Twist Knot Invariants and Volume Conjecture
283
where q-binomial is given by
n
k
q
=
(q;q) n
(q;q) k (q;q) n−k
. P n (K p<0 ; a, q, t) has the
same expression as P n (K p>0 ; a, q, t) but without the factor (−t) −n+1 . Following
the methodology in section 2, the Chern–Simons field theory invariant R [K p ]],
in terms of braiding eigenvalues and U q (sl N ) Racah coefficients, for twist knots K p
turns out to be
R [K p ]] =
s,s
R, ¯
R
s
d s
R, ¯
R
s
d s
λ
(−)
s
R, ¯
R
−2
a ss
R ¯
R
R ¯
R
λ
(−)
s
R, ¯
R
−2p
,
where s, s ∈ R ⊗ ¯
R. Comparing the superpolynomials at t = −1 with the
above invariant, we obtained U q (sl N ) Racah coefficients for SU (N) symmetric
representations up to rank n = 3 [16]. Comparing with the formula of the U q (sl 2 )
6j -symbols obtained by Kirillov and Reshetikhin [20], we conjectured a closed
form expression for the U q (sl N ) quantum Racah coefficients [21] for the following
two types:
a ts
R 1 ¯
R 2
R 3 ¯
R 4
; a ts
R 1 R 2
¯
R 3 ¯
R 4
,
where R 1 , R 2 , R 3 , R 4 are symmetric representations with single row in the Young
diagram.
4 Volume Conjectures and Super-A-Polynomial
We consider the asymptotic form of our conjectured formula of superpolynomials (20) for knot 5 2 and perform saddle point analysis to obtain classical
super-A-polynomials A super (K p=2 ; x, y; a, t). We introduce two variables z =
e ¯
hk , w = e ¯
hh and take the limits: q = e ¯
h → 1, a = fixed, t = fixed, x =
q n = fixed, in (20) and convert the two summation to integrals over z and w. Then
using the categorified volume conjecture, we have
P n (K p>0 ; a, q, t)
n→∞
¯
h→0
∼
dzdw e
1
¯
h
W(K p>0 ;z,w)+O( ¯
h)
∼ e
1
¯
h
log y
dx
x + ...
,
(21)
where the integral on RHS in Eq. (21) is over the zero locus of the classical superA-polynomial, i.e., A super (K p=2 ; x, y; a, t) = 0. Taking such limits replaces
q-Pochhammer into di-logarithms giving the superpotential
283
where q-binomial is given by
n
k
q
=
(q;q) n
(q;q) k (q;q) n−k
. P n (K p<0 ; a, q, t) has the
same expression as P n (K p>0 ; a, q, t) but without the factor (−t) −n+1 . Following
the methodology in section 2, the Chern–Simons field theory invariant R [K p ]],
in terms of braiding eigenvalues and U q (sl N ) Racah coefficients, for twist knots K p
turns out to be
R [K p ]] =
s,s
R, ¯
R
s
d s
R, ¯
R
s
d s
λ
(−)
s
R, ¯
R
−2
a ss
R ¯
R
R ¯
R
λ
(−)
s
R, ¯
R
−2p
,
where s, s ∈ R ⊗ ¯
R. Comparing the superpolynomials at t = −1 with the
above invariant, we obtained U q (sl N ) Racah coefficients for SU (N) symmetric
representations up to rank n = 3 [16]. Comparing with the formula of the U q (sl 2 )
6j -symbols obtained by Kirillov and Reshetikhin [20], we conjectured a closed
form expression for the U q (sl N ) quantum Racah coefficients [21] for the following
two types:
a ts
R 1 ¯
R 2
R 3 ¯
R 4
; a ts
R 1 R 2
¯
R 3 ¯
R 4
,
where R 1 , R 2 , R 3 , R 4 are symmetric representations with single row in the Young
diagram.
4 Volume Conjectures and Super-A-Polynomial
We consider the asymptotic form of our conjectured formula of superpolynomials (20) for knot 5 2 and perform saddle point analysis to obtain classical
super-A-polynomials A super (K p=2 ; x, y; a, t). We introduce two variables z =
e ¯
hk , w = e ¯
hh and take the limits: q = e ¯
h → 1, a = fixed, t = fixed, x =
q n = fixed, in (20) and convert the two summation to integrals over z and w. Then
using the categorified volume conjecture, we have
P n (K p>0 ; a, q, t)
n→∞
¯
h→0
∼
dzdw e
1
¯
h
W(K p>0 ;z,w)+O( ¯
h)
∼ e
1
¯
h
log y
dx
x + ...
,
(21)
where the integral on RHS in Eq. (21) is over the zero locus of the classical superA-polynomial, i.e., A super (K p=2 ; x, y; a, t) = 0. Taking such limits replaces
q-Pochhammer into di-logarithms giving the superpotential