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.
"Representations of the Kauffman bracket skein algebra, II: Punctured surfaces." Algebr. Geom. Topol. 17 (6) 3399 - 3434, 2017. https://doi.org/10.2140/agt.2017.17.3399