New identities for small hyperbolic surfaces

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 in Luo and Tan [‘A dilogarithm identity on moduli spaces of curves’, J.
Differential Geom., Preprint, 2011, arXiv:1102.2133[math.GT]]. 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