Knot polynomials, homological invariants & topological strings
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1 Knot polynomials, homological invariants & topological strings Ramadevi Pichai Department of Physics, IIT Bombay Talk at STRINGS 2015 Plan: (i) Knot polynomials from Chern-Simons gauge theory (ii) Our results on mutant knots (iii) Homological invariants (iv) Large N Chern-Simons & their closed topological string duals (v) Conclusions & challenging problems. 1
2 Knot polynomials from Chern-Simons theory Torus Knot :Trefoil 3 1 Hyperbolic Knot: Figure-Eight knot 4 1 Chern-Simons theory provides a natural framework for the study of knots
3 Chern-Simons action on S 3 based on gauge group G: S CS [A] = k (A 4π T r da + 23 ) S 3 A3 k is the coupling constant, A s are the gauge connections. Any knot K carrying representation R are described by expectation value of Wilson loop operators W R (K) = T r[p exp A]: VR G [K] = W R(K) = S 3 [DA] W R (K) exp(is CS [A]) Z[S 3 ] V G R where Z[S 3 ] = [K] are the knot invariants. S 3[DA] exp(is CS[A]) (partition function)
4 Knot invariant computations These knot invariants (VR G [K]) can be directly evaluated using two inputs:(kaul, Govindarajan,PR (1992)) 1) Relation between Chern-Simons theory to G k Wess-Zumino conformal field theory (Witten 1989). 2) Any knot can be obtained as a closure or platting of braid(alexander, Birman) For example, the trefoil can be redrawn as
5 V R [3 1 ] = ψ 0 ψ 3 = ψ 0 B 3 ψ 0 where B is the braiding operator. To write the polynomial form of the knot invariant: Expand the state ψ 0 in a suitable basis in which B is diagonal.
6 For the four-punctured S 2 boundary, the conformal block bases are: where t R 1 R 2 R 3 R 4 and s R 2 R 3 R 1 R 4. ] [ R1 R a 2 st is the duality matrix (fusion matrix) relating these two bases R 3 R 4 which is proportional to the quantum Wigner 6j symbols: [ ] R1 R a 2 ts R 3 R 4 { R1 R 2 } t R 3 R 4 s
7 For knots, two of the R i s will be R and the other two will be conjugate R depending on the orientation. In the braid diagram for trefoil, middle two strands are parallely oriented and they are braided. where µ s = Ψ 0 = s R R µ s ˆΦ s ( R, R, R, R) S 0s /S 00 dim q t (unknot normalisation) V R [3 1 ] = Ψ 0 B 3 Ψ 0 = s dim q s (λ s (R, R)) 3 where braiding eigenvalue for parallelly oriented right-handed half-twists is λ (+) t (R, R) = ( 1) ɛ t q 2C R C t /2, q = ek+cv 2πi where ɛ t = ±1
8 Antiparallel braiding eigenvalue will be λ ( ) s (R, R) = ( 1) ɛ sq Cs/2. We require them for figure-eight drawn as quasi-plat. opposite oriented boundary S 2 V R [4 1 ] = t,s R R [ ] R R dim q t dim q s a ts R R (λ ( ) t ) 2 (λ ( ) s ) 2
9 Knot invariants involves braiding eigenvalues & fusion matrices Fusion matrices proportional to quantum Wigner 6j (completely known for SU(2) (Kirillov, Reshetikhin) but not for other groups) For few R s, we determined using knot equivalence and properties of Wigner 6j-Kaul, Govindarajan,PR (1992), Zodinmawia,PR(2012) Hence the knot invariants VR G [K] can be written in variables dependent on k and rank of the group Well-known knot polynomials match with knot invariants (normalized by unknot) when R is fundamental representation
10 Gauge Group [n]-colored SU(2) Jones J[K;q] J n [K, q] SU(N) HOMFLY P [K; a = q N, q] P n [K; a, q] SO(N) Kauffman F [K; λ = q N 1, q] F n [K; λ, q] Jones polynomial for trefoil: J[T ; q] = V SU(2) [T ]/V SU(2) [U] = q + q 3 q 4. We can place any representation R (higher spins) of SU(2) on the knot and obtain colored Jones polynomials J n [K; q] where subscript n means spin (n 1)/2 or Young diagram single row with n 1 boxes. The colored Jones for 3 1 &4 1 are J n [3 1 ; q] = n 1 k=0 ( 1) k q k(k+3)/2+nk (q n 1, q) k (q n+1, q) k
11 where (z; q) k = k 1 j=0 (1 zqj ) is called q-pochhammer symbol. J n [4 1 ; q] = n 1 k=0 ( 1) k q nk (q n 1, q 1 ) k (q n+1, q) k These two colored Jones polynomial is sufficient to determine J n [K p ; q] for twist knots K p and with initial conditions p full twists Figure 1 p knots A K2 (l, m) = l 2 + l 3 + 2l 2 m 2 + lm 4 + 2l 2 m 4 lm 6 l 2 m 8 A K1 (l, m) = l + m 6, A K0 (l, m) = 1, +2lm 10 + l 2 m lm 12 + m 14 lm 14,
12 [n]-colored HOMFLY P n [K p ; a, q] (Nawata,Zodin, PR),2012 gave more data leading to conjecture a closed form expression for U q (sl N ) quantum Wigner 6j symbols for a class of R i s (Nawata,Zodin,PR),2013 There are two types of Wigner 6j for SU(N): Type I: n 1 n 2 k 1 n 1 n 2 + k 1 n 3 n 4 k 2 n 3 n 2 + k 2 where n 2 n 1 n 3, k 1 n 2 and k 2 n 1.
13 Type II n 1 n 2 k 1 n 1 + n 2 2k 1 n 3 n 4 k 2 n 2 n 3 + k 2 where n 1 n 2, k 2 min(n 1, n 3 ) and k 1 min(n 1, n 3, n 4 ). fusion rule requires n 1 + n 2 = n 3 + n 4. The
14 { λ1 λ 2 λ 12 λ 3 λ 4 λ 23 } = (1,2,12) (3,4,12) (1,4,23) (2,3,23) [N 1]! ( ) z z 0 [z + N 1]!C z {[z 1 2 λ 1 + λ 2 + λ 12, α 1 + α N 1 [z ]! 12 ] λ 3 + λ 4 + λ 12, α 1 + α N 1! [z 12 ] λ 1 + λ 4 + λ 23, α 1 + α N 1! [z 12 ] λ 2 + λ 3 + λ 23, α 1 + α N 1! [ 1 2 λ 1 + λ 2 + λ 3 + λ 4, α 1 + α N 1 z [ 1 2 λ 1 + λ 3 + λ 12 + λ 23, α 1 + α N 1 z! [ ] 1 2 λ 2 + λ 4 + λ 12 + λ 23, α 1 + α N 1 z!} 1, ]! ]
15 where (1,2,3) (λ 1, λ 2, λ 3 ) = { [ [ 1 2 λ 1 + λ 2 + λ 3, α 1 + α N 1 [ 1 2 λ 1 λ 2 + λ 3, α 1 + α N 1 [ 1 2 λ 1 + λ 2 λ 3, α 1 + α N 1! {[ 1 2 λ 1 + λ 2 + λ 3, α 1 + α N 1 + N 1 ] ]! ]! } 1/2 ]! }
16 [ N 2 + k2 i ] 1 for k 1 > k 2, C (I) z = δ z,zmin +i k 2 i [ N 2 + k1 i ] 1 for k 1 k 2,, δ z,zmin +i k 1 i [ N 2 + k2 i ] 1 for k 1 > k 2, C (II) z = δ z,zmax i k 2 i [ N 2 + k1 i ] 1 for k 1 k 2, δ z,zmax i k 1 i
17 OBSERVATIONS [n]-colored HOMFLY for knots drawn as quasi-plat of 4-strand braids can be obtained using this data. Our data is not sufficient to write the polynomial for knots obtained from quasi-plat of braids with more than 4-strands. (dual basis of 6-point or higher point conformal blocks requires Wigner 6-j beyond our conjectured class)
18 How do we obtain [n]-colored HOMFLY for 9 42 knot involving six braids? 9_42 knot
19 Knots 10 71, can be drawn gluing 3-manifolds involving three S 2 boundaries each with four-punctures: I IV II III
20 Requires the following building blocks to compute knot polynomials u r = t(dim q t) (1 r/2) φ (1) t... φ t (r)
21
22 We have redrawn many knots using these building blocks enabling evaluation of [n]-colored HOMFLY polynomials. These polynomials do not distinguish mutants.
23 Kinoshita-Terasaka & Conway mutants
24 Need to work out polynomials for mixed representations. The two types of U q (sl N ) Wigner 6j has been recently determined for [2, 1] (Gu,Jockers),2014-first mixed representation Wanted to check the power [2,1] colored HOMFLY for the mutant pair The two knots (mutant pairs) are indeed distinguished (Nawata,Singh,PR,2015) Enumerated class of mutants which can be distinguished but some pretzel mutants with antiparallel odd-braids cannot be distinguished (Mironov, Morozov,Morozov, Singh, PR),2015
25 Crucial input in the context of mixed representation: multiplicity (21; 0) (21; 0) = (42; 0) 0 (2 3 ; 0) 0 (31 3 ; 0) 0 (321; 0) 0 (321; 0) 1 (41 2 ; 0) 0 (3 2 ; 0) 0 ( ; 0) 0 We see appears twice. Multiplicity incorporated four-point conformal blocks: = φ (1) t,r 3 r 4 (R 1,..., R 4 ), = φ (2) s,r 1 r 2 (R 1,.., R 4 )
26 Table gives the three-manifolds obtained from surgery of framed knots Frame link knot 3-Manifold S 2 X S 1 S 3 RP 3 P L(P,1) a 1 =2 a 2 =3 a p 1 = a 2 1 q a 3 L(5,3) P 3
27 Framed links related by Kirby moves gives the same three-manifold Three-manifold invariant proportional to Chern-Simons partition function Z[M] respecting Kirby moves is(kaul,pr),2000 Z[M] R dim q R V SU(N) R [K]
28 Questions large N expansion logz[m]? For any knot, we observe Laurent series: P [K; a, q] = i,j c i,j a i q j where c i,j are integers. Topological meaning or reason? Why integers?- two parallel developments From physics (topological strings,bps states counting) Ooguri,Vafa (1999) From mathematics(homological chain complex) Khovanov(1999), Khovanov-Rosansky(2004) c i,j = k ( 1) k dimh i,j,k
29 Homological Invariants Introduce A and B slicing as shown in the diagram. Define n(s) = n B, j(s) = n B + n + n
30 C nj is the vector space with basis as states with n(s) = n and j(s) = j. J[K; q] = n,j( 1) n q j dim(c nj ) where the homology chain : C n,j C n+1,j, 2 = 0
31
32 C j : C 0j C 1j C 2j... The vector space H n (C j ) = ker( : C n,j C n+1, j) Image( : C n 1,j C n,j ) Kh(K; q, t) = n,j t n q j dim(h nj ). Taking t = 1 gives the Jones polynomial J[K; q]
33 superpolynomial Fundamnetal Representation colored superpolynomial Higher Rank Representation Khovanov-Rozansky Khovanov colored Khovanov-Rozansky colored Khovanov Refinement Refinement Refinement Refinement HOMFLY Jones colored HOMFLY colored Jones SU(N) SU(2) SU(N) SU(2) integer homology of bigraded vector space
34 Large N Chern-Simons &topological strings Gopakumar-Vafa duality gives A-model closed topological string on a resolved conifold from large N expansion of logz[s 3 ]. Ooguri-Vafa conjecture a form for reformulated knot invariants f R (q, λ). We verified Ooguri-Vafa conjecture for non-torus knots (Sarkar, PR 2000) and higher crossing knots(nawata,zodin,pr 2013) using our [n]-colored HOMFLY polynomials. Making use of these two duality conjectures, we attempted the N expansion of logz[m] for some three-manifolds M giving(p. Borhade, T. Sarkar,PR(2003)
35 logz 0 [M] = n=1 { Q g 1 n ( {l α } {s α } sinh dg s 2 ) 2g 2 ˆN (Rl1,s 1,...R l r,sr ),g,q ( 1) α s α ( 1) n α l αp α λ 1 2 n α l αp α ( λ n{q+ α ( l α 2 +s α )} Subtle Issues In the large k limit = λ n{q+1+ α ( l α 2 +s α )} ) } g,n,m 1 n (2 sinh ng s 2 )2g 2 n g,m e dmt Z[M] = c Z c [M]
36 t Hooft proposal requires whereas we find ln ( c ln Z c [M] = Closed String expansion Z c [M] ) = Closed String partition function Hence we cannot predict duality between Chern-Simons gauge theory on M with the A-model string theory with the n g,m s we have determined
37 Generalisation of the duality to SO gauge groups A-model closed strings on an orientifold of the resolved conifold is dual to SO/Sp Chern-Simons theory (Sinha and Vafa) lnz (SO) (CS) [S3 ] = 1 2 Z(or) + Z (unor) Incorporating Wilson loop observables ln Z({U α }, {V α }) = F G (V ) = 1 2 F(or) R (V ) + F(unor) (V ) Not clear how to seperate, we showed LHS (Pravina Borhade and PR) Z({U α }, {V α }) = exp n=1 {R α } g R1,R 2,...R r (q n, λ n ) 1 n r α=1 T r Rα V n α
38 where g R1,...R r (q, λ) = 1 Q,s (q 1/2 q 1/2 ) N (R 1,R 2...R r ),Q,s qs λ Q N (R1,...R r ),Q,s are integers-how to find oriented contribution? Composite representation invariants to extract oriented contribution (Marino) F (or) R (V ) = W (R,S) [C] (T r R V )(T r S V ) = R,S t R t [C]T r t V = d=1 R h R (q d, λ d )T r R V d where R and S are representation on the knot C in two Lagrangian submanifolds N ɛ and N ɛ related by orientifolding action. The composite representation (R, S) has highest weight Λ R + Λ S. Obtained invariants of torus knots & links for some composite representations and verified the form of h R (q, λ)(2010, C. Paul, P. Borhade,PR)
39 Using [r]-colored Kauffman polynomials for figure-eight knot from the study of structural properties of colored Kauffman homologies of knots (Nawata,Zodin,PR,2013) and Morton s result for (, ) composite invariant for figure-eight, we verified the integrality structure for h and g.
40 Summary Elegant computation of [n]-colored HOMFLY for many knots from Chern- Simons approach [2,1]-colored HOMFLY distinguishes many mutants including Kinoshita- Terasaka & Conway mutant pair. These invariants are definitely useful to verify integrality structures predicted by U(N) and SO topological string duality conjectures
41 Challenging problems (i) Obtaining closed form expression for SU(N) q fusion matrices for general R s which could be rectangular or mixed representation. (ii) Can we get a better understanding of polynomial variable t in homological invariants? (iii) The results on refined CS for [2, 2p + 1]-torus knots suggests that there could be a combinatorial definition of colored Khovanov invariants for non-torus knots. So far no success!
42 (iv) Any idea on getting # of solutions to elliptic Nahm equations accounting for integer coefficients in Khovanov invariant for trefoil? Gaitto,Witten (2011) (v) Appears Kevin Costello work may be applicable to the determine spectral parameter dependent R-matrix for N-vertex models using 4-d twisted supersymmetric field theory. Due to time constraint, I have not discussed volume conjecture, A-polynomials and their relations to (i) SL(2, C) 3-d gravity(gukov,2003) and (ii) their appearance in mirror Calabi-Yau geometry (Aganagic,Vafa(2012)-interesting developments.
43 Thank You
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