Representations of the Kauffman bracket skein algebra I: invariants and miraculous cancellations

We study finite-dimensional representations of the Kauffman skein algebra of a surface S. In particular, we construct invariants of such irreducible representations when the underlying parameter q is a root of unity. The main one of these invariants is a point in the character variety consisting of group homomorphisms from the fundamental group of S to SL_2(C), or in a twisted version of this character variety. The proof relies on certain miraculous cancellations that occur for the quantum trace homomorphism constructed by the authors. These miraculous cancellations also play a fundamental role in subsequent work of the authors, where novel examples of representations of the skein algebra are constructed.Comment: Version 3: Improvements in the exposition following referee reports. This version also fixes a small gap in the proof of the miraculous cancellations of Theorems 4 and 21, originally caused by an incorrect interpretation of the reference [Bu] used to create a shortcut in the computations; the results are unchanged, and the modifications to the proof very minima

Quantum traces for representations of surface groups in SL_2

We consider two different quantizations of the character variety consisting of all representations of surface groups in SL_2. One is the skein algebra considered by Przytycki-Sikora and Turaev. The other is the quantum Teichmuller space introduced by Chekhov-Fock and Kashaev. We construct a homomorphism from the skein algebra to the quantum Teichmuller space which, when restricted the classical case, corresponds to the equivalence between these two algebras through trace functions.Comment: 40 pages, 25 figures. Version 2: Fixed classical case. Misprints corrected. Version 3: More misprints corrected, including statement of Lemma 22. Added observation that the quantum trace homomorphism is injective. Version 4: Final corrections before submissio

The Witten-Reshetikhin-Turaev representation of the Kauffman skein algebra

For A a primitive 2N-root of unity with N odd, the Witten-Reshetikhin-Turaev topological quantum field theory provides a representation of the Kauffman skein algebra of a closed surface. We show that this representation is irreducible and we compute its classical shadow, in the sense of earlier work of the authors (arXiv:1206.1638).Comment: 14 pages. Version 2: Minor revisions prior to submissio

Representations of the Kauffman bracket skein algebra II: punctured surfaces

In earlier work, we constructed invariants of irreducible representations of the Kauffman 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 will be lifted in subsequent work of the authors that relies on this one. A step in the proof is of independent interest, and describes the algebraic structure of the Thurston intersection form on the space of integer weight systems for a train track.Comment: 22 pages. Version 2: The article was much reorganized, for compatibility with the subsequent article [BonWon6] in the same series; the results are unchanged. Version 3: This new version takes into account the possible impact of sign reversal symmetries, overlooked in the earlier versions, on the uniqueness properties needed for [BonWon6]; the manuscript is now ready for journal submissio

The development and year one implementation of the Local Justice Reinvestment Pilot

This report focuses on the initial findings from a process evaluation of the Local Justice Reinvestment pilot (commissioned by the Ministry of Justice), which examines the early development and implementation of the pilot in the first test year. The pilot is one of the Ministry of Justice Payment by Results (PbR) schemes. The methodology was primarily qualitative and included: interviews with strategic and operational managers; interviews and focus groups with front line staff; workshops to map partnership and criminal justice system changes and a focus on exemplar interventions at three sites