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

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Abstract

We study finite-dimensional representations of the Kauffman bracket skein algebra of a surface S. In particular, we construct invariants of such irreducible representations when the underlying parameter \(q={{\mathrm {e}}}^{2\pi {{\mathrm {i}}}\hbar }\) 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 \(\pi _1(S)\) to \({{\mathrm {SL}}}_2(\mathbb {C})\), 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.

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Notes

  1. However, see [20] for an earlier occurrence, as well as [22] for a related but different context.

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Acknowledgments

The authors are grateful to the referees for several useful suggestions. See in particular Remark 5.

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Authors and Affiliations

  1. Department of Mathematics, University of Southern California, Los Angeles, CA, 90089-2532, USA

    Francis Bonahon

  2. Department of Mathematics, Carleton College, Northfield, MN, 55057, USA

    Helen Wong

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  1. Francis Bonahon
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  2. Helen Wong
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Correspondence to Francis Bonahon.

Additional information

This research was partially supported by Grants DMS-0604866, DMS-1105402 and DMS-1105692 from the National Science Foundation, and by a mentoring grant from the Association for Women in Mathematics.

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Bonahon, F., Wong, H. Representations of the Kauffman bracket skein algebra I: invariants and miraculous cancellations. Invent. math. 204, 195–243 (2016). https://doi.org/10.1007/s00222-015-0611-y

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