On the relation between the A-polynomial and the Jones polynomial (Q2781276)

Let \(M\) be a three-manifold and \(K_{t}(M)\) be its Kauffman bracket skein module, that is, the \(\mathbb{C}[t,t^{-1}]\)-module generated by the isotopy classes of framed links in \(M\) modulo the relations of the Kauffman bracket. NEWLINENEWLINENEWLINEIf \(K\) is a knot in the three-sphere and \(M\) is its complement, then \(K_{t}(T^{2}\times{I})\) acts from the left on \(K_{t}(M)\) with \(T^{2}\) a torus. The peripheral ideal \(I_{t}(K)\) is defined to be the left ideal of \(K_{t}(T^{2}\times{I})\) annihilating the empty link \(\emptyset\) in \(M\). The \(A\)-ideal, which is shown to be a generalization of the \(A\)-polynomial, can be defined by \(I_{t}(K)\) [\textit{C. Frohman, R. Gelca}, and \textit{W. LoFaro}, Trans. Am. Math. Soc. 354, No. 2, 735-747 (2002; Zbl 0980.57002)]. Here the \(A\)-polynomial is a two-variable polynomial invariant of a knot defined by using the character variety of \(SL(2;\mathbb{C})\)-representations of \(\pi_{1}(M)\) [\textit{D. Cooper, M. Culler, H. Gillet, D. D. Long} and \textit{P. B. Shalen}, Invent. Math. 118, No. 1, 47-84 (1994; Zbl 0842.57013)]. NEWLINENEWLINENEWLINEA pairing \(K_{t}(D^{2}\times S^{1})\times K_{t}(M)\to\mathbb{C}[t,t^{-1}]\) is defined by glueing the solid torus \(D^{2}\times S^{1}\) to the knot complement \(M\). Let \(S_{n}(\alpha)\) be the skein obtained as a Chebyshev polynomial of \(\alpha=\{0\}\times S^{1}\subset D^{2}\times S^{1}\). Then \(\langle S_{n}(\alpha),\emptyset\rangle\) defines the \(n\)th colored Kauffman bracket, a version of the colored Jones polynomial. NEWLINENEWLINENEWLINEUsing these facts the author proves that for a knot \(K\) and a nonzero element \(\psi\in I_{t}(K)\) there exists a number \(\nu\) such that the first \(\nu\) colored Kauffman brackets of \(K\) and \(\psi\) determine all the other colored Kauffman brackets. He also gives a technical condition that the \(A\)-ideal of a knot determines all the Kauffman brackets. As an example a recursive formula for the colored Kauffman brackets of the trefoil knot is given.