Twist Knot Invariants and Volume Conjecture
281
Fig. 3 Types of braiding
Fig. 4 Two bases for four-point conformal blocks
v r =
t
d t
2−r
φ t
R 1 , ¯
R 1 , R 2 , R 2
(1) . . .
φ t
R r , ¯
R r , R 1 , R 1
(r) ,
(15)
where t ∈ (R 1 ⊗ ¯
R 1 ) ∩ . . . . ∩ (R r ⊗ ¯
R r ) and d R is the quantum dimension of a
representation R. The superscripts outside the four-point conformal blocks denote
the boundaries as indicated in Fig. 1. The 0 in φ 0 and ˆ
φ 0 represents the singlet
representation. Substituting the building blocks states for each three-ball, we can
write R [10 152 ]]. Basically, gluing the three-balls along oppositely oriented S 2
boundaries involves inner product of four-point conformal bases with its dual fourpoint conformal blocks. The final invariant can be written in terms of braiding
eigenvalues and the U q (sl N ) Racah coefficients [16]. In order to write the explicit
polynomial form in variables a = q N , q, we require the U q (sl N ) Racah coefficients
which is not known for general SU (N) representations. We will now review our
work on superpolynomials for twist knots leading us to conjecture some U q (sl N )
Racah coefficients.