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QUANTUM TEICHMÜLLER SPACES AND QUANTUM TRACE MAP

Part of: Low-dimensional topology Topological manifolds

Published online by Cambridge University Press:  06 April 2017

Thang T. Q. Lê
Thang T. Q. Lê*
Affiliation:
School of Mathematics, 686 Cherry Street, Georgia Tech, Atlanta, GA 30332, USA ([email protected])
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Abstract

We show how the quantum trace map of Bonahon and Wong can be constructed in a natural way using the skein algebra of Muller, which is an extension of the Kauffman bracket skein algebra of surfaces. We also show that the quantum Teichmüller space of a marked surface, defined by Chekhov–Fock (and Kashaev) in an abstract way, can be realized as a concrete subalgebra of the skew field of the skein algebra.

Keywords

Kauffman bracket skein modulequantum Teichmüller spacequantum trace map

MSC classification

Primary: 57N10: Topology of general $3$-manifolds 57M25: Knots and links in $S^3$
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 18 , Issue 2 , March 2019 , pp. 249 - 291
Copyright
© Cambridge University Press 2017 

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Footnotes

Supported in part by National Science Foundation.

References

Bai, H., A uniqueness property for the quantization of Teichmüller spaces, Geom. Dedicata 128 (2007), 1–16.Google Scholar
Blanchet, C., Habegger, N., Masbaum, G. and P., Vogel, Topological quantum field theories derived from the Kauffman bracket, Topology 34 (1995), 883–927.Google Scholar
Bonahon, F. and Liu, X., Representations of the quantum Teichmüller space and invariants of surface diffeomorphisms, Geom. Topol. 11 (2007), 889–937.Google Scholar
Bonahon, F. and Wong, H., Quantum traces for representations of surface groups in SL2(C), Geom. Topol. 15(3) (2011), 1569–1615.Google Scholar
Bonahon, F. and Wong, H., Representations of the Kauffman skein algebra I: invariants and miraculous cancellations, Invent. Math. 204 (2016), 195–243.Google Scholar
Bullock, D., Estimating a skein module with SL2(C) characters, Proc. Amer. Math. Soc. 125 (1997), 1835–1839.Google Scholar
Bullock, D., Rings of Sl 2(ℂ)-characters and the Kauffman bracket skein module, Comment. Math. Helv. 72(4) (1997), 521–542.Google Scholar
Bullock, D., Frohman, C. and Kania-Bartoszynska, J., Understanding the Kauffman bracket skein module, J. Knot Theory Ramifications 8(3) (1999), 265–277.Google Scholar
Chekhov, L. O. and Fock, V. V., Quantum Teichmüller spaces, Teoret. Mat. Fiz. 120(3) (1999), 511–528. (Russian); Theoret. Math. Phys. 120(3) (1999), 1245–1259 (translation).Google Scholar
Chekhov, L. O. and Fock, V. V., Observables in 3D gravity and geodesic algebras, in: Quantum groups and integrable systems (Prague, 2000, Czechoslovak), J. Phys. 50 (2000), 1201–1208.Google Scholar
Chekhov, L. O. and Penner, R. C., Introduction to Thurston’s quantum theory, Uspekhi Mat. Nauk 58 (2003), 93–138.Google Scholar
Fock, V. V., Dual Teichmüller spaces, unpublished, preprint, 1997, arXiv:Math/dg-ga/9702018.Google Scholar
Fomin, S., Shapiro, M. and Thurston, D., Cluster algebras and triangulated surfaces. I. Cluster complexes, Acta Math. 201 (2008), 83–146.Google Scholar
Freedman, M., Hass, J. and Scott, P., Closed geodesics on surfaces, Bull. Lond. Math. Soc. 14 (1982), 385–391.Google Scholar
Goodearl, K. R. and Warfield, R. B., An Introduction to Noncommutative Noetherian Rings, second edition, London Mathematical Society Student Texts, Volume 61 (Cambridge University Press, Cambridge, 2004).Google Scholar
Hiatt, C., Quantum traces in quantum Teichmüller theory, Algebr. Geom. Topol. 10 (2010), 1245–1283.Google Scholar
Kashaev, R., Quantization of Teichmüller spaces and the quantum dilogarithm, Lett. Math. Phys. 43(2) (1998), 105–115.Google Scholar
Kauffman, L., States models and the Jones polynomial, Topology 26 (1987), 395–407.Google Scholar
Lê, T. T. Q., The colored Jones polynomial and the A-polynomial of knots, Adv. Math. 207(2) (2006), 782–804.Google Scholar
Lê, T. T. Q. and Paprocki, J., to appear.Google Scholar
Lê, T. T. Q. and Tran, A., On the AJ conjecture for knots, Indiana Univ. Math. J. 64 (2015), 1103–1151.Google Scholar
Lê, T. T. Q. and Zhang, X., Character varieties, A-polynomials, and the AJ Conjecture, Algebr. Geom. Topol. 17(1) (2017), 157–188.Google Scholar
Liu, X., The quantum Teichmüller space as a noncommutative algebraic object, J. Knot Theory Ramifications 18 (2009), 705–726.Google Scholar
Muller, G., Skein algebras and cluster algebras of marked surfaces, Quantum Topol. 7 (2016), 435–503.Google Scholar
Penner, R. C., Decorated Teichmüller Theory, with a Foreword by Yuri I. Manin, QGM Master Class Series (European Mathematical Society, Zürich, 2012).Google Scholar
Przytycki, J., Fundamentals of Kauffman bracket skein modules, Kobe J. Math. 16 (1999), 45–66.Google Scholar
Przytycki, J. and Sikora, A., On the skein algebras and Sl 2(ℂ)-character varieties, Topology 39 (2000), 115–148.Google Scholar
Sikora, A. and Westbury, B. W., Confluence theory for graphs, Algebr. Geom. Topol. 7 (2007), 439–478.Google Scholar
Turaev, V., Skein quantization of Poisson algebras of loops on surfaces, Ann. Sci. Éc. Norm. Supér. (4) 24(6) (1991), 635–704.Google Scholar