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Representations of the Kauffman bracket skein algebra. II: Punctured surfaces. (English) Zbl 1422.57032

Summary: In Part I [the authors, Invent. Math. 204, No. 1, 195–243 (2016; Zbl 1383.57015)], 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.

MSC:

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57R56 Topological quantum field theories (aspects of differential topology)

Citations:

Zbl 1383.57015
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