On asymptotic behavior of the colored superpolynomials

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1 .... On asymptotic behavior of the colored superpolynomials Hiroyuki Fuji Nagoya University Collaboration with S. Gukov (Caltech& MPI-Bonn) and P. Su lkowski (Amsterdam& Caltech) 20th String-Math 2012 Papers: S.Gukov, H.F. and P.Sulkowski, arxiv: , [hep-th]. R.H.Dijkgraaf and H.F., Fortsch.Phys.57(2009) , arxiv: [hep-th] R.H.Dijkgraaf, H.F. and M.Manabe, Nucl.Phys.B849(2011)

Alexander polynomial, Jones polynomial, HOMFLY

2 . 1. Introduction The topological classification of knot diagrams Knot invariants A-polynomial, Alexander polynomial, Jones polynomial, HOMFLY polynomial, Kauffman polynomial, Superpolynomial,...

3 A-polynomials and quantum knot invariants Today s focus: A-polynomial A K (x, y) Via volume conjecture : [Kashaev][Murakami 2 ][Gukov] A-polynomial Colored Jones polynomial A K (x, y) J n (K; q) Refined generalization [F-Gukov-Su lkowski-awata] Refined A-polynomial Colored Khovanov invariant A ref K (x, y; t) Kh n(k; q, t) = i,j qi t j dimh (n) ij (K) sl N generalization [Aganagic-Vafa] Q-deformed A-polynomial Colored HOMFLY polynomial A Q def K (x, y; a) P n (K; a, q) (a = q N q 2 ) Homological genrelization [F.-Gukov-Su lkowski] Super-A-polynomial Colored superpolynomial A super K (x, y; a, t) P n (K; a, q, t)

4 . 2. A-polynomial A-polynomial has various aspects:..1 A-polynomial as the SL(2; C) flat connection..2 A-polynomial as the hyperbolic geometry..3 A-polynomial as the the colored Jone polynomial..4 A-polynomial as the q-difference equation..5 A-polynomial as the spectral curve In the following, these aspects will be overviewed.

5 (1) A-polynomial and flat connection [Cooper-Culler-Gillet-Long-Shallen] Defining equation of the moduli space of SL(2; C) flat connection M SL(2;C) flat on the knot complement (S 3 \K): M SL(2;C) flat (S 3 \K) = { } (x, y) C C A K (x, y) = 0. x, y: SL(2; C) holonomy eigenvalues M SL(2;C) (T 2 ) 3 S Lagrangian submfd. M SL(2;C) (S 3 \K) A 41 (x, y) = (y 1)(y + y 1 + (x 2 x 2 x 1 + x 2 ), A 31 (x, y) = (y 1)(y + x 3 ).

6 (2) A-polynomial and deformation [Neumann-Zagier][Hodgeson][Kirk-Klassen] Abel-Jacobi map Neumann-Zagier s potential function ϕ K (x) : ϕ K (x) = x 1 dx log y(x). x y(x): A solution of the algebraic equation A K (x, y) = 0 Hyperbolic Dehn filling 1-parameter x deformation of the hyperbolic structure: (p log x+q log y = 2πi) Vol(M x ) + ics(m x ) = Vol(S 3 \K) + ics(s 3 \K) + i log x log y 2 +ϕ K (x) π 2 [Length(γ) x + itorsion(γ) x ]

7 (3) A-polynomial and volume conjecture [Kashaev][Murakami 2 ][Gukov] Colored Jones polynomial: J n (K; q) n: dimension of the highest weight representation of su(2) alg. log J n (K; q = e 2πi/k ) 2π lim = S 0 (K; x) n,k k S 0 (K, x) = Vol(S 3 \K) + ics(s 3 \K) + ϕ K (x). Double scaling limit: x := e 2πin/k : fixed 3 S A-polynomial A K (x, y) describes the exponential growth behavior of the colored Jones polynomial in the asymptotic limit n, k x S 0(K; x) x = log y(x) A K (x, y)

8 (4) A-polynomial and quantum volume conjecture [Gukov][Garoufalidis-Le] Â K (ˆx, ŷ; q)j n (K; q) = 0, ŷf(n) = f(n + 1), ˆxf(n) = q n f(n). Â K (x, y; q = 1) A K (x, y). Quantization of the Chern-Simons gauge field: [Â a α (z), 4πi Âb β (w)] = k δab ϵ αβ δ 2 (z w), ŷˆx = qˆxŷ. Phase space M ph = {(q, p)} M flat Hamiltonian H A K (x, y) Quantum Hamiltonian Ĥ = Â ĤΨ = 0 : Schrödinger eq. ÂJ = 0

9 (5) A-polynomial and topological recursion [Dijkgraaf-F][Dijkgraaf-F-Manabe][Dimofte-Gukov-Lennells-Zagier][Gukov-Su lkowski][borot-eynard] WKB expansion k 1 0 of the colored Jones polynomial: [ 1 J n (K; q) = exp S 0(x) + δ ] log + l S l+1 (x), q := e 2 l=0 S l can be found iteratively via q-difference equation ÂJ = 0. Proposal in [Dijkgraaf-F.-Manabe] Eynard-Oranti s topological recursion on M flat (S 3 \K): 2g+h=l 1 h! F g,h(x,, x) = S l (x) [Borot-Eynard] showed that both sides agrees by including non-perturbative corrections in the topological recursion! x x x x h x x1 xh x 1 x xk x 1 i h 1 i j k j q q g = + Σ q g 1 l J g q l l

10 Generalization of A-polynomials [F-Gukov-Su lkowski][aganagic-vafa] Aspects of (3) and (4): A-polynomial is found from the asymptotic behavior of the quantum knot invariants Generalization of the A-polynomial can be discussed: Superpolynomial P n(k;a,q,t) Khovanov Kh n(k;q,t) HOMFLY P n(k;a,q) Jones J n(k;q)

11 Super-quantum volume-conjecture [F.-Gukov-Su lkowski] For any knots K, the following super-a-polynomials always exist and coincide : The saddle point value of the colored superpolynomial P n (K; a, q, t) (q = e 1, n = log x): P n (K; a, q, t) e 1 S0(K;x,a,t), y(x) = e x S0/ x, A super K (x, y; a, t) = 0. There exists a q-difference equation which annihilates the colored superpolynomial such as: Â super K (ˆx, ŷ; a, t, q)p n (K; a, q, t) = 0, ŷˆx = qˆxŷ, (x, y; a, t, q = 1) = A super K (x, y; a, t). Â super K

12 . 3. Superpolynomial The superpolynomial P(K; a, q, t) is the Poincaré polynomial for the triply-graded homology H ijk (K) : [Dunfield-Gukov-Rasmussen] P(K; a, q, t) = i,j,k a i q j t k dimh ijk (K). The triply-graded homology is specified with differentials d N : d N : H ijk (K) H i j k (K), d Nd M = d M d N. H i,j,k (K) d 0 N d 2 sln H i,j (K) Floer H i,j (K) sl2 H i,j (K) H ijk reduces to sl N homology by taking homology w.r.t. d N d (H, d N ) H sl N

13 Superpolynomial for trefoil [Dunfield-Gukov-Rasmussen][Gukov-Stosic] Example: Trefoil knot HOMFLY polynomial P and reduced (normalized) Khovanov invariant Kh : P (3 1 ; a, q) = aq 1 + aq a 2, Kh (3 1 ; q, t) = q + q 3 t 2 + q 4 t 3. a a q t a q t a q t a q t (H, d ±1 ) sl 1 homology: 1-dim. The extrapolating polynomial P(3 1 ; a, q, t) P (3 1 ; a, q, t) = aq 1 + aqt 2 + a 2 t 3. q

14 Colored superpolynomials [Gukov-Stosic] The triply-graded homology with symm. rep. R = S n 1 : (K) The colored superpolynomial P R=Sn 1 (K; a, q, t): H Sn 1 ijk P n (K; a, q, t) := P Sn 1 (K; a, q, t) = i,j,k a i q j t k dimh Sn 1 ijk (K). New differential: n-1 R=S : Symmetric rep. n-1 boxes in 1-row Colored differentials d Sn 1 r m, dsn 1 1 k r boxes m boxes Taking homology homology in smaller dimension m < r: ( ) H Sr, d Sn 1 r m H Sm.

Colored superpolynomials for symmetric representation Sn 1 [F.-Gukov-Sulkowski][Itoyama et.al.] Strategy Determine Pn from consistency of axioms of d s.

15 Colored superpolynomials for symmetric representation Sn 1 [F.-Gukov-Sulkowski][Itoyama et.al.] Strategy Determine Pn from consistency of axioms of d s. Pn (41 ; a, q, t) = n 1 ( atq 1, q)k k=0 (q, q)k ( 1)k a k t 2k q k(k 3)/2 (q1 n, q)k ( at3 qn 1, q)k. Pn (31 ; a, q, t) = n 1 an 1 t2k qn(k 1)+1 k=0 (x; q)n := n i=1 (1 (qn 1, q 1 )k ( atq 1, q)k (q, q)k. xqi 1 ) : q-pochhammer symbol

16 Refined topological string theory [Gukov-Schwarz-Vafa][Gukov-Iqbal-Kozcaz-Vafa][Aganagic-Shakirov] Physical computation of the superpolynomial Refined topological BPS counting. Z Z (T M,L ;q) open * BPS K = (T M,L ;q) open * top K Z (T M,L ;q,q ) open * BPS K 1 Z CS (M,K ;q) Z ref (M,K ;q,q ) 1 SU(N) Z open BPS (X; q 1, q 2 ) := Tr H open BPS ( 1)F q s 1 1 q s 2 2 e th Hol R (K) = P R (K; a, q, t) = i,j,k SU(N) a i q j t k dimh R ijk (K). 2 2 (A = q N 2, q 1, q 2 ) = ( at 3, q, qt 2 ).

17 Refined computation [Dunin-Barkowski-Mironov-Morozov-Sleptsov-Smirnov][F-Gukov-Su lkowsky] (2, 2p + 1)-torus knots T 2,2p+1 ψ 0 (R 1,R 2 ) B B B Σ R1 R1 R2 R2 ψ 0 (R 1,R 2 ) = P n (T 2,2p+1 ; a, q, t) (n = r + 1) r (qt 2 ; q) l ( at 3 ; q) r+l ( aq 1 t; q) r l (q; q) r (1 q 2l+1 t 2 ) (q; q) l=0 l (q 2 t 2 ; q) r+l (q; q) r l ( at 3 ; q) r (1 qt 2 ) [ ] ( 1) r a r 3r 2 q 2 l t rp l+ r 2 ( 1) l a r r 2 2p+1 l(l+1) 2 q 2 t 3r 2 l. This result passes tests by d s!.

18 . 5. Super-Quantum Volume Conjecture Super-A-polynomial Colored superpolynomial P n (K; a, q, t) Â super K (x, y; a, t) Physical motivations Refined CS does not have classical Lagrangian What kind of deformations for flat connection F(A) = 0? Does refined CS admit the quantum structure? (c.f. quantum str. in refined top. string [Aganagic-Cheng-Dijkgraaf-Krefl-Vafa] ) Quantum integrable structure of the superpolynomial Generalization of the B-model approach Study of the super-a-polynomial will be the first step toward the deeper understanding of the refined / homological theory! Super-quantum volume conjecture Via computer talk ( qzeil.m ) q-difference equations can be found explicitly.

19 q-difference equations for the colored Superpolynomials -Fig.8-3rd order q-difference equation for P n (4 1 ; a, q, t): Â super 4 1 (ˆx, ŷ; a, q, t) = a 0 + a 1 ŷ + a 2 ŷ 2 + a 3 ŷ 3, a 0 = at3 (1 ˆx)(1 qˆx)(1 + at 3 q 2ˆx 2 )(1 + at 3 q 3ˆx 2 ) q 3 (1 + at 3ˆx)(1 + at 3ˆx 2 )(1 + at 3 qˆx)(1 + at 3 q 1ˆx 2 ) (1 qˆx)(1 + at 3 q 3ˆx 2 ) a 1 = tq 3ˆx 2 (1 + at 3ˆx)(1 + at 3 qˆx)(1 + at 3 q 1ˆx 2 ) ( 1 t(t 1)qˆx + at 3 q 1 (1 + q 3 + qt + q 2 t)ˆx 2 at 4 (q + q 2 + t + q 3 t)ˆx 3 a 2 (t 1)t 6 qˆx 4 a 2 t 8 5) q 2ˆx (1 + at 3 q 2ˆx 2 ) a 2 = at 2 q 2ˆx 2 (1 + at 3ˆx 2 )(1 + at 3 qˆx) ( 1 at(t 1)ˆx + at 2 (q + q 2 + t + q 3 t)ˆx 2 a 3 = 1. +a 2 t 4 (1 + q 3 + qt + q 2 t)ˆx 3 + a 2 (t 1)t 5 q 3ˆx 4 + a 3 t 7 5) q 3ˆx

20 Super-A-polynomial -Fig.8- A super 4 1 (x, y; a, t) = a 2 t 5 (x 1) 2 x 2 +at(x 1)(1 + t(1 t)x + 2at 3 (t + 1)x 2 2at 4 (t + 1)x 3 +a 2 t 6 (1 t)x 4 a 2 t 8 x 5 )y (1 + at 3 x)(1 + at(1 t)x + 2at 2 (t + 1)x 2 +2a 2 t 4 (t + 1)x 3 + a 2 t 5 (t 1)x 4 + a 3 t 7 x 5 )y 2 +at 2 x 2 (1 + at 3 x) 2 y 3. This coincides with the saddle point analysis S 0 (4 1 ; x, a, t)!

21 Hierarchy of A-polynomials super A (x,y;a,t) a=1 t= 1 ref A (x,y;t) Q def A (x,y;a) t= 1 a=1 A(x,y) A-polynomials [Gukov][Aganagic-Vafa][F.-Gukov-Sulkowski] Jones polynomial (SU(2)): A-polynomial HOMFLY polynomial (SU(N)): Augmentation polynomial Superpolynomial (Categorification): Super-A-polynomial

x = 1 specialization: [Conjecture] A super K (x = 1, y; a, t) = y k + y k 1 P (K; a,

22 A property of the super-a-polynomial Super-A-polynomial has 4-parameters (x, y; a, t) various specializations will show the interesting properties. x = 1 specialization: [Conjecture] A super K (x = 1, y; a, t) = y k + y k 1 P (K; a, q = 1, t). Superpolynomial in R = appears! Strong knot inv.! Kinoshita-Terasaka knot Conway knot

23 . 6. Conclusion and Future Directions Conclusions Volume conjecture connects between the quantum knot invariants and A-polynomials Reformulation of the colored Jones polynomial by topological recursion. Quanum volume conjecture for the colored superpolynomial Super-A-polynomial A super K (x, y; a, t). Future Directions First step Enjoy with more properties of super-a! Property (1): Geometric/arithmetic meanings of the super-a-polynomial Property (2): Find a rule for decorated triangulation to realize (a, t)-dependence. Property (5): B-model / Matrix model reformulation for superpolynomial? Super knot contact homology?

24 Thank you for your attention!

25 Superpolynomial [Dunfield-Gukov-Rasmussen][Gukov-Stosic] Example: Trefoil knot HOMFLY polynomial P and reduced (normalized) Khovanov invariant Kh : P (3 1 ; a, q) = aq 1 + aq a 2, Kh (3 1 ; q, t) = q + q 3 t 2 + q 4 t 3. a a q t a q t a q t a q t (H, d ±1 ) sl 1 homology: 1-dim. The extrapolating polynomial P(3 1 ; a, q, t) P (3 1 ; a, q, t) = aq 1 + aqt 2 + a 2 t 3. q

Empirical computations of superpolynomials [Dunfield-Gukov-Rasmussen][Gukov-Stosic] The existence of the triply graded homology H ijk (K) is conjectural.

26 Empirical computations of superpolynomials [Dunfield-Gukov-Rasmussen][Gukov-Stosic] The existence of the triply graded homology H ijk (K) is conjectural. Determine P n from consistency of axioms of d s: t = 1 limit Colored HOMFLY polynomial P n (K; a, q) Cancelling Differentials d Sn 1 1 and d Sn 1 1 n 1-dim. Degree ijk of surviving gen. ( Rasmussen s invariant ) S(K)) Reduction to sl N homologies : H Sn 1, dn Sn 1 H sl N: Colored differentials relates S r and S m (m < r) homologies 4 a R= 6 3 3d d -2 d d q

27 Braiding computation [Dunin-Barkowski-Mironov-Morozov-Sleptsov-Smirnov][F-Gukov-Su lkowsky] (2, 2p + 1)-torus knots T 2,2p+1 Braiding computation in CS theory is effective. ψ 0 (R 1,R 2 ) B B B Σ R1 R1 R2 R2 ψ 0 (R 1,R 2 ) Z ref SU(N) (T2,2p+1 R = Z ref SU(N) (S3 ) ) = ψ 0 (R, R) (B 2p+1 ) RR ψ 0 (R, R) Q R R γ Q RR λ(+) Q (R, R)2p+1 N Q RR M Q(q ϱ 2 ). λ (+) Q (R 1, R 1 ): Eigenvalue of half-twist B T 1/2

28 Gamma factor and exact solution The braid eigenvalue must be corrected by gamma factor γ Q R 1 R 2. This gamma factor can be determined by consistency of p = 1. = P Sr (T 2,2p+1 ; a, q, t) r (qt 2 ; q) l ( at 3 ; q) r+l ( aq 1 t; q) r l (q; q) r (1 q 2l+1 t 2 ) (q; q) l=0 l (q 2 t 2 ; q) r+l (q; q) r l ( at 3 ; q) r (1 qt 2 ) [ ] ( 1) r a r 3r 2 q 2 l t rp l+ r 2 ( 1) l a r r 2 2p+1 l(l+1) 2 q 2 t 3r 2 l. Ad hoc procedure by gamma factor This solution passes the necessary conditions of the colored superpolynomial by various differentials!.

29 Asymptotic limit of colored superpolynomial for 4 1 Double scaling limit ( 0 ): P n (4 1 ; a, q, t) n 1 k=0 q = e, n := logx, k := logz = ( 1) k a k t 2k q k(k 3)/2 ( atq 1, q) k (q, q) k (q 1 n, q) k ( at 3 q n 1, q) k. Approximating q-dilog functions by (x; q) k e 1 (Li 2(x) Li 2 (xz)) : P n (K; a, q, t) dz e 1 W K (x,z;a,t)+o( 0).

30 Saddle point approximation 0 limit Saddle point value of z 0 becomes dominant. W(4 1 ; z, x) = πi log z π2 6 (log a + 2 log t) log z 1 (log z)2 2 +Li 2 (x 1 ) Li 2 (x 1 z) + Li 2 ( at) Li 2 ( atz) +Li 2 ( axt 3 ) Li 2 ( axt 3 z) Li 2 (z). Saddle point condition: z W K z = 0 Saddle point z = z 0. S 0 (4 1 ; x, a, t) = W 41 (x, z 0 ; a, t). Abel-Jacobi map as Neumann-Zagier s potential function: x S 0 (4 1 ; x, a, t)/ x = log y(x; a, t) A super 4 1 (x, y; a, t) = 0!

31 .Contents Superpolynomial

32 Superpolynomial [Dunfield-Gukov-Rasmussen][Gukov-Stosic] The superpolynomial P = P R= satisfies the following axioms: Superpolynomial is the Poincaré polynomial of the triply-graded homology H ijk (K) (dimh ijk < 0): P(K; a, q, t) = i,j,k a i q j t k H ijk (K). Differentials d N (i.e. Endmorphisms) d N : H ijk (K) H i 1,j+N,k 1, (N 0), H ijk (K) H i 1,j+N,k 3, (N < 0), d N d M = d M d N. The differential d N has gradings (GS grading): ( 1, N, 1) for N 0, ( 1, N, 3) for N < 0. Remark: DGR-grading is different convention: ( 2, 2N, 1) for N > 0, ( 2, 0, 3) for N > 0, ( 2, 2N, 1 + 2N) for N < 0.

33 Superpolynomial 2 [Dunfield-Gukov-Rasmussen][Gukov-Stosic] Example: Trefoil knot HOMFLY polynomial P and reduced (normalized) Khovanov invariant Kh : P (3 1 ; a, q) = aq 1 + aq a 2, Kh (3 1 ; q, t) = q + q 3 t 2 + q 4 t 3. a a q t a q t a q t q The extrapolating polynomial P(3 1 ; a, q, t) satisfies the axiom of superpolynomial: P(3 1 ; a, q, t) = aq 1 + aqt 2 + a 2 t 3.

34 Colored superpolynomial 1 [Gukov-Stosic] For symmetric/anti-symmetric representations R = S r /Λ r, the colored superpolynomial P R is also defined by adding the universal colored differntial d Sr r m ) ( H Sr, d r m to the axiom: H Sm, r > m. Modding out the homology elements H Sr which are related by d r m, one finds the homology for S m. The previous differentials d N d Sr N has degrees deg(d Sr N )=( 1, N, 1) for N 1 r deg(d Sr N )=( 1, N, 3) for N r. The differentials d Sr 1 and dsr r are called cancelling differentials : Only one generator survivies modded by cancelling diffs. ( ) ( ) deg H Sr, d Sr 1 = (rs, rs, 0), deg H Sr, d Sr r = (rs, rs, 0). S: Rasumssen s S-invariant

35 Colored superpolynomial 2 [Gukov-Stosic] Conjecture For each knot K, there exist a triply-graded homology H Sr ijk (K) and colored superpolynomial PSr (K; a, q, t) which satisfies the axioms of differentials below. Differentials d Sr N : Restriction to SL(N) homology deg(d Sr N )=( 1, N, 1) for N 1 r deg(d Sr N )=( 1, N, 3) for N r. 2 special classes: A. Cancelling differential: Restriction to SL(1) homology dim ( H Sr, ) ) dsr 1 = 1 =dim (H Sr, dsr r B. Vertical differentials d Sr (1 k r 1): ( ) 1 k H Sr, dsr 1 k q=1 H Sk : doubly graded homology (a, t) Colored differentials: d r m : Restriction to S m homology ( H Sr, d r m ) H Sm, r > m.

36 Example: Trefoil (S = 1) P S2 (3 1 ; a, q) = a 2 q 2 + a 2 q + a 2 q 2 + a 2 q 4 a 3 a 3 q + q 3 q 3 a 3 q 4 + a 4 q 3, KhR S2 2 (3 1; q, t) = q 2 + q 5 t 2 + q 6 t 2 + q 6 t 3 + q 7 t 3 4 a +q 8 t 4 + q 9 t 5 + q 10 t 5 + q 11 t q P S2 (3 1 ; a, q, t) = a 2 (q 2 + qt 2 + q 2 t 2 + q 4 t 4 ) +a 3 (t 3 + qt 3 + q 3 t 5 + q 4 t 5 ) + a 4 q 3 t 6.

37 Consistency of differentials: Cancelling differentials d S2 1 and ds2 2 : Reduction to sl(1) homology Only 1-dimensional. d S2 1 : Mod out elements connected by blue arrows : a2 q 2 t 0 d S2 2 : Mod out elements connected by red arrows : a2 q 4 t 4 Vertical colored differentials d S2 0 : Mod out by green dotted arrows : Reduction to S 1 homology ( ) P H S2, ds2 0 = a 2 q 2 + a 3 q 0 t 3 + a 2 q 2 t 2. ( ) P H S2, ds2 0 = ap S1 (3 1 ; a, q 2, t). Universal colored differential d 2 1 : Mod out by purple arrows : Reduction to S 1 homology ( ) P H S2, d 2 1 = a 2 q 2 + a 4 q 3 t 6 + a 2 q 4 t 4 = P S1 (3 1 ; a 2, q 2, t 2 q), where P S1 (3 1 ; a, q, t) = aq 1 + aqt 2 + a 2 t 3.

38 Quantum A-polynomial of Figure 8 knot [Garoufalidis-Le] Â 41 (ˆx, ŷ; q) = 3 a j (ˆx; q)ŷ j. j=0 a 0 (ˆx; q) = q 5ˆx 2 ( q 3 + q 3ˆx 2 ) (q 2 + q 3ˆx 2 )( q 5 + q 6ˆx 4 ), a 3(ˆx; q) = q4ˆx 2 ( 1 + q 3ˆx 2 ) (q + q 3ˆx 2 )(q q 6ˆx 4 ) q 2 q 3ˆx 2 a 1 (ˆx; q) = q 5ˆx 2 (q + q 3ˆx 2 )(q 5 q 6ˆx 4 ), (q 8 2q 9ˆx 2 + q 10ˆx 2 q 9ˆx 4 + q 10ˆx 4 q 11ˆx 4 + q 10ˆx 6 2q 11ˆx 6 + q 12ˆx 8 ), a 2 (ˆx; q) = q + q 3ˆx 2 q 4ˆx 2 (q 2 + q 3ˆx 2 )( q + q 6ˆx 4 ) (q 4 + q 5ˆx 2 2q 6ˆx 2 q 7ˆx 4 + q 8ˆx 4 q 9ˆx 4 2q 10ˆx 6 + q 11ˆx 6 + q 12ˆx 8 ), Classical limit q 1 : Â 41 (ˆx, ŷ; q) A 41 (x, y).

Super-A-polynomials of Twist Knots

Super-A-polynomials of Twist Knots joint work with Ramadevi and Zodinmawia to appear soon Satoshi Nawata Perimeter Institute for Theoretical Physics Aug 28 2012 Satoshi Nawata (Perimeter) Super-A-poly