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We conjecture formulae of the colored superpolynomials for a class of twist knots K p where p denotes the number of full twists. The validity of the formulae is checked by applying differentials and taking special limits. Using the formulae, we compute both the classical and quantum super-A-polynomials for the twist knots with small values of p. The results support the categorified versions of the generalized volume conjecture and the quantum volume conjecture. Furthermore, we obtain the evidence that the Q-deformed Apolynomials can be identified with the augmentation polynomials of knot contact homology in the case of the twist knots.

In this paper, we elaborate on aspects of the recently introduced BMS bootstrap programme. We consider two-dimensional (2d) field theories with BMS 3 symmetry and extensively use highest weight representations to uncover the BMS version of crossing symmetry in 4-point functions that are constrained by symmetry. The BMS bootstrap equation is formulated and then analytic expressions for BMS blocks are constructed by looking at the limit of large central charges. These results are also applicable to 2d Galilean Conformal Field Theories through the isomorphism between the BMS 3 and 2d Galilean Conformal Algebras. We recover our previously obtained results in the non-relativistic limit of the corresponding ones in 2d relativistic CFTs. This provides a comprehensive check of our previous analysis. We also explore the chiral limit of BMS 3 where the BMS algebra reduces to a single copy of the Virasoro algebra and show that our analysis is consistent with earlier work in this direction.arXiv:1705.05890v1 [hep-th] 16 May 2017 C Level 2 analysis of coefficients in the Chiral limit 44The constraint that both sides of the OPE transform the same way under the two quadratic Casimirs gives two differential equations for K hp,hp (z,z) z 2 (1 − z)∂ 2 z − z 2 ∂ z K hp,hp (z,z) = h p (h p + 1)K hp,hp (z,z),

We elaborate the Chern-Simons field theoretic method to obtain colored HOMFLY invariants of knots and links. Using multiplicity-free quantum 6j-symbols for Uq(sl N ), we present explicit evaluations of the HOMFLY invariants colored by symmetric representations for a variety of knots, two-component links and three-component links. out explicit computations in general. Even in mathematics, although the definition [19,17] of colored HOMFLY polynomials was provided, explicit calculations for non-trivial knots and links are far from under control.Nevertheless, there have been spectacular developments on computations of colored HOMFLY polynomials in recent years. For torus knots and links, the HOMFLY invariants colored by arbitrary representations can be, in principle, computed by using the generalizations [17,25,4] of the Rosso-Jones formulae [24]. In addition, Kawagoe has lately formulated a mathematically rigorous procedure based on the linear skein theory to calculate HOMFLY invariants colored by symmetric representations for some non-torus knots and links [16]. Furthermore, the explicit closed formulae of the colored HOMFLY polynomials P [n] (K; a, q) with symmetric representations (R = ) were provided for the (2, 2p + 1)-torus knots [8] and the twist knots [11,21,7,16].In this paper, we shall demonstrate the computations of the HOMFLY polynomials colored by symmetric representations in the framework of Chern-Simons theory. Exploiting the connection between Chern-Simons theory and the twodimensional Wess-Zumino-Novikov-Witten (WZNW) model, the prescription to evaluate expectation values of Wilson loops was formulated entirely in terms of the fusion and braid operations on conformal blocks of the WZNW model [14,15,22]. Therefore, the procedure inevitably involves the SU(N ) quantum Racah coefficients (the quantum 6j-symbols for U q (sl N )), which makes explicit computations hard. The first step along this direction has been made in [28]: using the properties the SU(N ) quantum Racah coefficients should obey, the explicit expressions involving first few symmetric representations are determined. This result as well as the closed formulae of the twist knots motivated us to explore a closed form expression for the SU(N ) quantum Racah coefficients. We succeeded in writing the expression for multiplicity-free representations [20] which enables us to compute the colored HOMFLY polynomials carrying symmetric representations. To consider more complicated knots and links than the ones treated in [28], we make use of the method developed in topological quantum field theory [14,15].With this method, the expressions of the twist knots, the Whitehead links, the twist links and the Borromean rings [16,21,9] have been reproduced up to four boxes. Even apart from these classes of knots and links, the validity of our procedure is checked from the complete agreement with the results obtained in [12,13]. Furthermore, the explicit evaluations of multi-colored link invariants shed a new light on the general properties of colored HOMFL...