  Exceptional cycles in a triangulated category $\mathcal T$ with Serre
duality, introduced by N. Broomhead, D. Pauksztello, and D. Ploog, have a
notable impact on the global structure of $\mathcal T$. In this paper we show
that if $\mathcal T$ is homotopy-like, then any exceptional $1$-cycle is
indecomposable and at the mouth; and any object in an exceptional $n$-cycle
with $n\ge 3$ is at the mouth. Let $A$ be an indecomposable gentle $k$-algebra
with $A\ne k$. The Hom spaces between string complexes at the mouth are
explicitly determined. The main result classifies "almost all" the exceptional
cycles in $K^b(A\mbox{-}{\rm proj})$, using characteristic components and their
AG-invariants, except those exceptional $1$-cycles which are band complexes.
Namely, the mouth of a characteristic component $C$ of $K^b(A\mbox{-}{\rm
proj})$ forms a unique exceptional cycle in $C$, up to an equivalent relation
$\approx$; if the quiver of $A$ is not of type $A_3$, this gives all the
exceptional $n$-cycle in $K^b(A\mbox{-}{\rm proj})$ with $n\ge 2$, up to
$\approx$; and a string complex is an exceptional $1$-cycle if and only if it
is at the mouth of a characteristic component with {\rm AG}-invariant $(1, m)$.
However, a band complex at the mouth is possibly not an exceptional $1$-cycle.
