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Representations of the Kauffman bracket skein algebra, II: Punctured surfaces

Francis Bonahon and Helen Wong

Algebraic & Geometric Topology 17 (2017) 3399–3434

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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Kauffman bracket, skein algebra, quantum Teichmüller space
Mathematical Subject Classification 2010
Primary: 57M27, 57R56
Secondary: 57M27
Received: 8 March 2016
Revised: 27 September 2016
Accepted: 25 April 2017
Published: 4 October 2017
Francis Bonahon
Department of Mathematics
University of Southern California
Los Angeles, CA
United States
Helen Wong
Department of Mathematics
Carleton College
Northfield, MN
United States