A topological description of colored Alexander invariant Tetsuya Ito (RIMS) 2015 March 26 Low dimensional topology and number theory VII Tetsuya Ito (RIMS) Colored Alexnader invariant 2015 March 1 / 27
Alexander polynomial K S 3 : Oriented knot The Alexander polynomial is one the most fundamental invariant of knots, admitting various interpretations: Seifert matrix Infinite cyclic covering Skein relation Kauffman s state-sum Alexander polynomial K (t) U q (sl 2 ) or, U q (sl(1 1)) Tetsuya Ito (RIMS) Colored Alexnader invariant 2015 March 2 / 27
Generalizations Expoloiting various interpretation of Alexander polynomial, a lot of generalization of K (t) are known: Alexander modules Alexander polynomial K (t) Heegaard Floer homology Twisted Alexander polynomial HOMFLY, Kauffman polynomial Colored Alexander invariant Tetsuya Ito (RIMS) Colored Alexnader invariant 2015 March 3 / 27
Arithmetic topology and Alexander polynomial The philosophy of Arithmetic topology Topology of knot complement S 3 K ( Knot K S 3 ) Analogy Spec(F p ) Spec(Z) { } (p : prime number) In this point of view, one uses (Abelian) Covering Correspondence Galois group Alexander polynomial Analogy Iwasawa polynomial So more topological (covering) interpretation of generalization of Alexander polynomial would be fruitful. Tetsuya Ito (RIMS) Colored Alexnader invariant 2015 March 4 / 27
Main questions Motivating quetsion What is the topological content of quantum invariants? Witten s solution, (Chern-Simons functional (path integral) interpretation) is not easy to use for studying classical topology (naive problems) for example, consider a question like Does the Jones polynomial gives an estimate of knot genus? In today s talk we address the problem Main Problem Are colored Alexander polynomial (first defined by Akutsu-Deguchi-Ohtsuki in 91 by using state-sum) really generalizations of Alexander polynomial (in a topological point of view)? Tetsuya Ito (RIMS) Colored Alexnader invariant 2015 March 5 / 27
Summary of result @@ 1-parameter central deformation of U q (sl 2 ) representation of B n at 2N-th root of unity Key Theorem Homological representation of B n Akutsu-Deguchi-Ohtsuki J.Murakami Colored Alexander invariant This leads to: Main Theorem: More topological formula New (more topological) formula of Kasahev s invariant More similarity between Alexander polynomial and colored Alexander invariants: Our formula generalizes the formula of Alexander polynomial from reduced Burau representation). Tetsuya Ito (RIMS) Colored Alexnader invariant 2015 March 6 / 27
Quantum 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) Colored Alexnader invariant 2015 March 7 / 27
Generic quantum sl 2 -representation (after Jackson-Kerler) By using a universal R-matrix of U q (sl 2 ) we have a braid group representation 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 n 1 F i,j,n (q) (q λ k j q λ+k+j )v j+n v i n k=0 [ ] Here F i,j,n (q) = q 2(i n)(j+n) q n(n 1) n + j 2. j q We call ϕ Vλ,q generic quantum sl 2 -representation. Tetsuya Ito (RIMS) Colored Alexnader invariant 2015 March 8 / 27
Colored Alexander invariant Put q = ζ = exp( 2π 1 2N ) (N = 2, 3,...). Then U N (λ) = span{v 0, v 1,..., v N 1 } V λ,q is an N-dimensional irreducible U ζ (sl 2 )-module (parametrized by λ). 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). U N (λ) K β (N)-Colored Alexander invariant β U N (λ) Φ K N (λ) id Tetsuya Ito (RIMS) Colored Alexnader invariant 2015 March 9 / 27
Colored Alexander invariant Remark 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-Conway polynomial). Strictly speaking, we need to consider framing corrections (throughout this talk, we always assume that every knot is represented so that the blackboard framing is equal to the zero framing). Tetsuya Ito (RIMS) Colored Alexnader invariant 2015 March 10 / 27
Quantum sl 2 -representation and knot invariants V λ,q (generic module) q=ζ=exp( π 1 N ) :non-generic λ-deformation of N- dim irreducible U ζ (sl 2 ) module λ= N 1 2 : non-generic N-dim irreducible U q (sl 2 ) module (Hyperbolic Volume) quantum trace Colored Jones polynomial V N K (q) q=exp( π 1 N ) quantum(1,1) trace (Volume conjecture) N 1 Colored Alexander invariant Φ N 2 λ= :non-generic K (λ) Kashaev s invariant Tetsuya Ito (RIMS) Colored Alexnader invariant 2015 March 11 / 27
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) Colored Alexnader invariant 2015 March 12 / 27
Homological representation of braid groups (2) 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) Colored Alexnader invariant 2015 March 13 / 27
Homological representation (3) H m ( C n,m ; Z) has rich structures: By regarding Z 2 = x, d as deck translation, 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 induces ( ( ) ) ϕ : B n GL H m ( C n,m ; Z)) n + m 2 = GL( ; Z[x ±1, d ±1 ] m (Homological representation) 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) Tetsuya Ito (RIMS) Colored Alexnader invariant 2015 March 14 / 27
Lawrence s representation We express homology class of H lf m( C n,m ; Z)) geometrically by using multifork (introduced by Krammar and Bigelow) [F] H lf m ( C n,m ; Z) Multifork F lift of edge C n,m D n C n,m Tetsuya Ito (RIMS) Colored Alexnader invariant 2015 March 15 / 27
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 The subspace H n,m Hm( lf C n,m ; Z)) spanned by standard multiforks provides 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) Colored Alexnader invariant 2015 March 16 / 27
Lawrence s representation (3) Remark For generic x and d (over the quotient filed Q of Z[x ±1, d ±1 ])), H m ( C n,m ; Z), H lf m( C n,m ; Z) and H n,m are all isomorphic as the braid group representation (Kohno 12). At non-generic x and d they are different (Paoluzzi-Paris 02) Conjecture (Bigelow) H n,m = H m ( C n,m, ν; Z) for some ν C n,m. Equivalently, H n,m will be regarded as the homology of a certain compactification of the configuration space C n,m. (c.f. configuration space integral construction of finite type invariants) Tetsuya Ito (RIMS) Colored Alexnader invariant 2015 March 17 / 27
Lawrence s representation (4) L n,m are known to be nice representations of B n. Remark L n,1 = Reduced Burau representation. L n,2 = Lawrence-Krammer-Bigelow representation. L n,m d= 1 = m-th symmetric power of L n,1 L n,m are d -deformation of symmetric powers of the reduced Burau representation. Theorem (I.-Wiest (Geom. Topol. to appear), I.( 15)) For m > 1, L n,m detects the dual Garside normal form of braids (behaves nicely with respect to algebraic/combinatorial structure of braids). In particular, it is faithful. Tetsuya Ito (RIMS) Colored Alexnader invariant 2015 March 18 / 27
Correspondence between homological and quantum representation: generic case Lemma-Definition (Jackson-Kerler 11) An infinite dimensional quantum representation V n λ,q V n λ,q = V n,m = m=0 m=0 k=0 m F k W n,m k = split as 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 n + m 2 dim V n,m =, dim W n,m =. m m Tetsuya Ito (RIMS) Colored Alexnader invariant 2015 March 19 / 27
Correspondence between homological and quantum representation: generic case Theorem (Kohno 12, (c.f. I. 15)) For generic x, d, q, λ, there is an isomorphism of the braid group representation As an application: Theorem (I.) (quantum) W n,m q 2λ =x, q 2 =d = H n,m (topological) The loop expansion of the colored Jones polynomial (expansion of U q (sl 2 )-invariant by variable z = q λ 1 near = 0 (q = e ) is given by CJ β(, 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 Tetsuya Ito (RIMS) Colored Alexnader invariant 2015 March 20 / 27
Correspondence between homological and quantum representation: non-generic q(d) case We consider the non-generic case q = ζ = exp( π 1 N ). However, we assume that λ is generic with respect to ζ, in the sense [λ + n] ζ 0 for all Z. Proposition-Definition A quantum representation U N (λ) n split as U N (λ) n = X n,m = m=0 m=0 k=0 m F k Y n,m k = 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) Colored Alexnader invariant 2015 March 21 / 27
Correspondence between homological and quantum representation: non-generic q(d) case 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) Colored Alexnader invariant 2015 March 22 / 27
Correspondence between homological and quantum representation: non-generic q(d) case Key Theorem (I.) At d = ζ 2 = exp( 2π 1 N ) = q 2, there is an isomorphism of the braid group representation 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) Colored Alexnader invariant 2015 March 23 / 27
Formula of colored Alexander invariant Using the Key theorem, we express the colored Alexander invariants by using truncated Lawrence s representation: Main Theorem (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 j=0 N 1 Ci N (λ, µ) = ζ 2j ζ jλ µ j 1 k=0 [(µ 2i) k] ζ j 1 k=0 [(λ + µ 2i) k] ζ j=0 Thus, Φ N K (λ) is a linear combination of the traces of l N n,i (β). Tetsuya Ito (RIMS) Colored Alexnader invariant 2015 March 24 / 27
N = 2 case Example: N = 2 case 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 x n 2 x n 2 j=0 n 1 trace ln,2j+1(β) 2 j=0 Moreover, l 2 n,k (β) = k l 2 n,1 (β) = Therefore k Ln,1 (β) = k (Reduced Burau representation) Φ 2 K (λ) = x 1 2 x 1 2 x n 2 x det(i ϕ n Burau (β)) (ζ 2λ = x) 2 = (Alexander polynomial) Tetsuya Ito (RIMS) Colored Alexnader invariant 2015 March 25 / 27
Concluding remarks on Kashaev s invariant Kashaev s invariant is obtained from the colored Alexander invariant Φ N N 1 K (λ) by putting λ = 2. Kashaev s invariant is a (weighted) sum of traces of truncated Lawrence s representation at λ = N 1 2. Kashaev s invariant is a (weighted) sum of eigenvalues of truncated Lawrence s representation at λ = N 1 2. The eigenvalues of truncated Lawrence s representation at λ = N 1 2 are (essentially) 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) Colored Alexnader invariant 2015 March 26 / 27
Summary Again 1-parameter central deformation of U q (sl 2 ) representation of B n at 2N-th root of unity Key Theorem Truncated Lawrence s representation deformation Akutsu-Deguchi-Ohtsuki J.Murakami Colored Alexander invariant Main Theorem generalization ( -of) Reduced Burau representation classical Alexander polynomial Future and further question: Finite covering of configuration space and Kasahev s invariant Structure of truncated Lawrence s representation Interpretation from ambient space S 3 K (relation to Seifert surface) Tetsuya Ito (RIMS) Colored Alexnader invariant 2015 March 27 / 27