  This paper presents, by example, an index theory appropriate to algebras
without trace. Whilst we work exclusively with the Cuntz algebras the
exposition is designed to indicate how to develop a general theory. Our main
result is an index theorem (formulated in terms of spectral flow) using a
twisted cyclic cocycle where the twisting comes from the modular automorphism
group for the canonical gauge action on the Cuntz algebra. We introduce a
modified $K_1$-group of the Cuntz algebra so as to pair with this twisted
cocycle. As a corollary we obtain a noncommutative geometry interpretation for
Araki's notion of relative entropy in this example. We also note the connection
of this example to the theory of noncommutative manifolds.
