Classical A-polynomials $A(\ell ,m)$ define constraints on coordinates $\ell$ and $m$ in $SL(2,\mathbb{C})$ [a complexification of $SU(2)$] character varieties associated to knot complements $S^3\backslash K$. Quantum A-polynomials $\widehat{A}(\widehat{\ell },\widehat{m})$ are difference operators annihilating Jones polynomials believed to represent wave functions of 3D Chern-Simons theory with gauge group $SU(2)$ on a toroidal pipe surrounding the knot $K$ strand—a boundary of the knot complement $S^3\backslash K$. In this note we suggest a construction of classical shaded A-polynomials $A_a({\ell }_b,m_c)$ associated to Lie groups $SU(N)$, assuming that color $N$ adds some extra “depth” and “shade” to the classical notion of A-polynomials. We exploit a formalism of Clebsch-Gordan (CG) chords, where indices $a$, $b$, $c$ run over $1,\dots ,N-1$. CG chords have a natural interpretation in terms of 2D conformal field theories of Wess–Zumino–Witten type, or, alternatively, in terms of quantum group $U_q({\mathfrak{su}}_N)$. In the case of $\mathfrak{s}{\mathfrak{u}}_2$, CG chords could be associated to Reeb chords in a knot contact homology (KCH) framework. KCH suggests its own analog of A-polynomials, known as augmentation polynomials, which is allowed to have extra spurious roots in principle. Yet the CG chord formalism could be easily extended to arbitrary ${\mathfrak{su}}_N$ allowing us to generalize the construction of A(ugmentation)-polynomials to arbitrary ${\mathfrak{su}}_N$ and arbitrary representations as well. Here we primarily aim at classical A-polynomials by considering a double scaling limit in which $q=e^{\hslash }$, $\hslash \to 0$ and the representations are huge: highest weight vector components $w_i\to \infty$ while combinations $\hslash w_i\sim m_i$ remain finite. Still, we expect the presented techniques will be helpful in deriving quantum A-polynomials for arbitrary Lie (super)algebras $\mathfrak{g}$. Also, we discuss explicit examples of A-polynomials for knots $3_1$, $4_1$ and $5_1$ for $\mathfrak{g}=\mathfrak{s}{\mathfrak{u}}_3$.

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• Chern-Simons gauge theory
• Strings & branes
• Topological field theories
• Conformal symmetry
• Quantum group symmetries
• Group theory
• Topology