\(L^{2}\)-Alexander invariant for torus knots (Q611163)

In this short note the authors explicitly calculate the \(L^2\)-Alexander invariant of torus knots \(T(p,q)\), see [\textit{W. Li} and \textit{W. Zhang}, Commun. Contemp. Math. 8, No.~2, 167--187 (2006; Zbl 1104.57008); in: Ge, Mo-Lin (ed.) et al., Differential geometry and physics. Proceedings of the 23rd international conference of differential geometric methods in theoretical physics, Tianjin, China, August 20--26, 2005. Hackensack, NJ: World Scientific. Nankai Tracts in Mathematics 10, 303--312 (2006); Zbl 1131.57015)] for an introduction to \(L^2\)-Alexander invariants. The key result in a forthcoming paper by the authors is that the calculation can be done by using the standard presentation of the fundamental group of the \(T(p,q)\) torus knot complement \((x,y|x^p=y^q)\) which is not a Wirtinger presentation of the a diagram of \(T(p,q)\).