Trivalent graphs, volume conjectures and character varieties

The generalized volume conjecture and the AJ conjecture (a.k.a. the quantum volume conjecture) are extended to U_q(\fraksl_2) colored quantum invariants of the theta and tetrahedron graph. The \SL(2,\bC) character variety of the fundamental group of the complement of a trivalent graph with $E$ edges in $S^3$ is a Lagrangian subvariety of the Hitchin moduli space over the Riemann surface of genus $g=E/3+1$. For the theta and tetrahedron graph, we conjecture that the configuration of the character variety is locally determined by large color asymptotics of the quantum invariants of the trivalent graph in terms of complex Fenchel-Nielsen coordinates. Moreover, the $q$-holonomic difference equation of the quantum invariants provides the quantization of the character variety.Comment: 11 pages, 2 figure

Multiplicity-free quantum 6j-symbols for U_q(sl_N)

We conjecture a closed form expression for the simplest class of multiplicity-free quantum 6j-symbols for U_q(sl_N). The expression is a natural generalization of the quantum 6j-symbols for U_q(sl_2) obtained by Kirillov and Reshetikhin. Our conjectured form enables computation of colored HOMFLY polynomials for various knots and links carrying arbitrary symmetric representations.Comment: 8 pages; v2 typos corrected; v3 minor corrections and reference adde

Colored HOMFLY polynomials from Chern-Simons theory

We elaborate the Chern-Simons field theoretic method to obtain colored HOMFLY invariants of knots and links. Using multiplicity-free quantum 6j-symbols for U_q(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.Comment: 40 pages, 23 figures, a Mathematica notebook linked on the right as an ancillary file; v2 typos corrected; v3 corrections in section 4.2 and cosmetic changes; v4 corrections in two-component link

Super-A-polynomials for Twist Knots

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-polynomial 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 A-polynomials can be identified with the augmentation polynomials of knot contact homology in the case of the twist knots.Comment: 22+16 pages, 16 tables and 5 figures; with a Maple program by Xinyu Sun and a Mathematica notebook in the ancillary files linked on the right; v2 change in appendix B, typos corrected and references added; v3 change in section 3.3; v4 corrections in Ooguri-Vafa polynomials and quantum super-A-polynomials for 7_2 and 8_1 are adde

Challenges of beta-deformation

A brief review of problems, arising in the study of the beta-deformation, also known as "refinement", which appears as a central difficult element in a number of related modern subjects: beta \neq 1 is responsible for deviation from free fermions in 2d conformal theories, from symmetric omega-backgrounds with epsilon_2 = - epsilon_1 in instanton sums in 4d SYM theories, from eigenvalue matrix models to beta-ensembles, from HOMFLY to super-polynomials in Chern-Simons theory, from quantum groups to elliptic and hyperbolic algebras etc. The main attention is paid to the context of AGT relation and its possible generalizations.Comment: 20 page

SU(N) quantum Racah coefficients and non-torus links

It is well known that the SU(2) quantum Racah coefficients or the Wigner 6j symbols have a closed form expression which enables the evaluation of any knot or link polynomials in SU(2) Chem-Simons field theory. Using isotopy equivalence of SU(N) Chem-Simons functional integrals over three-balls with one or more S-2 boundaries with punctures, we obtain identities to be satisfied by the SU(N) quantum Racah coefficients. This enables evaluation of the coefficients for a class of SU (N) representations. Using these coefficients, we can compute the polynomials for some non-torus knots and two-component links. These results are useful for verifying conjectures in topological string theory. (C) 2013 Elsevier B.V. All rights reserved