A variation of McShane's identity for 2-bridge links (Q357736)

In 1991, G.\ McShane found a remarkable identity concerning the lengths of closed geodesics on a hyperbolic once-punctured torus, which is now called McShane's identity. The identity holds for every complete hyperbolic metric on a once-punctured torus. It has a form as follows: NEWLINE\[NEWLINE\sum_{\gamma \in \mathcal{C}} \frac{1}{1+\exp{l(\gamma)}}=\frac{1}{2},NEWLINE\]NEWLINE where \(\mathcal C\) denotes the set of isotopy classes of non-peripheral simple closed on once-punctured torus, and \(l(\gamma)\) denotes the geodesic length with respect to the given hyperbolic metric. This identity has been generalised to fit into various settings by McShane himself, Bowditch, Mirzakhani, Tan-Wong-Zhang, and Akiyoshi-Miyachi-Sakuma.NEWLINENEWLINEIn the present paper, the authors find a variation of McShane's identity for a hyperbolic two-bridge link complement. The identity for a hyperbolic manifold which is the complement of a two-bridge link of slope \(r\) looks like this: NEWLINE\[NEWLINE\begin{multlined} 2 \sum_{s \in \hat{{\mathbf Q}} \cap \roman{int} I_1(r)}\frac{1}{1+\exp(l_{\rho_r}(\beta_s))}+2\sum_{s \in \hat{{\mathbf Q}} \cap \roman{int} I_2(r)} \frac{1}{1+\exp(l_{\rho_r}(\beta_s))}\\ +\sum_{s \in \partial I_1(r) \cup \partial I_2(r)}\frac{1}{1+\exp(l_{\rho_r}(\beta_s))}=-1.\end{multlined}NEWLINE\]NEWLINE Here is an explanation of symbols: \(l_{\rho_r}(\beta_s)\) denotes the complex length of the closed geodesic in the hyperbolic manifold which is homotopic to a simple closed curve with slope \(s\) on the Conway sphere, and \(I_1(r),I_2(r)\) are intervals in \(\hat{\mathbf Q}\) which are proved to constitute representatives for homotopy classes of closed curves in the hyperbolic manifold homotopic to simple closed curves on the Conway sphere. The authors also give a formula which describes the cusp form of the manifold in terms of the complex lengths as appeared in the above identity.