278
P. Ramadevi and Zodinmawia
A(K; ˆ
x, ˆ
y; q)J n (K; q) = 0,
(7)
which is also equivalent to the q-difference equation
k
m=0 a m ( ˆ
x, q)J n+m (K; q) of
minimal order. We expect to recover the classical A-polynomial from the quantum
A-polynomial by taking the classical limit q = 1:
A(K; ˆ
x, ˆ
y; q = 1) = A(K; x, y).
(8)
The above assertion is known as the quantum volume conjecture [11], or, the AJ
conjecture [12, 13].
The generalized volume conjecture and the quantum volume conjecture (AJ conjecture) were further categorified for superpolynomials in [14, 15] by including the
two-parameters (a, t). In this categorified version, one defines the classical super-Apolynomial, A super (K; x, y; a, t), to be an (a, t) deformation of A(K; x, y) which
can be obtained by substituting P n (K; a, q, t) for J n (K; q) in (5). Likewise, the
quantum super-A-polynomial, ˆ
A super (K; ˆ
x, ˆ
y; a, q, t), is also an (a, t) deformation
of
A(K; ˆ
x, ˆ
y; q). Basically ˆ
x, ˆ
y are defined as
ˆ
xP n (K; a, q, t) = q
n
P n (K; a, q, t), ˆ
yP n (K; a, q, t) = P n+1 (K; a, q, t),
and the corresponding quantum super-A-polynomial obeys
ˆ
A
super (K; ˆ
x, ˆ
y; a, q, t)P n (K; a, q, t) = 0.
(9)
The categorified version of the quantum volume conjecture states that
ˆ
A
super (K; ˆ
x, ˆ
y; a, q = 1, t)) = A
super (K; x, y; a, t).
(10)
This note is organized as follows. In Sect. 2, we briefly review how to use
the correspondence between Chern–Simons theory and Wess–Zumino–Novikov–
Witten (WZNW) model given in [1] to calculate knot invariants. This method
explicitly requires the knowledge of the quantum 6j -symbols or the quantum
group U q (sl N ) to write colored HOMFL-PT polynomials. In Sect. 3, we focus on
a class of knots called twist knots K p . Particularly, motivated by the structure of
colored Jones polynomials J n (K p , q) for the twist knots, we conjectured the colored
superpolynomials for the twist knots. Comparing P n (K p ; a, q) with the formal
Chern–Simons knot invariant, we obtained a closed form algebraic expression for
the U q (sl N ) quantum 6j -symbols for a class of multiplicity-free representation.
Using our conjectured superpolynomials, we find the classical super-A-polynomial
and the quantum super-A-polynomials for the 5 2 twist knot in Sect. 4 and test the
categorified quantum volume conjecture.
P. Ramadevi and Zodinmawia
A(K; ˆ
x, ˆ
y; q)J n (K; q) = 0,
(7)
which is also equivalent to the q-difference equation
k
m=0 a m ( ˆ
x, q)J n+m (K; q) of
minimal order. We expect to recover the classical A-polynomial from the quantum
A-polynomial by taking the classical limit q = 1:
A(K; ˆ
x, ˆ
y; q = 1) = A(K; x, y).
(8)
The above assertion is known as the quantum volume conjecture [11], or, the AJ
conjecture [12, 13].
The generalized volume conjecture and the quantum volume conjecture (AJ conjecture) were further categorified for superpolynomials in [14, 15] by including the
two-parameters (a, t). In this categorified version, one defines the classical super-Apolynomial, A super (K; x, y; a, t), to be an (a, t) deformation of A(K; x, y) which
can be obtained by substituting P n (K; a, q, t) for J n (K; q) in (5). Likewise, the
quantum super-A-polynomial, ˆ
A super (K; ˆ
x, ˆ
y; a, q, t), is also an (a, t) deformation
of
A(K; ˆ
x, ˆ
y; q). Basically ˆ
x, ˆ
y are defined as
ˆ
xP n (K; a, q, t) = q
n
P n (K; a, q, t), ˆ
yP n (K; a, q, t) = P n+1 (K; a, q, t),
and the corresponding quantum super-A-polynomial obeys
ˆ
A
super (K; ˆ
x, ˆ
y; a, q, t)P n (K; a, q, t) = 0.
(9)
The categorified version of the quantum volume conjecture states that
ˆ
A
super (K; ˆ
x, ˆ
y; a, q = 1, t)) = A
super (K; x, y; a, t).
(10)
This note is organized as follows. In Sect. 2, we briefly review how to use
the correspondence between Chern–Simons theory and Wess–Zumino–Novikov–
Witten (WZNW) model given in [1] to calculate knot invariants. This method
explicitly requires the knowledge of the quantum 6j -symbols or the quantum
group U q (sl N ) to write colored HOMFL-PT polynomials. In Sect. 3, we focus on
a class of knots called twist knots K p . Particularly, motivated by the structure of
colored Jones polynomials J n (K p , q) for the twist knots, we conjectured the colored
superpolynomials for the twist knots. Comparing P n (K p ; a, q) with the formal
Chern–Simons knot invariant, we obtained a closed form algebraic expression for
the U q (sl N ) quantum 6j -symbols for a class of multiplicity-free representation.
Using our conjectured superpolynomials, we find the classical super-A-polynomial
and the quantum super-A-polynomials for the 5 2 twist knot in Sect. 4 and test the
categorified quantum volume conjecture.