On the volume conjecture for quantum 6 j symbols Jun Murakami Waseda University July 27, 2016 Workshop on Teichmüller and Grothendieck-Teichmüller theories Chern Institute of Mathematics, Nankai University
Quantum Invariants Volume Conjecture Quantum 6 j symbol Contents Quantum Invariants Volume Conjecture Quantum 6 j symbol 2/22
Quantum Invariants Volume Conjecture Quantum 6 j symbol Overview Quantum Invariants Volume Conjecture Quantum 6 j symbol 3/22
Quantum Invariants Volume Conjecture Quantum 6 j symbol Jones Polynomial ▶ Jones polynomial (1984) ` U q ( sl 2 ) t 1 = 2 ` t ` 1 = 2 ” t ` 1 V K ` ( t ) ` t V K + ( t ) = “ V K 0 K + : K ` : K 0 : j K 1 j : Cf. Alexander polynomial r K + ( z ) `r K ` ( z ) = z r K 0 ( z ) ▶ HOMFLY-PT polynomial (1987) ` U q ( sl n ) a ` 1 P K ` ( t ) ` a P K + ( t ) = t 1 = 2 ` t ` 1 = 2 ” “ P K 0 ▶ Kauffman polynomial (1987) ` U q ( so n ) ; U q ( sp 2 n ) a ` 1 F j K ` j ( t ) ` a F j K + j ( t ) = t 1 = 2 ` t ` 1 = 2 ” “ “ ” F j K 0 j ` F j K 1 j 4/22
Quantum Invariants Volume Conjecture Quantum 6 j symbol Overview Quantum Invariants Volume Conjecture Quantum 6 j symbol 5/22
Quantum Invariants Volume Conjecture Quantum 6 j symbol Kashaev’s conjecture Let L be a knot and K N ( L ) be the quantum invariant introduced by Kashaev. Then 2 ı N log j K N ( L ) j = Vol ( S 3 n L ) : lim N !1 N ` 1 k X j ( q ) k j 2 , Y ( 1 ` q j ) K N ( 4 1 ) = ( q ) k = 4 1 : j = 1 k = 0 ` ! 2.02988321. . . N !1 q ` k ( l + 1 ) = 2 ( q ) 2 X K N ( 5 2 ) = l ( q ` 1 ) k 5 2 : k » l ` ! 2.82812208. . . N !1 q ( m ` k ` l )( m ` k + 1 ) = 2 j ( q ) m j 2 X K N ( 6 1 ) = 6 1 : ( q ) k ( q ` 1 ) l k + l » m ` ! 3.16396322. . . N !1 K N ( L ) is equal to the colored Jones inv. V N L ( q ) for N -dim. rep. of U q ( sl 2 ) at q = exp 2 ı p` 1 = N . ( H. M urakami -J.M. ) 6/22
Quantum Invariants Volume Conjecture Quantum 6 j symbol Generalizations of Kashaev’s Conjecture s N = exp ( ı p` 1 = N ) , q N = s 2 N . Volume Conjecture. ( H. M urakami -J.M. ) L : a knot 2 ı ˛ = Vol Gr ( S 3 n L ) ; ˛ ˛ ˛ V N lim N log L ( q N ) ˛ ˛ N !1 where Vol Gr is Gromov’s simplicial volume. Complesified Volume Conjecture ( H.M urakami -J.M.-M.O kamoto -T.T akata -Y.Y okota ) L : hyperbolic knot 2 ı L ( q N ) = Vol ( S 3 n L ) + q ` 1 CS ( S 3 n L ) : N log V N lim N !1 7/22
Quantum Invariants Volume Conjecture Quantum 6 j symbol Proof for the figure-eight knot ( E kholm ) s N = exp ( ı p` 1 = N ) , q N = s 2 K : figure-eight knot, N . N ` 1 N ` 1 j j 4 sin 2 ı k X X Y Y ( s N ` k ` s ` N + k )( s N + k ` s ` N ` k V N K ( s N ) = ) = N : N N N N k = 1 k = 1 j = 0 j = 0 Q j k = 1 4 sin 2 ı k Let a j = N and a j max be the max. term of a j . Since a j max » V N K ( s N ) » N a j max , 2 ı log V N K ( s N ) 2 ı log a j max 2 ı log ( N a j max ) 2 ı log ( a j max ) lim » lim » lim = lim : N N N N N !1 N !1 N !1 N !1 a j is decreasing for small j ’s, increasing for middle j ’s and then decreasing for large j ’s. It takes the maximal at j ≒ 5 N 6 . Since a 0 = 1, a N ` 1 = N 2 , a j is maximum at j ≒ 5 N 6 . Therefore P 5 N = 6 4 ı k = 1 log ( 2 sin ı k = N ) 2 ı log ( a j max ) lim = lim = N !1 N N !1 N 5 ı 5 ı ! Z 6 4 log ( 2 sinx ) dx = ` 4 ˜ = 2 : 02988321 ::: 6 0 8/22
Quantum Invariants Volume Conjecture Quantum 6 j symbol Volume potential function ▶ Dilogarithm function Analytic continuation of Z x du = x + x 2 2 2 + x 3 log ( 1 ` u ) Li 2 ( x ) = ` 3 2 + ´ ´ ´ ( 0 < x < 1 ) : u 0 It is a multi-valued function and its branches are q ` 1 log z + 4 l ı 2 li 2 ( z ) = Li 2 ( z )+ 2 k ı ( k ; l 2 Z ) ▶ Volume potential function U ( x 1 ; x 2 ; : : : ) The function obtained by replacing ( q ) k in U q ( sl 2 ) -invariant by Li 2 ( x ) ( x = q k N ). ▶ Saddle points of the volume potential function Points satisfying @ U = 0 ( i = 1 ; 2 ; : : : ). @ x i These equations correspond to the glueing equation of the tetrahedral decomposition ( Y okota ). 9/22
Quantum Invariants Volume Conjecture Quantum 6 j symbol Hyperbolic volume from the saddle point K : hyperbolic knot, U : Volume potential function of V N K ( q N ) (colored Jones) ▶ Hyperbolic volume @ U ! ( 0 ) ( 0 ) For some x , x , : : : satisfying exp = 1, 1 2 @ x i ˛ @ U ( 0 ) ( 0 ) X ˛ ( 0 ) U ( x ; x ; : : : ) ` log x ˛ 1 2 i ˛ @ x i ˛ x 1 = x ( 0 ) ; ´´´ i 1 is the hyperbolic volume of the complement of K ( Y okota , J. C ho ). ▶ Optimistic calculation ( H.M urakami , arXiv:math/0005289 ) M : 3-manifold ， fi N ( M ) : W itten -R eshetikhin - T uraev invariant Check that fi N ( M ) ` ! hyperoblic volume of M for M obtained by surgery along the figure-eight knot. ▶ Optimistic conjecture U q ( sl 2 ) inariant ` ! hyperbolic volume 10/22
Quantum Invariants Volume Conjecture Quantum 6 j symbol ! q 2 q N ` N Attention! ▶ The U q ( sl 2 ) -invariant grows exponentially only for Kashaev’s invariant and its deformation. ▶ For other U q ( sl 2 ) -invariants, they does not grow exponentially. Renovation by Q. C hen -T. Y ang arXiv:1503.02547 ▶ Replace q N = exp ( 2 ı p` 1 = N ) by q 2 N . ▶ Various U q ( sl 2 ) -invariants grow exponentially and the growth rates are given by the hyperbolic volume of the corresponding geometric objects. Including: WRT invariant, Turaev-Viro inv. for 3-mfds, Kirillov-Reshetikhin invariant for knotted graphs 11/22
Quantum Invariants Volume Conjecture Quantum 6 j symbol Overview Quantum Invariants Volume Conjecture Quantum 6 j symbol 12/22
Quantum Invariants Volume Conjecture Quantum 6 j symbol Quantum 6 j symbol The quantum 6 j symbol is introduced to express 0 1 ȷ i 0 1 V m , ! V m , ! ff j l X A = V l ˙ V k , ! V i ˙ V n , ! @ @ A m k n V i ˙ V j ˙ V k q V i ˙ V j ˙ V k n for representations of U q ( sl 2 ) by Kirillov-Reshetikhin . f k g = f q 1 = 2 ` q ` 1 = 2 g ; f k g ! = f k g f k ` 1 g : : : f 1 g ; f c 1 + c 2 + c 3 + 1 g ! W 1 ( e ) = f c + 1 g 2 ; W 2 ( f ) = ; f 1 g f 1 g f ` c 1 + c 2 + c 3 g ! f c 1 ` c 2 + c 3 g ! f c 1 + c 2 ` c 3 g ! 2 2 2 ȷ c 1 ff c 2 c 5 „ c 1 « c 2 c 5 ff RW c 4 c 3 c 6 ȷ c 1 c 4 c 3 c 6 c 2 c 5 q = p = p W 2 ( 1 ; 2 ; 5 ) W 2 ( 1 ; 3 ; 6 ) W 2 ( 2 ; 4 ; 6 ) W 2 ( 3 ; 4 ; 5 ) ; c 4 c 3 c 6 W 1 ( c 5 ) W 1 ( c 6 ) q min ( A 1 ;:::; A 3 ) ( ` 1 ) k f k + 1 g ! „ c 1 « c 2 c 5 X = f 1 g f A 1 ` k g ! : : : f A 3 ` k g ! f k ` B 1 g ! : : : f k ` B 4 g ! ; c 4 c 3 c 6 k = max ( B 1 ; ´´´ ; B 4 ) where 2 A 1 = c 1 + c 2 + c 3 + c 4 ; 2 B 1 = c 1 + c 2 + c 5 ; 2 A 2 = c 1 + c 4 + c 5 + c 6 ; 2 B 2 = c 1 + c 3 + c 6 ; 2 A 3 = c 2 + c 3 + c 5 + c 6 ; 2 B 3 = c 2 + c 4 + c 6 ; 2 B 4 = c 3 + c 4 + c 5 : Here we assume c 1 , ´ ´ ´ , c 6 are admissible, i.e. A i , B j , A i ` B j 2 Z – 0 . 13/22
Quantum Invariants Volume Conjecture Quantum 6 j symbol Turaev-Viro invariant State sum on a tetrahedral decomposition ı p N ` s ` k s k ` 1 Let s N = exp ( ) , [ k ] = N , [ k ]! = [ k ][ k ` 1 ] ´ ´ ´ [ 1 ] , N s N ` s ` 1 N I N = f 0 ; 1 ; : : : ; N ` 2 g . ▶ Turaev-Viro invariant Let M be a closed 3-manifold and let ´ be a tetrahedral decomposition of M . Let T , F , E be the set of tetrahedrons, faces, edges of ´ . Then X Y Y W 2 ( f ) ` 1 Y TV N ( M ) = W 1 ( e ) W 3 ( t ) : e 2 E f 2 F t 2 T ’ : E ! I N admissible TV ! potential fun. ! saddle pt. ! hyp. volume ▶ Quantum 6 j symbol quantum 6 j ! potential function ! saddle point ! hyperbolic volume symbol of tetrahedron ( J. M.-M. Y ano ) 14/22
Quantum Invariants Volume Conjecture Quantum 6 j symbol ! q 2 Chen-Yang’s invariant and q ` Let s N = exp ( ı p` 1 = N ) and q N = s 2 N . ▶ Chen-Yang’s invariant Extend TV-invariant for 3-manifolds with boundary by using ideal tetrahedrons and truncated tetrahedrons. Conjecture (Q.Chen-T.Yang, arXiv:1503.02547) Let M be a 3-manifold ， N be a positive odd integer, CY N ( M ) be Chen-Yang’s invariant, fi N ( M ) be the WRT invariant, and TV N ( M ) be the Turaev-Viro invariant of M . Then we have „ « q 2 ı 1. lim log CY N ( M ) j s N ! s 2 = Vol ( M ) + ` 1 CS ( M ) N N !1 N „ « q 4 ı 2. lim log fi N ( M ) j s N ! s 2 = Vol ( M ) + ` 1 CS ( M ) N N !1 N „ « 2 ı 3. lim log TV N ( M ) j s N ! s 2 = Vol ( M ) ; N N !1 N since 2. and the fact that TV N ( M ) = j fi N ( M ) j 2 . 15/22
Quantum Invariants Volume Conjecture Quantum 6 j symbol Volume conjecture for the quantum 6 j symbol Conjecture T : hyperbolic tetrahedron with dihedral angles „ 1 , : : : , „ 6 , N : positive odd integer, ( N ) ( N ) 2 ı : admissible sequences with lim = ı ` „ i , a N a i i N !1 (1 » i » 6) Then ˛ ˛ RW 8 ( N ) ( N ) ( N ) 9 ˛ ˛ 2 ı a a a ˛ < = ˛ 1 2 5 lim log = Vol ( T ) : ˛ ˛ ( N ) ( N ) ( N ) ˛ ˛ N !1 N a a a ˛ : ; ˛ 4 3 6 q 2 ˛ ˛ N Theorem The above conjecture is true if all the vertices are truncated, i.e. the sum of the three dihedral angles sharing a vertex is less than ı . 16/22
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