A vanishing identity of adjoint Reidemeister torsions of twist knots (Q2133954)

\textit{J. Porti} [Torsion de Reidemeister pour les variétés hyperboliques. Providence, RI: American Mathematical Society (AMS) (1997; Zbl 0881.57020)] defined for a (hyperbolic) \(3\)-manifold \(M\) with one cusp and a fixed non homologically-trivial boundary curve \(\gamma\), an \emph{adjoint torsion}, \(\tau_\gamma(\chi_\rho)\), associated to a generic irreducible \(SL_2\)-representation \(\rho\), with character \(\chi_\rho\). This is basically the Reidemeister torsion twisted by the adjoint representation and can be seen as a meromorphic function on the one-dimesional components of the character variety corresponding to irreducible representations. \par In the paper under review the following conjecture is discussed. Assuming that the character variety of a hyperbolic manifold \(M\) is one-dimensional, for a generic value \(C\), the sum over all characters \(\chi_\rho\) such that \(\chi_\rho(\gamma)=C\) of the inverses \(1/\tau_\gamma(\chi_\rho)\) of the adjoint torsions vanishes. \par The author shows that the conjecture holds true for all hyperbolic twists knots. It should be noted that the hypothesis of the conjecture is fulfilled by the larger class of all (hyperbolic) \(2\)-bridge knots. The author remarks that the hyperbolicity hypothesis is necessary, as the conjecture fails for the \((2,3)\)-torus knot which is a non-hyperbolic twist knot. \par The proof is carried out by direct computations exploiting Jacobi's residue theorem and the fact that, for twist knots, the character variety can be explicitely computed in terms of Chebyshev polynomials. \par Finally, the author observes that his result might be obtained as a corollary of the global residue theorem of complex analysis, provided one could prove a genericity condition for a family of pairs of rational fuctions in two variables.