On the integral of \(\log x\frac{dy}{y}-\log y\frac{dx}{x}\) over the A-polynomial curves (Q2390681)

\textit{D. W. Boyd} [in: Bennett, M. A. (ed.) et al., Number theory for the millennium I. Proceedings of the millennial conference on number theory, Urbana-Champaign, IL, USA, May 21-26, 2000. Natick, MA: A K Peters. 127-143 (2002; Zbl 1030.11055)] and \textit{D. W. Boyd} and \textit{F. Rodriguez-Villegas} [Mahler's Measure and the Dilogarithm. II. arXiv:math/0308041v2 (math.NT)] (with an appendix by N. Dunfield)] showed connections between the volume of some hyperbolic knot complements and the logarithmic Mahler measure of the \(A\)-polynomial, an invariant defined by \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)]. The author proposes to study the arithmetic properties of the Chern-Simons invariant, a complex counterpart of the hyperbolic 3-volume. More precisely, the author considers the integration of \(\log x \frac{dy}{y}-\log y \frac{dx}{x}\) over certain paths in the \(A\)-polynomial curve. This number should correspond to a complex companion for the Mahler measure. The author uses his result on the Godbillon-Vey invariant, cf. \textit{Vu The Khoi} [Math. Ann. 326, No.~4, 759--801 (2003; Zbl 1035.57008)], to compute this integral in some cases. The numbers that appear typically are rational multiples of \(\pi^2\).