Homological representation formula of some quantum sl 2 invariants Tetsuya Ito (RIMS) 2015 Jul 20 First Encounter to Quantum Topology: School and Workshop Tetsuya Ito (RIMS) Homological representation formula 2015 Jul 1 / 34
I: Introduction Tetsuya Ito (RIMS) Homological representation formula 2015 Jul 2 / 34
Quantum invariants and braids The braid group representation plays an important and fundamental role in quantum topology. U q (g): Quantum group of semi-simple lie algebra g V : U q (g)-module(s) = φ V : B n GL(V n ): quantum representation {Braids} Closure {(Oriented) Links } Surgery {Closed 3-manifolds} φ V Quantum representation Q V GL(V n ) quantumtrace C[q, q 1 ] Quantum invariant q=e 2π 1 r Take linear sums τ g r C Quantum invariant Tetsuya Ito (RIMS) Homological representation formula 2015 Jul 3 / 34
Main questions Motivating quetsion What is the topological content of quantum invariants? Tetsuya Ito (RIMS) Homological representation formula 2015 Jul 4 / 34
Main questions Motivating quetsion What is the topological content of quantum invariants? Witten s (Physical) solution, Chern-Simons functional (path integral) interpretation, is not easy to use (to study classical topology). Can estimate knot genus? Can detect fiberedness? One of the most common (and easiest) way to manipulate quantum invariants is to use diagram. Good: We can treat everything combinatorially. Bad: It makes difficult to understand underlying topology. Tetsuya Ito (RIMS) Homological representation formula 2015 Jul 4 / 34
Model case: Alexander polynomial (I) The Alexander polynomial K (t) is well-understood in terms of topology having various applications. Seifert matrix Infinite cyclic covering Skein relation State-sum U q (sl 2 ) or, U q (sl(1 1)) A lot of equivalent definitions! Alexander polynomial K (t) Tetsuya Ito (RIMS) Homological representation formula 2015 Jul 5 / 34
Model case: Alexander polynomial (II) In a braid group point of view, Alexander polynomial arise from homological representation (reduced Burau representation) Classical Theorem ρ Burau : B n GL(n 1; Z[x, x 1 ]) = GL(H 1 ( D n ); Z). For a closed n-braid K = β, 1. K (x) = x 1 2 x 1 2 x n 2 x n 2 det(ρ Burau(β) I ). 2. (ρ Burau (β) I ) is equal to the Alexander matrix (presentation matrix of Alexander module) = Homological nature of braid group representation is useful. Tetsuya Ito (RIMS) Homological representation formula 2015 Jul 6 / 34
Summary of result I: Quantum sl 2 representations as homological representations By choosing a weight λ C {0}, we have a U q (sl 2 )-module V λ,q. When λ (and q) are generic, Generic U q (sl 2 ) representations (from V λ,q ) q=exp( 2π 1 2N ) (q:non-generic) λ-deformation of N- dim irreducible representation of U q (sl 2 ) Theorem A Theorem A Lawrence s representation quotient q=exp( 2π 1 2N ) truncated Lawrence s representations Tetsuya Ito (RIMS) Homological representation formula 2015 Jul 7 / 34
Summary of result (II) II: Homology representation formula of the loop expansion of quantum sl 2 invariants Generic U q (sl 2 ) representation (from V quite different! N-dim irreducible representation of U λ,q ) generic vs. Non-generic q (sl 2 ) (quantum) trace sum of trace Theorem A Colored Jones polynomial Lawrence s (homological) representation sum of trace Theorem B lim q 1,N Loop expansion of sl 2 invariant Tetsuya Ito (RIMS) Homological representation formula 2015 Jul 8 / 34
Summary of result (II) II: Homology representation formula of the loop expansion of quantum sl 2 invariants Generic U q (sl 2 ) representation (from V quite different! N-dim irreducible representation of U λ,q ) generic vs. Non-generic q (sl 2 ) (quantum) trace sum of trace Theorem A Colored Jones polynomial Lawrence s (homological) representation sum of trace Theorem B lim q 1,N Loop expansion of sl 2 invariant = more topological proof of Melvin-Morton-Rozansky conjecture Tetsuya Ito (RIMS) Homological representation formula 2015 Jul 8 / 34
Summary of result (III) III: Homology representation formula of colored Alexander invariant At q = exp( 2π 1 2N ), λ-deformation of N-dim irr. representation of U q (sl 2 ) (quantum) N-Colored Alexander trace invariant N=2 sum of trace Theorem A Alexander polynomial Theorem C Classical formula (Exterior powers of) N=2 Burau representation Truncated Lawrence s representation Tetsuya Ito (RIMS) Homological representation formula 2015 Jul 9 / 34
Summary of result (III) III: Homology representation formula of colored Alexander invariant At q = exp( 2π 1 2N ), λ-deformation of N-dim irr. representation of U q (sl 2 ) (quantum) N-Colored Alexander trace invariant N=2 sum of trace Theorem A Alexander polynomial Theorem C Classical formula (Exterior powers of) N=2 Burau representation Truncated Lawrence s representation = generalization of Burau representation formula of the Alexander polynomial Tetsuya Ito (RIMS) Homological representation formula 2015 Jul 9 / 34
II: Quantum versus homological braid representations. Tetsuya Ito (RIMS) Homological representation formula 2015 Jul 10 / 34
Quantum sl 2 representation Quantum group U q (sl 2 ): K, K U q (sl 2 ) = 1 E, F KK 1 =K 1 K =1, KEK 1 = q 2 E, K K 1 q q 1 [E, F ] = KFK 1 = q 2 F By choosing a weight λ C {0}, we have a U q (sl 2 )-module V λ,q = span{v 0, v 1, v 2,...} with U q (sl 2 )-action Kv i = q λ q 2i v i Ev i = [λ + i 1] q v i 1 Fv i = [i + 1] q v i+1. (Here [a] q = qa q a q q 1 ) : quantum integer Tetsuya Ito (RIMS) Homological representation formula 2015 Jul 11 / 34
Generic quantum sl 2 -representation (after Jackson-Kerler) By using a universal R-matrix of U q (sl 2 ) we have φ Vλ,q : B n GL(V n λ,q ), σ i id (i 1) R id (n i 1). where R : V λ,q V λ,q V λ,q V λ,q is given by R(v i v j ) = q λ(i+j) i n=0 Here F i,j,n (q) = q 2(i n)(j+n) q n(n 1) 2 n 1 F i,j,n (q) (q λ k j q λ+k+j )v j+n v i n k=0 [ n + j j Definition We view λ, q as variables, and call φ Vλ,q generic quantum sl 2 -representation. ] q. Tetsuya Ito (RIMS) Homological representation formula 2015 Jul 12 / 34
Generic quantum sl 2 -representation (after Jackson-Kerler) Remark Generic quantum sl 2 -representation at λ Z does not equal to usual quantum sl 2 -representation representation (from finite dim module) Lemma-Definition (Jackson-Kerler 11) The braid group representation V n λ,q splits as V n λ,q = m V n,m = F k W n,m k = m=0 m=0 k=0 m=0 k=0 m W n,m k. where V n,m = span{v e1 v en V n λ,q e 1 + + e n = m}, and W n,m = Ker E V n,m. In particular, ( ) n + m 1 dim V n,m =, dim W n,m = m ( n + m 2 m ). Tetsuya Ito (RIMS) Homological representation formula 2015 Jul 13 / 34
1-parameter deformation Put q = ζ = exp( 2π 1 2N ) (N = 2, 3,...). Definition (λ-deformation of irreducible N-dim module) U N (λ) = span{v 0, v 1,..., v N 1 } V λ,q U N (λ) is an N-dimensional irreducible U ζ (sl 2 )-module and we have Proposition-Definition φ UN (λ) : B n GL(U n (λ) n ) The braid group representation U N (λ) n splits as m U N (λ) n = X n,m = F k Y n,m k = m=0 m=0 k=0 m=0 k=0 m Y n,m k. where X n,m = span{v e1 v en U N (λ) n e 1 + + e n = m}, and Y n,m = Ker E Y n,m. Tetsuya Ito (RIMS) Homological representation formula 2015 Jul 14 / 34
Homological representation of braid groups Regard the braid group B n as: B n = MCG(Dn ) = {Mapping class group of the n-punctured disc D n } = {f : D n Homeo D n f Dn = id}/{isotopy} σ i Half Dehn-twist swapping p i and p i+1. Ø ¼ Ð Ò¹ØÛ Ø Ø ½ Tetsuya Ito (RIMS) Homological representation formula 2015 Jul 15 / 34
Lawrence s construction To get more detailed information of the action B n D n, we use C n,m = (Unordered) Configuration space of m points of D n = {(z 1,..., z n ) Dn m z i z j if i j}/s m. Then for m > 1, H 1 (C n,m ) = Z n Z = meridian of {z 1 =puncture} meridian of i j{z i =z j } Let C n,m be the Z 2 -cover associated to Ker α : π 1 (C n,m ) Z 2, given by α : π 1 (C n,m ) Hurewicz H 1 (C n,m ; Z) = Z n Z C Z Z= x d Tetsuya Ito (RIMS) Homological representation formula 2015 Jul 16 / 34
Lawrence s construction H m ( C n,m ; Z) has rich structures: By regarding Z 2 = x, d as deck translations, H m ( C n,m ; Z) is a free Z[x ±1, d ±1 ]-module of rank ( ) n+m 2 m. The action of B n on D n gives rise to ( ) φ : B n GL(H m ( C n,m ; Z)) n + m 2 = GL( ; Z[x ±1, d ±1 ]) m In the case m = 1, C n,1 = D n, and the variable d does not appear. Then, φ : B n GL(H 1 ( D n ; Z)) = GL(n 1; Z[x ±1 ]) is nothing but the reduced Burau representation. Tetsuya Ito (RIMS) Homological representation formula 2015 Jul 17 / 34
Lawrence s representation Technical, but important point We will actually use a geometric variant: Action on a free Z[x ±1, d ±1 ] sub-module H n,m H lf m( C n,m ; Z)) (homology of locally finite chains), spanned by multifork. ¾ ÀÑ Ð ÒÑ µ ÅÙÐØÓÖ ÐØ Ó ÒÑ Ò ÒÑ Tetsuya Ito (RIMS) Homological representation formula 2015 Jul 18 / 34
Lawrence s representation (2) For e = (e 1,..., e n 1 ) Z n 1 0, e 1 + + e n 1 = m, we assign standard multifork F e. ½ Ò ½ µ ½ ¾ Ò ¾ Ò ½ Definition-Proposition H n,m Hm( lf C n,m ; Z)) is freely spanned by standard multiforks and we have a braid group representation (Lawrence s representation.) (( ) ) L n,m : B n GL(H n,m ) n + m 2 = GL ; Z[x ±1, d ±1 ] m Tetsuya Ito (RIMS) Homological representation formula 2015 Jul 19 / 34
Truncated Lawrence s representation We consider sepcial case where d is put as d = ζ 2 = exp( 2π 1 N ). Proposition-Definition (I.) Let us put d = ζ 2 = exp( 2π 1 N ). 1. The subspace H N n,m = span{f (e1,...,e n 1 ) e 1 + e n 1 = m, e i N for some i} is B n -invariant. 2. Put H n,m = H n,m /H N n,m. Then we have a linear representation l N n,m : B n GL(H n,m ) which we call truncated Lawrence s representation. Tetsuya Ito (RIMS) Homological representation formula 2015 Jul 20 / 34
Correspondence between homological and quantum representation: generic case Theorem A (Kohno 12, (c.f. I. 15)) For generic x, d, q, λ (i.e. when we regard them as variables), (quantum) W n,m q 2λ =x, q 2 =d = H n,m (topological) (Actually, we have a explicit formula of a basis of W n,m that corresponds to standard multifork basis.) Tetsuya Ito (RIMS) Homological representation formula 2015 Jul 21 / 34
Correspondence between homological and quantum representation: generic case Theorem A (Kohno 12, (c.f. I. 15)) For generic x, d, q, λ (i.e. when we regard them as variables), (quantum) W n,m q 2λ =x, q 2 =d = H n,m (topological) (Actually, we have a explicit formula of a basis of W n,m that corresponds to standard multifork basis.) Caution, again This is not sufficient to get homology representation formula of usual sl 2 invariants (Jones polynomial), since corresponding representation is not generic. Tetsuya Ito (RIMS) Homological representation formula 2015 Jul 21 / 34
Correspondence between homological and quantum representation: q non-generic case Theorem A (I.) Put d = ζ 2 = exp( 2π 1 N ) = q 2. Then for generic λ, x (i.e. when we continue to treat them as variables) ] Remark (quantum) Y n,m ζ 2λ =x = H n,m (topological) Informally, H n,m is seen as H m ( C n,m, ( Fat discriminants); Z): We regard a collision of more than N points = 0. Tetsuya Ito (RIMS) Homological representation formula 2015 Jul 22 / 34
Rough idea of proof Quantum representation Drinfel d-kohno Theorem Holomony representation via KZ connection (KZ equation) over configuration space Expression of solutions of KZ equation by integration over cycle (with respect to local system coefficient) local system homology = space of solutions of KZ equation Tetsuya Ito (RIMS) Homological representation formula 2015 Jul 23 / 34
III: Applications. homological representation formula Tetsuya Ito (RIMS) Homological representation formula 2015 Jul 24 / 34
Application I: Loop expansion JK α (q): α-colored Joens polynomial =(The quantum invariant from α-dim irreducible U q (sl 2 )-module) Proposition-Definition (Melvin-Morton 95) Let q = e ħ. By regarding α as variable, J α (K)(q) is expanded as a power series of ħ and z = q α = e ħα : CJ K (ħ, z) = JK α (eħ ) = V (k) K (z)ħk We call CJ K (ħ, z) loop expansion of quantum sl 2 invariants. Loop expansion is an expansion of lim α J α (K)(e ħ ), near ħ = 0 q = 1 Tetsuya Ito (RIMS) Homological representation formula 2015 Jul 25 / 34
Application I: Loop expansion Theorem (I.) Assume that K is a closure of an n-braid β. Then the loop expansion of a is given by CJ K (ħ, z) = z n q 1 n 2m z n q 1+n+2m z z 1 trace L n,m (β) x=qz 1,d= q. m=0 Proof is mostly a calculation, noticing: In the situation α, α-dim irreducible U q (sl 2 )-module (not generic!) approaches to generic module V α 1,q. To compute CJ K, we can use generic quantum representation = Lawrence s representation Tetsuya Ito (RIMS) Homological representation formula 2015 Jul 26 / 34
Application I: Loop expansion Corollary (Melvin-Morton-Rozansky conjecture, proven by Bar-Natan, Garoufalidis 96) V (0) K (z) = 1/ K (z) i.e., the 0th coefficient of the loop expansion is the Alexander polynomial. Proof: By putting ħ = 0 (i.e, q = 1), Lemma V (0) K (z) = zn z n z z 1 trace L n,m (β) x=z 1,d= 1 m=0 L n,m (β) x=z 1,d= 1 = Sym k L n,1 (β) x=z 1 = k-th Symmetric power of reduced Burau representation Tetsuya Ito (RIMS) Homological representation formula 2015 Jul 27 / 34
Application I: Loop expansion Thus V (0) K (z) = zn z n z z 1 trace Sym k L n,1 (β) x=z 1. m=0 By the formula (generalization of X i = 1 X 1 ), trace Sym k A = m=0 1 det(a I ) V (0) K (z) = zn z n 1 z z 1 det(l n,m (β) I ) 1 = K (z). = More topological explanation why Alexander polynomial appears. Tetsuya Ito (RIMS) Homological representation formula 2015 Jul 28 / 34
Application II: Colored Alexander invariant At q = ζ = exp( 2π 1 2N ) (2N-th root of unity), we have different kind of knot invariant other than colored Jones polynomials. Definition (J. Murakami s formulation 08) (N-)Colored Alexander invariant of a knot K Def = Quantum (Operator) invariant of (1, 1)-tangle from K and U N (λ). Í Æ µ à Ƶ¹ÓÐÓÖ ÐÜÒÖ ÒÚÖÒØ Í Æ µ Ã Æ µ Tetsuya Ito (RIMS) Homological representation formula 2015 Jul 29 / 34
Colored Alexander invariant The original definition of colored Alexander invariant (Akutsu-Deguchi-Ohtsuki 91) uses state-sum method, inspired from physics. The usual quantum trace of the quantum representation (usual definition of quantum invariant) φ UN (λ) : B n GL(U N (λ) n ) vanish. (This is a reason why we need to use quantum (1,1)-trace). In the case N = 2, Φ N K (λ) ζ 2λ =t = K (t) (Alexander polynomial) This justifies the name colored Alexander. Tetsuya Ito (RIMS) Homological representation formula 2015 Jul 30 / 34
Formula of colored Alexander invariant Theorem C (I.) Let K be a knot represented as a closure of an n-braid β. Then N 1 Φ N K (λ) = n 1 (λ, (n 1)λ) trace ln,i+nj N (β) x=ζ 2λ i=0 C N i where Ci N (λ, µ) is given by C N i (λ, µ) = j=0 [λ] ζ [λ N + 2] ζ, [λ] ζ = ζλ ζ λ [λ + µ 2i] ζ [λ + µ 2i N + 2] ζ ζ ζ 1. (Note that unlike loop expansion formula, now the summation is finite.) Tetsuya Ito (RIMS) Homological representation formula 2015 Jul 31 / 34
N = 2 case This formula gives an explanation that Φ N K Alexander polynomial. For N = 2, if we put x = ζ 2λ, is a generalization of the hence C0 2 (λ, (n 1)λ) = x 1 2 x 1 2 x n 2 x, C n 1 2 (λ, (n 1)λ) = x 1 2 x 1 2 2 x n 2 x n 2 Φ 2 K (λ) = x 1 2 x 1 2 x n 2 x n 2 n 1 trace ln,2j(β) 2 x 1 2 x 1 2 n 1 x n 2 x trace ln,2j+1(β) 2 n 2 j=0 j=0 Tetsuya Ito (RIMS) Homological representation formula 2015 Jul 32 / 34
N = 2 case Lemma l 2 n,k (β) = k l 2 n,1 (β) = k Ln,1 (β) = k (Reduced Burau representation) (cf. L n,m (β) d= 1 = Sym k (Reduced Burau representation) Therefore Φ 2 K (λ) = x 1 2 x 1 2 x n 2 x n 2 ( trace = x 1 2 x 1 2 x n 2 x det(i φ n Burau (β)) 2 = (Alexander polynomial) ) even odd Ln,1 (β) trace Ln,1 (β) Tetsuya Ito (RIMS) Homological representation formula 2015 Jul 33 / 34
Remarks on Kashaev s invariant (volume conjecture) Kashaev s invariant is obtained from the colored Alexander invariant Φ N K (λ) by putting λ = N 1. We have a more topology-flavored description of Kashaev s invariant. Our formula says Kashaev s invariant is a (weighted) sum of eigenvalues of truncated Lawrence s representation at λ = N 1. By definition these eigenvalues are equal to the eigenvalues of the action on the finite abelian covering of C n,m. Tetsuya Ito (RIMS) Homological representation formula 2015 Jul 34 / 34
Remarks on Kashaev s invariant (volume conjecture) Kashaev s invariant is obtained from the colored Alexander invariant Φ N K (λ) by putting λ = N 1. We have a more topology-flavored description of Kashaev s invariant. Our formula says Kashaev s invariant is a (weighted) sum of eigenvalues of truncated Lawrence s representation at λ = N 1. By definition these eigenvalues are equal to the eigenvalues of the action on the finite abelian covering of C n,m. Implication Question Kashaev s invariant is related to a finite covering of configuration space. Can we provide more direct formula? Tetsuya Ito (RIMS) Homological representation formula 2015 Jul 34 / 34