Super-A-polynomials of Twist Knots

Transcription

1 Super-A-polynomials of Twist Knots joint work with Ramadevi and Zodinmawia to appear soon Satoshi Nawata Perimeter Institute for Theoretical Physics Aug Satoshi Nawata (Perimeter) Super-A-poly of Twist Knots Aug / 21

2 Polynomial knot invariants Alexander polynomials (K; q) = q 1/2 q 1/2 For trefoil, = q 1 + q 1 Jones polynomials J(K; q) q 1/2 J q 1/2 J For trefoil, J = q + q 3 q 4 = q 1/2 q 1/2 J HOMFLY polynomials P(K; a, q) a 1/2 P a 1/2 = q 1/2 q 1/2 P For trefoil, P = aq 1 + aq a 2 Satoshi Nawata (Perimeter) Super-A-poly of Twist Knots Aug / 21

3 Categoifications the Poincaré polynomial of colored Khovanov homology H sl 2,R i,j [Khovanov 00] Kh R (K; q, t) = X i,j t j q i dim H sl 2,R i,j (K), q-graded Euler characteristic gives colored Jones polynomial: J R (K; q) = Kh(q, t = 1) = X i,j ( 1) j q i dim H sl 2,R i,j (K). The Poincaré polynomial of the Khovanov-Rozansky homology [Khovanov-Rozansky 04] KhR R (K; q, t) = X i,j t i q j dim H sl N,R i,j (K). is related to the colored HOMFLY polynomial via KhR R (K; q, t = 1) = P R (K; a = q N, q) Satoshi Nawata (Perimeter) Super-A-poly of Twist Knots Aug / 21

4 Colored superpolynomials the Poincaré polynomial of triply-graded homology H sl 2,R i,j [Dunfield-Gukov-Rasmussen 05] P R (K; a, q, t) = X i,j,k a i q j t k dim H R i,j,k (K). The (a, q)-graded Euler characteristic of the triply-graded homology theory is equivalent to the colored HOMFLY polynomial P R (K; a, q) = X i,j,k( 1) k a i q j dim H R i,j,k (K). Jones J n (K; q) t = 1 colored sl 1 homology (a/q) (n 1)s(K) d 1 d 0 colored HOMFLY homology P n (K; a, q, t) d 2 Knot Floer homology HFK(K; q, t) d N t = 1 colored sl 2 homology colored sl N homology Pn sl2 (K; q, t) Pn sln (K; q, t) t = 1 HOMFLY P n (K; a, q) t = 1 Alexander (K; q) a = q N SU(N) CSinv. Pn(K; a=q N,q) Satoshi Nawata (Perimeter) Super-A-poly of Twist Knots Aug / 21

5 Twist knot K p p full twist p knots The correspondence between the twist number and the knots in Rolfsen s table Colored Jones polynomials and A-polynomials are known Quantum A-polynomials were already computed for p = Satoshi Nawata (Perimeter) Super-A-poly of Twist Knots Aug / 21

6 Colored Jones polynomials of twist knots The double-sum expressions [Habiro 03][Masbaum 03] J n(k p; q) = X kx q k (q 1 n ; q) k (q n+1 ; q) k k=0 l=0 ( 1) l q l(l+1)p+l(l 1)/2 (1 q 2l+1 (q; q) k ). (q; q) k+l+1 (q; q) k l The multi-sum expressions J n(k p>0 ; q) = J n(k p<0 ; q) = X q `q sp 1 n ; q `q1+n p 1 Y» ; q s p q s s i (s i +1) si+1 p s i s p s 1 0 i=1 q X ( 1) s p q s p (s p +1) `q1 n 2 ; `q1+n ; q q s p s p s 1 0 p 1 Y» q s i (s i+1 +1) si+1 s i i=1 q s p Satoshi Nawata (Perimeter) Super-A-poly of Twist Knots Aug / 21

7 Colored superpolynomials of trefoil and figure-8 Colored superpolynomials of trefoil and figure-8 are known [Fuji Gukov Sulkowski 12][Itoyama, Mironov, Morozov 2 12] X P n(3 1 ; a, q, t) = ( t) n+1 q k ( atq 1 ; q) k (q 1 n ; q) k ( at 3 q n 1 ; q) k, (q; q) k=0 k X P n(4 1 ; a, q, t) = ( 1) k a k t 2k q k(k 3)/2 ( atq 1 ; q) k (q 1 n ; q) k ( at 3 q n 1 ; q) k. (q; q) k=0 k Colored superpolynomials P n(a, q, t) for 5 2 and 6 1 are also known up to n = 3 [Gukov Stosic 11] One can do educated guess on colored superpolynomials of twist knots Satoshi Nawata (Perimeter) Super-A-poly of Twist Knots Aug / 21

8 Colored superpolynomials of twist knots The double-sum expressions P n(k p; a, q, t) = X kx q k ( atq 1 ; q) k (q 1 n ; q) k ( at 3 q n 1 ; q) k (q; q) k=0 l=0 k ( 1) l a pl t 2pl q (p+1/2)l(l 1) 1 at 2 q 2l 1 (at 2 q l 1 ; q) k+1» k l. q The multi-sum expressions X P n(k p>0 ; a, q, t) = ( t) n+1 q sp ( atq 1 ; q) sp (q 1 n ; q) sp ( at 3 q n 1 ; q) sp (q; q) s p s 1 0 sp p 1 Y» (at 2 ) s i q s i (s i 1) si+1 s i i=1 q P n(k p<0 ; a, q, t) X = ( 1) k a s p t 2s p q s p (s p 3)/2 ( atq 1 ; q) s p (q 1 n ; q) s p ( at 3 q n 1 ; q) s p (q; q) s p s p s 1 0 p 1 Y» (at 2 ) s i q s i (s i+1 1) si+1. s i i=1 q Satoshi Nawata (Perimeter) Super-A-poly of Twist Knots Aug / 21

9 Colored Jones polynomials of twist knots The double-sum expressions [Habiro 03][Masbaum 03] J n(k p; q) = X kx q k (q 1 n ; q) k (q n+1 ; q) k k=0 l=0 ( 1) l q l(l+1)p+l(l 1)/2 (1 q 2l+1 (q; q) k ). (q; q) k+l+1 (q; q) k l The multi-sum expressions J n(k p>0 ; q) = J n(k p<0 ; q) = X q `q sp 1 n ; q `q1+n p 1 Y» ; q s p q s s i (s i +1) si+1 p s i s p s 1 0 i=1 q X ( 1) s p q s p (s p +1) `q1 n 2 ; `q1+n ; q q s p s p s 1 0 p 1 Y» q s i (s i+1 +1) si+1 s i i=1 q s p Satoshi Nawata (Perimeter) Super-A-poly of Twist Knots Aug / 21

10 Colored superpolynomials of twist knots The double-sum expressions P n(k p; a, q, t) = X kx q k ( atq 1 ; q) k (q 1 n ; q) k ( at 3 q n 1 ; q) k (q; q) k=0 l=0 k ( 1) l a pl t 2pl q (p+1/2)l(l 1) 1 at 2 q 2l 1 (at 2 q l 1 ; q) k+1» k l. q The multi-sum expressions X P n(k p>0 ; a, q, t) = ( t) n+1 q sp ( atq 1 ; q) sp (q 1 n ; q) sp ( at 3 q n 1 ; q) sp (q; q) s p s 1 0 sp p 1 Y» (at 2 ) s i q s i (s i 1) si+1 s i i=1 q P n(k p<0 ; a, q, t) X = ( 1) k a s p t 2s p q s p (s p 3)/2 ( atq 1 ; q) s p (q 1 n ; q) s p ( at 3 q n 1 ; q) s p (q; q) s p s p s 1 0 p 1 Y» (at 2 ) s i q s i (s i+1 1) si+1. s i i=1 q Satoshi Nawata (Perimeter) Super-A-poly of Twist Knots Aug / 21

11 Checks For a = q 2 and t = 1, the above formulae reduce to the colored Jones polynomials For t = 1, they reduce to the colored HOMFLY polynomials. We checked they agree with the colored HOMFLY polynomials computed by SU(N) Chern-Simons theory up to 10 crossings. The colored HOMFLY polynomials can be reformulated into the Ooguri-Vafa polynomials. We checked that the Ooguri-Vafa polynomials have interger coefficients up to specific factors. We checked that the special polynomials which are the limits q 1 of the colored HOMFLY polynomials have the property,» n 1 lim Pn(Kp; a, q) = lim P 2(K p; a, q). q 1 q 1 Satoshi Nawata (Perimeter) Super-A-poly of Twist Knots Aug / 21

12 Cancelling differentials and Rasmussen s-invarinats the action of the differential d 1 on the colored superpolynomials P n+1 (K p>0 ; a, q, t) = a n q n + (1 + a 1 qt 1 )Q sl 1 n+1 (K p>0; a, q, t), P n+1 (K p<0 ; a, q, t) = 1 + (1 + a 1 qt 1 )Q sl 1 n+1 (K p<0; a, q, t), the action of the differential d n on the colored superpolynomials P n+1 (K p>0 ; a, q, t) = a n q n2 t 2n + (1 + a 1 q n t 3 )Q n+1 (K p>0 ; a, q, t), P n+1 (K p<0 ; a, q, t) = 1 + (1 + a 1 q n t 3 )Q n+1 (K p<0 ; a, q, t). The exponents of the remaining monomials are consistent with the Rassmusen s s-invariant of the twist knots, s(k p>0 ) = 1 and s(k p<0 ) = 0 [Rasmussen 04] deg H,, Sn (K), d 1 = n s(k), n s(k), 0, deg H,, (K), Sn d n ) = `n s(k), n 2 s(k), 2n s(k), Satoshi Nawata (Perimeter) Super-A-poly of Twist Knots Aug / 21

13 Volume conjecture and A-polynomials The volume conjecture relates quantum invariants of knots to classical 3d topology [Kashaev 99][Murakami 2 00] 2π lim Jn(K; n n log q = e 2πi n ) = Vol(S 3 \K). The relation b/w volume conjecture and A-polynomial [Gukov 03] log y = x d dx lim n,k e iπn/k =x 1 2πi log Jn(K; q = e k ), k gives the zero locus of the A-polynomial A(K; x, y) of the knot K. A-polynomial A(K; x, y) is a character variety of SL(2, C)-representation of the fundamental group of the knot complement M flat (SL(2, C), T 2 ) Lag.sub. M flat (SL(2, C), S 3 \K) = {(x, y) C C A(K; x, y) = 0} Satoshi Nawata (Perimeter) Super-A-poly of Twist Knots Aug / 21

14 AJ conjecture Quantum version of the volume conjecture AJ conjecture [Gukov 03][Garoufalidis 03] ba(k; ˆx, ŷ, q)j n(k; q) = 0 where action of ˆx and ŷ on the set of colored Jones polynomials as ˆxJ n(k; q n ) = q n J n(k; q n ), ŷj n(k; q) = J n+1 (K; q). Find difference equations of colored Jones polynomials a k J n+k (K; q) a 1 J n+1 (K; q) + a 0 J n(k; q) = 0 where a k = a k (K; bx, q) and ba(k; bx, by; q) = X a i (K; bx, q)by i Taking the classical limit q = e 1, quantum (non-commutative) A-polynomials reduces to ordinary A-polynomials ba(k; bx, by; q) A(K; x, y) as q 1 Satoshi Nawata (Perimeter) Super-A-poly of Twist Knots Aug / 21

15 Super-A-polynomials Refinement of quantum and classical A-polynomials [Fuji Gukov Sulkowski 12] Quantum operator provides recursion for classical limit ba super (bx, by; a, q, t) colored superpolynomial A super (x, y; a, t) ba ref (bx, by; q, t) colored Khovanov-Rozansky homology A ref (x, y, t) ba Q def (bx, by; a, q) colored HOMFLY A Q def (x, y; a) ba(bx, by; q) colored Jones A(x, y) Satoshi Nawata (Perimeter) Super-A-poly of Twist Knots Aug / 21

16 Classical super-a-polynomials Refinement of volume conjecture [Fuji Gukov Sulkowski 12] log y = x d dx lim n,k e iπn/k =x 1 2πi log Pn(K; a, q = e k, t), k gives the zero locus of the super-a-polynomial A(K; x, y; a, q, t) of the knot K. Satoshi Nawata (Perimeter) Super-A-poly of Twist Knots Aug / 21

17 Quantum super-a-polynomials Refinement of AJ conjecture ba super (K; ˆx, ŷ; a, q, t)p n(k; a, q, t) = 0 where the operators ˆx and ŷ acts on the set of colored superpolynomials as ˆxP n(k; a, q, t) = q n P n(k; a, q, t), ŷp n(k; q) = P n+1 (K; a, q, t). Find difference equations of colored superpolynomials a k P n+k (K; q) a 1 P n+1 (K; q) + a 0 P n(k; q) = 0 where a k = a k (K; bx; a, q, t) and ba super (K; bx, by; a, q, t) = X a i (K; bx; a, q, t)by i Taking the classical limit q = e 1, quantum (non-commutative) super-a-polynomials reduces to classical super-a-polynomials ba super (K; bx, by; a, q, t) A super (K; x, y; a, q, t) as q 1 Satoshi Nawata (Perimeter) Super-A-poly of Twist Knots Aug / 21

18 Quantum super-a-polynomials Knot Âsuper (K; ˆx, ŷ; a, q, t) 5 2 q 6 t 4 ( 1 + at 3ˆx ) ( 1 + aqt 3ˆx ) ( 1 + aq 2 t 3ˆx ) ( 1 + at 3ˆx 2) ( q + at 3ˆx 2) ( 1 + aqt 3ˆx 2) ŷ 4 aq 5 t 4 (1 + at 3ˆx)(1 + aqt 3ˆx)(1 + at 3ˆx 2 )(q + at 3ˆx 2 )(1 + aq 4 t 3ˆx 2 )(1 + q q 3ˆx + q 3 tˆx q 2 t 2ˆx q 3 t 2ˆx + q 4 t 2ˆx 2 +aq 4 t 2ˆx 2 +q 5 t 2ˆx 2 +q 6 t 2ˆx 2 +aqt 3ˆx 2 +aq 2 t 3ˆx 2 +aq 5 t 3ˆx 2 +aq 6 t 3ˆx 2 aq 4 t 3ˆx 3 aq 8 t 3ˆx 3 +aq 4 t 4ˆx 3 + aq 8 t 4ˆx 3 +aq 5 t 5ˆx 3 +aq 6 t 5ˆx 3 +a 2 q 5 t 5ˆx 4 aq 8 t 5ˆx 4 +a 2 q 9 t 5ˆx 4 +a 2 q 6 t 6ˆx 4 +a 2 q 7 t 6ˆx 4 a 2 q 9 t 6ˆx 5 +a 2 q 9 t 7ˆx 5 + a 3 q 10 t 8ˆx 6 )ŷ 3 a 2 q 5 t 4 ( 1+q 2ˆx)(1+at 3ˆx)(q+at 3ˆx 2 )(1+aq 2 t 3ˆx 2 )(1+aq 5 t 3ˆx 2 )(1+q 2 tˆx qt 2ˆx q 2 t 2ˆx+q 3 t 2ˆx 2 +q 4 t 2ˆx 2 + at 3ˆx 2 + aqt 3ˆx 2 q 3 t 3ˆx 2 + aq 3 t 3ˆx 2 q 4 t 3ˆx 2 + aq 4 t 3ˆx 2 + q 3 t 4ˆx 2 + aq 2 t 4ˆx 3 q 4 t 4ˆx 3 aq 4 t 4ˆx 3 q 5 t 4ˆx 3 q 6 t 4ˆx 3 +aq 6 t 4ˆx 3 aqt 5ˆx 3 aq 2 t 5ˆx 3 +aq 3 t 5ˆx 3 +aq 4 t 5ˆx 3 aq 5 t 5ˆx 3 aq 6 t 5ˆx 3 +aq 3 t 5ˆx 4 +aq 4 t 5ˆx 4 +aq 7 t 5ˆx 4 + aq 8 t 5ˆx 4 +a 2 qt 6ˆx 4 aq 3 t 6ˆx 4 +a 2 q 3 t 6ˆx 4 aq 4 t 6ˆx 4 +2a 2 q 4 t 6ˆx 4 +a 2 q 5 t 6ˆx 4 aq 7 t 6ˆx 4 +a 2 q 7 t 6ˆx 4 aq 8 t 6ˆx 4 aq 4 t 7ˆx 4 2aq 5 t 7ˆx 4 aq 6 t 7ˆx 4 a 2 q 4 t 7ˆx 5 +aq 6 t 7ˆx 5 +a 2 q 6 t 7ˆx 5 +aq 7 t 7ˆx 5 +aq 8 t 7ˆx 5 a 2 q 8 t 7ˆx 5 +a 2 q 3 t 8ˆx 5 + a 2 q 4 t 8ˆx 5 a 2 q 5 t 8ˆx 5 a 2 q 6 t 8ˆx 5 + a 2 q 7 t 8ˆx 5 + a 2 q 8 t 8ˆx 5 + a 2 q 7 t 8ˆx 6 + a 2 q 8 t 8ˆx 6 + a 3 q 4 t 9ˆx 6 + a 3 q 5 t 9ˆx 6 a 2 q 7 t 9ˆx 6 +a 3 q 7 t 9ˆx 6 a 2 q 8 t 9ˆx 6 +a 3 q 8 t 9ˆx 6 +a 2 q 7 t 10ˆx 6 a 3 q 8 t 10ˆx 7 +a 3 q 7 t 11ˆx 7 +a 3 q 8 t 11ˆx 7 +a 4 q 8 t 12ˆx 8 )ŷ 2 +a 3 q 7 t 7ˆx 2 ( 1 + qˆx)( 1 + q 2ˆx)(1 + at 3ˆx 2 )(1 + aq 4 t 3ˆx 2 )(1 + aq 5 t 3ˆx 2 )(q + q 2 tˆx q 2 t 2ˆx + at 3ˆx 2 q 3 t 3ˆx 2 + aq 4 t 3ˆx 2 + aqt 4ˆx 2 + aq 2 t 4ˆx 2 + aqt 4ˆx 3 + aq 5 t 4ˆx 3 aqt 5ˆx 3 aq 5 t 5ˆx 3 aq 2 t 6ˆx 3 aq 3 t 6ˆx 3 + aq 3 t 6ˆx 4 + a 2 q 3 t 6ˆx 4 +aq 4 t 6ˆx 4 +aq 5 t 6ˆx 4 +a 2 t 7ˆx 4 +a 2 qt 7ˆx 4 +a 2 q 4 t 7ˆx 4 +a 2 q 5 t 7ˆx 4 +a 2 q 4 t 7ˆx 5 a 2 q 4 t 8ˆx 5 +a 2 q 3 t 9ˆx 5 + a 2 q 4 t 9ˆx 5 + a 3 q 3 t 10ˆx 6 + a 3 q 4 t 10ˆx 6 )ŷ a 5 q 8 t 15 ( 1 + ˆx)ˆx 7 ( 1 + qˆx)( 1 + q 2ˆx)(1 + aq 3 t 3ˆx 2 )(1 + aq 4 t 3ˆx 2 )(1 + aq 5 t 3ˆx 2 ) Table 5: Quantum super-a-polynomial of 5 2 knot. first used the Mathematica package qmultisum.m to obtain q-difference equations for 5 2 Satoshi Nawata (Perimeter) Super-A-poly of Twist Knots Aug / 21

19 Augmentation polynomials of knot contact homology Q-deformed A-polynomial is equivalent to augmentation polynomial of knot contact homology [Ng 10][Aganagic Vafa 12] A super K p>0 ; x = µ, y = 1 + µ «λ; a = U, t = Uµ = ( 1)p 1 (1 + µ) (2p 1) 1 + Uµ A super K p<0 ; x = µ, y = 1 + µ λ; a = U, t = Uµ Aug(K p>0 ; µ, λ; U, V = 1), «= ( 1)p (1 + µ) 2p Aug(K p<0 ; µ, λ; U, V = 1), 1 + Uµ Satoshi Nawata (Perimeter) Super-A-poly of Twist Knots Aug / 21

20 Prospects and Future Directions Anti-symmetric representation. Mirror symmetry Quantum 6j-symbolos for U q (sl N ) and their refinement Colored superpolynomials of other non-torus knots SU(N) analogue of WRT invariants and one-parameter deformations of Mock modular forms. refinement of colored Kaufmann polynomials Satoshi Nawata (Perimeter) Super-A-poly of Twist Knots Aug / 21

21 Thank you Satoshi Nawata (Perimeter) Super-A-poly of Twist Knots Aug / 21