Proof of a stronger version of the AJ conjecture for torus knots (Q1945744)

For a knot \(K\) in the three-sphere \(S^3\), let \(J_K(n)\in\mathbb{Z}[t,t^{-1}]\) be the colored Jones polynomial associated with the \(n\)-dimensional irreducible representation of \(sl_2(\mathbb{C})\). We can regard \(J_K\) as a function \(J_K:\mathbb{Z}\to\mathbb{C}[t,t^{-1}]\). Put \(\mathcal{T}:=\mathbb{C}[t,t^{-1}]\langle{M,M^{-1},L,L^{-1}}\rangle/(LM-t^2ML)\) and define \(\mathcal{A}_K:=\{P\in\mathcal{T}\mid PJ_K=0\}\), where \(L\) and \(M\) act on a function \(f:\mathbb{Z}\to\mathbb{C}[t,t^{-1}]\) as \((Lf)(n):=f(n+1)\) and \((Mf)(n):=t^{2n}f(n)\). It is called the recurrence ideal of \(K\). By adding inverses of polynomials in \(t\) and \(M\), \(\mathcal{T}\) becomes a principal left ideal domain \(\tilde{\mathcal{T}}\). Then we can choose a generator \(\alpha_K(t;M,L)\in\mathcal{A}_K\) of \(\tilde{\mathcal{T}}\mathcal{A}_K\) so that its degree in \(L\) is minimal, and that the coefficients of \(L^i\) are in \(\mathbb{Z}[t,t^{-1},M]\) and coprime. We call \(\alpha_K(t;M,L)\) the recurrence polynomial of \(K\) and it is defined up to a polynomial in \(\mathbb{Z}[t,t^{-1},M]\). Note that \(\alpha_K(t;M,L)\) gives a recurrence relation for \(J_K(n)\). Now the AJ conjecture, cf. \textit{S. Garoufalidis} [Coventry: Geometry and Topology Publications. Geometry and Topology Monographs 7, 291-309 (2004; Zbl 1080.57014)], states that \(\varepsilon\bigl(\alpha_K(t;M,L)\bigr)\) would equal the A-polynomial, cf. \textit{D. Cooper} et al., [Invent. Math. 118, No. 1, 47--84 (1994; Zbl 0842.57013)], up to a polynomial in \(M\), where \(\varepsilon\) is the evaluation map at \(t=-1\). So far it has been proved for some classes of two-bridge knots and pretzel knots, \textit{T. T. Q. Lê} [Adv. Math. 207, No. 2, 782--804 (2006; Zbl 1114.57014)], \textit{T. T. Q. Lê} and \textit{A. T. Tran} [``On the AJ conjecture for knots'', to appear in Trans. Am. Math. Soc., \url{arXiv:1111.5258}]. Sikora proposes a stronger version of the AJ conjecture as follows [\textit{A. S. Sikora}, ``Quantizations of Character Varieties and Quantum Knot Invariants'', \url{arXiv:0807.0943}]. Let \(\sigma\) be the involution on \(\mathcal{T}\) defined as \(\sigma(M^kL^l):=M^{-k}L^{-l}\). Denote by \(\mathfrak{p}\) the kernel of the map \(\mathbb{C}\left[\chi\bigl(\partial(S^3\setminus{K})\bigr)\right]\to\mathbb{C}\left[\chi(S^3\setminus{K})\right]\) induced by the inclusion, where \(\chi(Y)\) is the \(SL_2(\mathbb{C})\)-character variety of \(\pi_1(Y)\). Note that \(\mathfrak{p}\) determines the A-polynomial. Then Sikora conjectures that \(\mathfrak{p}\) coincides with the radical of \(\varepsilon\left(\mathcal{A}_K(t;M,L)^{\sigma}\right)\) in \(\mathbb{C}\left[\chi\bigl(\partial(S^3\setminus{K})\bigr)\right]\), which is the \(\sigma\)-invariant subspace of \(\mathbb{C}[M,M^{-1},L,L^{-1}]\). Here \(\mathcal{A}_K(t;M,L)^{\sigma}\) is the \(\sigma\)-invariant part of the recurrence ideal \(\mathcal{A}_K(t;M,L)\). Sikora proves that his conjecture implies the AJ conjecture and that his conjecture holds for the unknot and the trefoil. In the paper under review the author proves Sikora's conjecture for torus knots. Note that the AJ conjecture for torus knots was checked by \textit{K. Hikami} [Int. J. Math. 15, No. 9, 959-965 (2004; Zbl 1060.57012)] though a full proof was not given.