An amalgamation or self-amalgamation of handlebodies along an incompressible surface is a handlebody exactly when a primitive dual curve-disk system exists.
Gordon, On primitive sets of loops in the boundary of a handlebody
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Amalgamations along surfaces with boundary in a handlebody
An amalgamation or self-amalgamation of handlebodies along an incompressible surface is a handlebody exactly when a primitive dual curve-disk system exists.