The main result classifies “almost all” the exceptional n-cycles in Kb(A-proj), using characteristic components and their AG-invariants, except those exceptional 1-cycles which are band complexes. The Hom spaces between string complexes at the mouth are explicitly determined. Let A be an indecomposable gentle k-algebra with A≠k. In this paper we show that if T is homotopy-like, then any exceptional 1-cycle is indecomposable and at the mouth (i.e., the middle term of the Auslander-Reiten triangle ending at it is indecomposable) and any object in an exceptional n-cycle with n≥3 is at the mouth. Ploog, have a notable impact on the global structure of T. Exceptional cycles in a triangulated category T with Serre duality, introduced by N.
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