, denote (a/b) = ab Z/Z, whee b b 1 (mod ) Notice that if a/b = a 1 /b 1 and (b 1,) = 1, then ab = a 1 b 1 in Z/Z Theoem Let > 1 be odd and C = (,), then ( e 2 e 2 )e (3s(q,)) τ (L(,q)) = ( 1) 1 c 1 /c 2 2 ( e (/c) (q+q η ) c e 2 e 2 e 2η(/c) c, if c = 1 ) 1 )(e 12s(q,) /c )(q(1 )/ c ǫ(c) cη, if c > 1, c q +η e 2 e 2 and ±1 (mod ) 0, if c > 1 and c /q ±1 whee η = 1 o 1, +q q = 1 with 0 < q <, 2 2 2(2) 1 (mod ), ( a b ) is the Jacobi symbol, (/c) /c+(/c) /c = 1, and ǫ(c) = { 1 if c 1 (mod ) i if c 1 (mod ) 2
Poof By the fomulas in [] (see also [5]), we have τ νθ (M) = ( sinπ ) (M,±ie ) fo ±1 (mod ) and ξ (M,A) = 2 ν Θ (M, A) fo 3 1 (mod 2), whee ν is the fist Betti numbe Since fo lens saces, ν = 0, we have, fo ±1 (mod ), τ (L(,q)) = Θ (L(,q),±ie ) = ξ (L(,q), ie ) = ξ (L(,q),e 1 ), Case 1 c = 1 By Theoem 1 in [5] Since e 12s(q,) e (q+q ) 1 in [5]), it follows that ξ (L(,q),e ) = ( )e 12s(q,) e (q+q ) e 2 e 2 e 2 e 2 = e a fo some a Z (by checking the oof of Theoem τ (L(,q)) = ( )e 12s(q,)1 e (q+q ) 1 e 1 2 e 1 2 e 1 2 e 1 2 fo ±1 (mod ) The simle elations 1 = 2 and 2 = (2) in Z/Z imly 2 that e 1 2 e 1 2 e 1 2 e 1 2 It is well known that (cf [8] o [5]) = e2 e 2 e 2 e 2 12s(q,) = b q +q fo some b Z Hence 3s(q,) has a fom of m/ fo some m Z Now (,) = 1, so thee ae integes P and R such that P +R = 1 Thus e 12s(q,)1 e (q+q ) 1 = e m (P+R)(1 ) = e mp e (q+q ) 1 e 1 q+q (12s(q,)R ) 3
Now 12s(q,)R q +q = ( b+ q +q )R q +q Since R 1 (mod ), R It is concluded that 12s(q,)R q +q 0 (mod ) Because m = (3s(q,)), the oof is comlete fo case 1 Case 2 c > 1 and c q +η fo η = 1 o 1 Then ξ (L(,q),e ) = ( 1) 1 c 1 /c 2 2 ( /c )(q c )e12s(q,) = br+(r ) q +q e (/c) (q+q η ) c e 2η(/c) c Since c is odd and ( 1 ) = 1, it follows fom Theoem 22 in [5] that c ǫ(c) c = j=1 e j2 c = j=1 e c j2 ǫ(c) cη e 2 e 2 Hence, by the well-known fomula fo Gaussian sum (cf [10] o [5]), if ±1 (mod ), e 1 c j2 = j=1 j=1 By the oof of Theoem 1 in [5], e 1 j2 c = ǫ(c)( (1 )/ ) c c e 12s(q,) e (/c) (q+q η ) c e 2η(/c) c = e k fo some k Z Again fom e 1 2 it is concluded that if ±1 (mod ) τ (L(,q)) = ξ (L(,q),e 1 = e 1 2 = e 2, ) = ( 1) 1 c 1 /c 2 2 ( /c )(q(1 )/ c )e 1 k ǫ(c) cη e 2 Case 3 c > 1 and c /q ± 1 Since ξ (L(,q),e ) = 0, τ (L(,q)) = 0 The theoem is oved Coollay Fo any lens saces L(, q), Ohtsuki s invaiant τ(l(,q)) = t 3s(q,)t 1 2 t 1 2 t 1 2 t 1 2 e 2
Poof Po52 in [6] fo odd is now also tue fo even by the above theoem Hence fo even, τ(l(,q)) has the same fom as fo odd Remak 1 Lemma 5 in [6] concening odd is also tue fo even Theefoe, Remak 53 in [6] fo odd is tue all Remak 2 In [6], Ohtsuki egaded τ as SO(3) invaiants, while in [11], SO(3) invaiants wee efeed to those dived fom the Kauffman module Z[A,A 1 ][z 2 ] The latte includes the fome and something moe Refeences [1] Reshetikhin, NYu, Tuaev, VG: Invaiants of 3-manifolds via link olynomials and quantum gous, Invent Math 103 (1991), 57-597 [2] Kiby, R, Melvin, P: The 3-manifold invaiants of Witten and Reshetikhin- Tuaev fo sl(2, C), Invent Math 105 (1991), 73-55 [3] Blanchet, C, Habegge, N, Masbaum, G, Vogel, P: Thee- manifold invainats deived fom the Kauffman backet, Toology, 31 (1992), 685-699 [] Li, BH, Relations among Chen-Simons-Witten-Jones invaiants, Science in China, seies A, 38 (1995), 129-16 [5] Li, BH, Li, TJ: Genealized Gaussian Sums and Chen-Simons- Witten-Jones invaiants of Lens saces, J Knot theoy and its Ramifications, vol5 No 2 (1996) 183-22 [6] Ohtsuki, T, A olynomial invaiant of ational homology 3-shees, Inv Math 123 (1996), 21-257 [7] Kiby, R, Melvin, P: Quantum invaiants of Lens saces and a Dehn sugey fomula, Abstact Ame Math Soc 12 (1991), 35 [8] Gaoufalidis, S, Relation among 3-manifold invaiants, PhD Thesis, Univ of Chicago, 1992 [9] Hickeson, D, Continued faction and density esults, J Reine Angew Math 290 (1977), 113-116 5
[10] Lang, S, Algebaic Numbe Theoy, Singe, New Yok, 1986 [11] Li, BH, Li, TJ: SO(3) thee-manifold invaiants fom the Kauffman backet, Poc of the Confeence on Quantum Toology, Kansas, 199, 27-258 Autho s addess Bang-He Li Tian-Jun Li Institute of Systems Science, School of math Academia Sinica IAS Beijnig 100080 Pinceton NJ 0850 P R China USA Libh@iss06issaccn Tjli@IASedu 6