Higher even dimensional Reidemeister torsion for torus knot exteriors (Q2845625)

This is a paper on the Reidemeister torsion of torus knots with representations in \(\mathrm{SL}(2 n,\mathbb C)\), more precisely on its asymptotic behavior as \(n\to\infty\).NEWLINENEWLINETo motivate this research, we mention a result of \textit{W. Müller} [Progress in Mathematics 297, 317--352 (2012; Zbl 1264.58026)]. Müller composes (a lift) of the holonomy representation of a closed orientable hyperbolic three manifold in \(\mathrm{SL}(2,\mathbb C)\) with the \(n\)-th symmetric power, hence obtaining a representation in \(\mathrm{SL}(n+1,\mathbb C)\), and studies the behaviour of the modulus of its torsion as \(n\) goes to infinity. He proves that the logarithm of the modulus of this torsion divided by \(n^2\) equals precisely the volume divided by \(-4\pi\). Pere Menal-Ferrer and the reviewer proved that the formula of Müller also applies to non compact hyperbolic three manifolds of finite volume, in particular to hyperbolic knot exteriors [to appear in J. Topol. {\texttt{DOI:10.1112/jtopol/jtt024}}].NEWLINENEWLINEThe paper under review proves that the formula of Müller can be extended to torus knot exteriors, by replacing the volume by zero, because torus knot exteriors are Seifert fibered. For torus knots there is a distinguished representation in \(\mathrm{PSL}(2,\mathbb C)\), the holonomy of the basis 2-orbifold, that lifts to \(\mathrm{SL}(2,\mathbb C)\). Thus the author considers symmetric powers of this representation. To simplify, only odd symmetric powers are considered, namely only representations in \(\mathrm{SL}(2 n,\mathbb C)\).NEWLINENEWLINEThe formula is proved by explicit computations of the Reidemeister torsions. These might be compared to previous computations by other authors, like \textit{T. Kitano} in [Kobe J. Math. 13, No. 2, 133--144 (1996; Zbl 0876.57034)], or \textit{D. S. Freed} in [J. Reine Angew. Math. 429, 75--89 (1992; Zbl 0743.57015)]