  Luo and Tan gave a new identity for hyperbolic surfaces with/without geodesic
boundary in terms of dilogarithms of the lengths of simple closed geodesics on
embedded three-holed spheres or one-holed tori. However, the identity was
trivial for a hyperbolic one-holed torus with geodesic boundary. In this paper
we adapt the argument from Luo and Tan to give an identity for hyperbolic tori
with one geodesic boundary or cusp in terms of dilogarithm functions on the set
of lengths of simple closed geodesics on the torus. As a corollary, we are also
able to express the Luo-Tan identity as a sum over all immersed three-holed
spheres $P$ which are embeddings when restricted to the interior of $P$.

Geometric Topology;Algebraic Geometry;Complex Variables;Differential Geometry