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${\rm SL}_2$ quantum trace in quantum Teichmüller theory via writhe

Hyun Kyu Kim, Thang T Q Lê and Miri Son

Algebraic & Geometric Topology 23 (2023) 339–418
Bibliography
1 D G L Allegretti, The geometry of cluster varieties from surfaces, PhD thesis, Yale University (2016) MR3553669
2 D G L Allegretti, H K Kim, A duality map for quantum cluster varieties from surfaces, Adv. Math. 306 (2017) 1164 MR3581328
3 F Bonahon, X Liu, Representations of the quantum Teichmüller space and invariants of surface diffeomorphisms, Geom. Topol. 11 (2007) 889 MR2326938
4 F Bonahon, H Wong, Quantum traces for representations of surface groups in SL2(), Geom. Topol. 15 (2011) 1569 MR2851072
5 F Bonahon, H Wong, Representations of the Kauffman bracket skein algebra II : Punctured surfaces, Algebr. Geom. Topol. 17 (2017) 3399 MR3709650
6 K A Brown, K R Goodearl, Lectures on algebraic quantum groups, Birkhäuser (2002) MR1898492
7 S Y Cho, H Kim, H K Kim, D Oh, Laurent positivity of quantized canonical bases for quantum cluster varieties from surfaces, Comm. Math. Phys. 373 (2020) 655 MR4056646
8 P M Cohn, Skew fields: theory of general division rings, 57, Cambridge Univ. Press (1995) MR1349108
9 F Costantino, T T Q Lê, Stated skein algebras of surfaces, J. Eur. Math. Soc. 24 (2022) 4063 MR4493620
10 V V Fock, Dual Teichmüller spaces, preprint (1997) arXiv:dg-ga/9702018
11 V V Fock, L O Chekhov, Quantum Teichmüller spaces, Teoret. Mat. Fiz. 120 (1999) 511 MR1737362
12 V Fock, A Goncharov, Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes Études Sci. 103 (2006) 1 MR2233852
13 V V Fock, A B Goncharov, Dual Teichmüller and lamination spaces, from: "Handbook of Teichmüller theory, I" (editor A Papadopoulos), IRMA Lect. Math. Theor. Phys. 11, Eur. Math. Soc. (2007) 647 MR2349682
14 V V Fock, A B Goncharov, The quantum dilogarithm and representations of quantum cluster varieties, Invent. Math. 175 (2009) 223 MR2470108
15 S Fomin, M Shapiro, D Thurston, Cluster algebras and triangulated surfaces, I : Cluster complexes, Acta Math. 201 (2008) 83 MR2448067
16 M Gabella, Quantum holonomies from spectral networks and framed BPS states, Comm. Math. Phys. 351 (2017) 563 MR3613514
17 D Gaiotto, Surface operators in 𝒩 = 2 4d gauge theories, J. High Energy Phys. (2012) 90 MR3036474
18 D Gaiotto, G W Moore, A Neitzke, Wall-crossing in coupled 2d-4d systems, J. High Energy Phys. (2012) 82 MR3045271
19 D Gaiotto, G W Moore, A Neitzke, Spectral networks, Ann. Henri Poincaré 14 (2013) 1643 MR3115984
20 D Gaiotto, G W Moore, A Neitzke, Spectral networks and snakes, Ann. Henri Poincaré 15 (2014) 61 MR3147409
21 D Galakhov, P Longhi, G W Moore, Spectral networks with spin, Comm. Math. Phys. 340 (2015) 171 MR3395151
22 M Gross, P Hacking, S Keel, Birational geometry of cluster algebras, Algebr. Geom. 2 (2015) 137 MR3350154
23 C Hiatt, Quantum traces in quantum Teichmüller theory, Algebr. Geom. Topol. 10 (2010) 1245 MR2661526
24 R M Kashaev, Quantization of Teichmüller spaces and the quantum dilogarithm, Lett. Math. Phys. 43 (1998) 105 MR1607296
25 H K Kim, C Scarinci, A quantization of moduli spaces of 3–dimensional gravity, preprint (2021) arXiv:2112.13329
26 H K Kim, M Son, SL2 quantum trace in quantum Teichmüller theory via writhe, (2018) arXiv:1812.11628v1
27 T T Q Lê, Triangular decomposition of skein algebras, Quantum Topol. 9 (2018) 591 MR3827810
28 T T Q Lê, Quantum Teichmüller spaces and quantum trace map, J. Inst. Math. Jussieu 18 (2019) 249 MR3915288
29 X Liu, The quantum Teichmüller space as a noncommutative algebraic object, J. Knot Theory Ramifications 18 (2009) 705 MR2527682
30 T Ohtsuki, Quantum invariants : a study of knots, 3–manifolds, and their sets, 29, World Sci. (2002) MR1881401
31 R C Penner, Decorated Teichmüller theory, Eur. Math. Soc. (2012) MR3052157
32 J H Przytycki, Fundamentals of Kauffman bracket skein modules, Kobe J. Math. 16 (1999) 45 MR1723531
33 J H Przytycki, A S Sikora, On skein algebras and Sl2(C)–character varieties, Topology 39 (2000) 115 MR1710996
34 N Y Reshetikhin, V G Turaev, Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127 (1990) 1 MR1036112
35 L Shen, Duals of semisimple Poisson–Lie groups and cluster theory of moduli spaces of G-local systems, Int. Math. Res. Not. (2022) 14295 MR4485958
36 M Son, Quantum coordinate change map for Chekhov–Fock square root algebras, master’s thesis, Ewha Womans University (2020)
37 W P Thurston, The geometry and topology of three-manifolds, lecture notes (1980)
38 V G Turaev, Algebras of loops on surfaces, algebras of knots, and quantization, from: "Braid group, knot theory and statistical mechanics" (editors C N Yang, M L Ge), Adv. Ser. Math. Phys. 9, World Sci. (1989) 59 MR1062423
39 V G Turaev, Quantum invariants of knots and 3–manifolds, 18, de Gruyter (2016) MR3617439