In part I, we constructed invariants of irreducible finite-dimensional representations
of the Kauffman bracket skein algebra of a surface. We introduce here an inverse
construction, which to a set of possible invariants associates an irreducible
representation that realizes these invariants. The current article is restricted to
surfaces with at least one puncture, a condition that is lifted in subsequent work
relying on this one. A step in the proof is of independent interest, and describes the
arithmetic structure of the Thurston intersection form on the space of integer weight
systems for a train track.
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