Twist knot orbifold Chern-Simons

Size: px

Start display at page:

Download "Twist knot orbifold Chern-Simons"

きょういちかんけ
4 years ago
Views:

1 Twist knot orbifold Chern-Simons

2 1 3 M π F : F (M) M ω = {ω ij }, Ω = {Ω ij }, cs := 1 4π 2 (ω 12 ω 13 ω 23 + ω 12 Ω 12 + ω 13 Ω 13 + ω 23 Ω 23 ) M Chern-Simons., S. Chern J. Simons, F (M) Pontrjagin 2., M Riemann, Levi-Civita, π F s : M F (M), cs mod 1 s(m) s. M Chern-Simons, CS(M). 3 Chern-Simons,., PSL(2, C) Γ, Γ ω ω : H 3 (sl(2, C)) H 3 DR(PSL(2, C) Γ PSL(2, C))., PSL(2, C) Γ PSL(2, C) Γ PSL(2, C) PSL(2, C)., 3 M = H 3 /Γ, 1 ω (A) = Vol(M) + 1 2π 2 CS(M) mod 1 2π 2 (1.1) 4 s(m)., A H 3 (sl(2, C)), s : M F (M) π F., Chern-Simons, M Vol(M). (1.1) (complex volume), CV(M). Chern-Simons, Meyerhoff [6], Hilden-Lozano- Montesinos [5] orbifold., S 3 twist knot orbifold Chern-Simons. [3],, dilogarithm Li 2 (z) =,. z 0 log(1 w) dw w 2 orbifold Chern-Simons, [5] S 3 Σ orbifold Chern-Simons, Yoshida [10, Definition 3.2]., Σ α (S 3, Σ α ) Σ(α), Σ α meridian holonomy 1

3 e u(α), Σ α holonomy e v(α). Σ α meridian m [6, Section 3.1], I(Σ(α)) := 1 2 s α : Σ(α) \ (Σ α m) F (Σ(α) \ (Σ α m)) s α(σ(α)\(σ α m)) cs + 1 4π s α(m) ω 23 1 Re u(α)v(α) (2.1) 2π2., s α s α Σ(α), ω ij F (Σ(α)) Levi- Civita, cs Σ(α) Chern-Simons. [5], orbifold Σ(2π/n) Chern-Simons CS(Σ(2π/n)) := I(Σ(2π/n)). mod 1 n (2.2) Yoshida, S 3 \ Σ,, f(u(α)) := s α (Σ(α)\Σ α m)) C 1 (θ 1 1ω 23 ) 2π s α (m)., C F (M(α)) 1 θ i Levi-Civita, C := 1 4π 2 (4θ 1 θ 2 θ 3 d(θ 1 ω 23 + θ 2 ω 31 + θ 3 ω 12 )) 1cs., α, Re f(u(α)) = 1 {Vol (Σ(α)) + Im u(α)v(α)}, π2 Im f(u(α)) = 2I(Σ(α)) + Re 1 π 2 u(α)v(α)., α = 2π/n, (2.2), Im f (u(2π/n)) = 1 π 2 { 2π 2 CS (Σ(2π/n)) + Re u(2π/n)v(2π/n) } mod 2 n., f(u(2π/n)) = 1 π 2 { CV(Σ(2π/n)) 1 u(2π/n)v(2π/n) } mod 2 n 1 (2.3). 3 Neumann-Zagier, Σ S 3., Neumann-Zagier [7] S 3 \ Σ Neumann-Zagier., Σ meridian holonomy e u, Σ holonomy e v., Φ(u) := u 0 (vdu udv) + uv (3.1) 2

4 1: x 1,..., x k 1, Φ, v = 1 Φ, Φ(0) = 0 (3.2) 2 u., Neumann-Zagier., [5, Theorem 3.4], S 3 \ Σ Yoshida f, 1 u (vdu udv) = f(u) 1 π 2 CV(S3 \ Σ) (3.3). π Hyperbolic twist knot k, 1 S 3 twist knot, T k. k > 1, S 3 \ T k. 4.1 [1], S 3 \T k Chern-Simons. 4.1 S 3 \ T k V 1 (x 1,..., x k 1 ), V (x 1,..., x k 1 ) := Li 2 (1/x 1 ) + Li 2 (x 1 ) k 1 + {Li 2 (x i ) + Li 2 (1/x i ) Li 2 (x i 1 /x i )} π2 (k 2) 6 i=2 3

5 ., x 1,..., x k 1 T k ([1]) { } V exp x 1 x 1 { } { V = exp x 2 = = exp x 2 V x k 1 x k 1 } = 1, S 3 \ T k., S 3 \ T k. (ξ 1,..., ξ k 1 ) 4.2 (ξ 1,..., ξ k 1 ) S 3 \ T k geometric solution., i, [ ] V x i = 2π x i (x1 1r i,...,x k 1 )=(ξ 1,...,ξ k 1 ) r i Z. 4.3 ([9, Theorem 2.6]) S 3 \ T k Ṽ (x 1,..., x k 1 ) := V (x 1,..., x k 1 ) 2π k 1 1 r i log x i i=1, Ṽ (ξ 1,..., ξ k 1 ) = 2π 2 CS(S 3 \ T k ) + 1Vol(S 3 \ T k ) mod π 2., Ṽ (ξ 1,..., ξ k 1 ) = 1 CV(S 3 \ T k ) mod π , [8], S 3 \ T k. 4.4 S 3 \ T k V (x 1,..., x k 1, u), V (x 1,..., x k 1, u) := Li 2 (1/x 1 e u ) + Li 2 (x 1 /e u ) k 1 + {Li 2 (x i /e u ) + Li 2 (e u /x i ) Li 2 (x i 1 /x i )} i=2 k 1 + 2u ( 1) i+1+k log x i i=1 π2 6 (k 2) 3 { 1 ( 1) k} u 2., x 1,..., x k 1, e u 2. 4

6 2: x 1,..., x k 1, e u, u = 0, ([8]) V V V x 1 = x 2 = = x k 1 = 0 x 1 x 2 x k 1 S 3 \ T k., e u T k meridian., S 3 \ T k. (ξ 1 (u),..., ξ k 1 (u)) 4.5 (ξ 1 (u),..., ξ k 1 (u)) S 3 \ T k geometric solution. 4.6 ([8]) { } V exp = e 5+( 1)k 7 k 1 { } V u (1 1/x 1 e u ) 2 exp x i x 2( 1)i+1+k i x i i=1, T k longitude holonomy e v, 1 V 2 u = v + 2π 1r (4.1) (x1,...,x k 1,u)=(ξ 1 (u),...,ξ k 1 (u),u) r Z. 5

7 , (3.2). 4.7 V (x 1,..., x k 1, u) Ṽ (x 1,..., x k 1, u) := V (x 1,..., x k 1, u) 2π k 1 1 r i log x i + 2π 1 ru (4.2) i=1, Φ(u) = Ṽ (ξ 1(u),..., ξ k 1 (u), u) + 1CV(S 3 \ T k ) mod π Twist knot orbifold Chern-Simons, T k α T k (α).,. 4.8 ( ) n, Twist knot orbifold T k (2π/n) Chern-Simons CS(T k (2π/n)), : n, CS (T k (2π/n)) = 1 2π 2 Re Ṽ (ξ 1(u(2π/n)),..., ξ k 1 (u(2π/n)), u(2π/n)) mod 1 n, n, CS (T k (2π/n)) = 1 2π 2 Re Ṽ (ξ 1(u(2π/n)),..., ξ k 1 (u(2π/n)), u(2π/n)) mod 1. (3.3) 4.7, Ṽ (ξ 1 (u(α)),..., ξ k 1 (u(α)), u(α)) = 1π 2 f(u(α)) mod π 2., α = 2π/n, (2.3), N, M Z, Ṽ (ξ 1 (u(2π/n)),..., ξ k 1 (u(2π/n)), u(2π/n)) = 1CV(T k (2π/n)) +., nn 2M π 2 (4.3) n CS(T k (2π/n)) = 1 2π 2 Re Ṽ (ξ 1(u(2π/n)),..., ξ k 1 (u(2π/n)), u(2π/n)) mod 1., n = 2l (l N), (4.3) π 2, nn 2M n = 2(lN M) 2l = ln M l., n, CS(T k (2π/n)) = 1 2π 2 Re Ṽ (ξ 1(u(2π/n)),..., ξ k 1 (u(2π/n)), u(2π/n)) mod 1 n. 6

8 5 Chern-Simons 4.5, T k (α) geometric solution ξ 1 (u(α)),..., ξ m 1 (u(α)), 1 ξ 2 (u(α)) = 1 ( ) 1 e ( 1)1+k2u(α) e {( 1)1+k 1}2u(α) x 1 ( + e {2( 1)1+k 1}u(α) 1 ξ i+1 (α) = e (2( 1)i+k +1)u(α) ξ k 1 (u(α)) = ξ k 2 (u(α)) e 3u(α) ) ξ 1 (α) 2, ( 1 ξ i 1(u(α)) ξ i (u(α)) ) + 1 ξ i (u(α)) (2 i k 2),.,, 4.8 T k (2π/n) n M n (T k ) Chern- Simons M n (T k ). T 3 (2π/n) n :even CS (T 3 (2π/n)) mod 1 n CS (M n (T 3 )) mod 1 Vol(M n (T 3 )) n :odd CS (T 3 (2π/n)) mod 1 CS (M n (T 3 )) mod 1 2 Vol (M n (T 3 )

9 T 4 (2π/n) n : even CS (T 4 (2π/n)) mod 1 n CS (M n (T 4 )) mod 1 Vol (M n (T 4 )) n : odd CS (T 4 (2π/n)) mod 1 CS (M n (T 4 )) mod 1 2 Vol (M n (T 4 )) T 5 (2π/n) n : even CS (T 5 (2π/n)) mod 1 n CS (M n (T 5 )) mod 1 Vol (M n (T 5 ))

10 n : odd CS (T 5 (2π/n)) mod 1 CS (M n (T 5 )) mod 1 2 Vol (M n (T 5 )) T 6 (2π/n) n : even CS (T 6 (2π/n)) mod 1 n CS (M n (T 6 )) mod 1 Vol (M n (T 6 )) n : odd CS (T 6 (2π/n)) mod 1 CS (M n (T 6 )) mod 1 2 Vol (M n (T 6 ))

11 T 7 (2π/n) n : even CS (T 7 (2π/n)) mod 1 n CS (M n (T 7 )) mod 1 Vol (M n (T 7 )) n : odd CS (T 7 (2π/n)) mod 1 CS (M n (T 7 )) mod 1 2 Vol (M n (T 7 )) T 8 (2π/n) n : even CS (T 8 (2π/n)) mod 1 n CS (M n (T 8 )) mod 1 Vol (M n (T 8 ))

12 n : odd CS (T 8 (2π/n)) mod 1 CS (M n (T 8 )) mod 1 2 Vol (M n (T 8 )) T 9 (2π/n) n : even CS (T 9 (2π/n)) mod 1 n CS (M n (T 9 )) mod 1 Vol (M n (T 9 )) n : odd CS (T 9 (2π/n)) mod 1 CS (M n (T 9 )) mod 1 2 Vol (M n (T 9 ))

13 T 10 (2π/n) n : even CS(T 10 (2π/n)) mod 1 n CS (M n (T 10 )) mod 1 Vol (M n (T 10 )) n : odd CS(T 10 (2π/n)) mod 1 CS (M n (T 10 )) mod 1 2 Vol (M n (T 10 )) ,,. [1] J. Cho, J. Murakami, and Y. Yokota, The complex volumes of twist knots, Proc. American Mathematical Society 137 (2009) [2], 3,, 2002 [3] Ji-young Ham, H. Kim, and J. Lee, Explicit formulae for Chern-Simons invariants of the twist knot orbifolds and Edge polynomials of twist knots, arxiv: vl [4] H. M. Hilden, M. T. Lozano, and J. M. Montesinos-Amilibia, Volumes and Chern-Simons invariants of cyclic coverings over rational knots, The 37th Taniguchi Symposium on Topology and Teichmuller spaces (Finland, July 1995) ed. by Sadayoshi Kojima et al., 1996 World Scientific Pub Co.,

14 [5] H. M. Hilden, M. T. Lozano, and J. M. Montesinos-Amilibia, On volumes and Chern- Simons invariants of geometric 3-manifolds, J. Math. Sci. Univ. Tokyo 3 (1996), [6] R. Meyerhoff, Density of the Chern-Simons invariant for hyperbolic 3-manifolds, Lowdimensional topology and Kleinian groups, London (D. B. A. Epstein, eds.), London Math. Soc. Lect. Notes 112 (Cambridge University Press, 1987), [7] W. D. Neumann and D. Zagier, Volumes of heyperbolic three-manifolds, Topology 24 (1985), [8] Y. Yokota, From the Jones polynomial to the A-polynomial of hyperbolic knots, Interdisplinary Information Sciences 9 (2003), [9] Y. Yokota, On the complex volume of hyperbolic knots, Journal of Knot Theory and Its Ramifications 20 (2011), [10] T. Yoshida, The η-invariant of hyperbolic 3-manifolds, Invent. math. 81 (1985),

Chern-Simons Jones 3 Chern-Simons 1 - Chern-Simons - Jones J(K; q) [1] Jones q 1 J (K + ; q) qj (K ; q) = (q 1/2 q

Chern-Simons Jones 3 Chern-Simons 1 - Chern-Simons - Jones J(K; q) [1] Jones q 1 J (K + ; q) qj (K ; q) = (q 1/2 q Chern-Simons E-mail: fuji@th.phys.nagoya-u.ac.jp Jones 3 Chern-Simons - Chern-Simons - Jones J(K; q) []Jones q J (K + ; q) qj (K ; q) = (q /2 q /2 )J (K 0 ; q), () J( ; q) =. (2) K Figure : K +, K, K 0