A variation of McShane's identity for 2–bridge links

Abstract

We give a variation of McShane’s identity, which describes the cusp shape of a hyperbolic $2$–bridge link in terms of the complex translation lengths of simple loops on the bridge sphere. We also explicitly determine the set of end invariants of $SL\left(2,ℂ\right)$–characters of the once-punctured torus corresponding to the holonomy representations of the complete hyperbolic structures of $2$–bridge link complements.