Let $ M $ be a complete finite-volume hyperbolic 3-manifold
which fibres over the circle with fibre a punctured torus.
Let $ S $ be the set of closed geodesics in $ M $ which are homotopic
to a simple closed curve in the fibre.
We show that
$ \sum_{s \in S} {1 \over 1+e^{l(s)}} = 0 $
where $ l(s) $ denotes the complex length of the closed geodesic $ s $
(i.e. the real hyperbolic length plus $ i $ times the rotational part).
The set $ S $ can be naturally partitioned into two disjoint subsets,
depending on how a curve in $ S $ crosses the stable and unstable foliations
of the fibre.
If we restrict the above sum to one of these subsets, we obtain
(up to sign) the complex modulus of the cusp of $ M $.
These formulae are variations of McShane's identity for a puctured torus,
and are proved here using Markoff triples.
Topology 36 (1997) 325-334.