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Skein algebras of surfaces. (English) Zbl 1464.57028

In this paper the authors prove that the Kauffman bracket skein algebra of an oriented surface has no zero divisors. It is important to note that the surface may have marked points on its boundary, and that the center of the skein algebra of the surface is generated by knots which are parallel to the unmarked components of the boundary of the surface. Finally, the authors prove that the skein algebras are Noetherian and Ore.

MSC:

57K31 Invariants of 3-manifolds (including skein modules, character varieties)
57M99 General low-dimensional topology
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[1] Abdiel, Nel; Frohman, Charles, The localized skein algebra is Frobenius, Algebr. Geom. Topol., 17, 6, 3341-3373 (2017) · Zbl 1421.57015
[2] Benyash-Krivets, V. V.; Chernousov, V. I., Varieties of representations of fundamental groups of compact nonoriented surfaces, Mat. Sb.. Sb. Math., 188 188, 7, 997-1039 (1997) · Zbl 0960.57001
[3] Bonahon, Francis; Wong, Helen, Quantum traces for representations of surface groups in \({\rm SL}_2(\mathbb{C})\), Geom. Topol., 15, 3, 1569-1615 (2011) · Zbl 1227.57003
[4] Bonahon, Francis; Wong, Helen, Kauffman brackets, character varieties and triangulations of surfaces. Topology and geometry in dimension three, Contemp. Math. 560, 179-194 (2011), Amer. Math. Soc., Providence, RI · Zbl 1333.57022
[5] F. Bonahon, H. Wong, Representations of the Kauffman bracket skein algebra III: closed surfaces and naturality, 2016, arXiv:1505.01522. · Zbl 1447.57017
[6] Bullock, Doug, Rings of \({\rm SL}_2({\bf C})\)-characters and the Kauffman bracket skein module, Comment. Math. Helv., 72, 4, 521-542 (1997) · Zbl 0907.57010
[7] Bullock, Doug; Frohman, Charles; Kania-Bartoszy\'nska, Joanna, Understanding the Kauffman bracket skein module, J. Knot Theory Ramifications, 8, 3, 265-277 (1999) · Zbl 0932.57015
[8] Brown, Ken A.; Goodearl, Ken R., Lectures on algebraic quantum groups, Advanced Courses in Mathematics. CRM Barcelona, x+348 pp. (2002), Birkh\"auser Verlag, Basel · Zbl 1027.17010
[9] Bullock, Doug; Przytycki, J\'ozef H., Multiplicative structure of Kauffman bracket skein module quantizations, Proc. Amer. Math. Soc., 128, 3, 923-931 (2000) · Zbl 0971.57021
[10] Fok, V. V.; Chekhov, L. O., Quantum Teichm\"uller spaces, Teoret. Mat. Fiz.. Theoret. and Math. Phys., 120 120, 3, 1245-1259 (1999) · Zbl 0986.32007
[11] Charles, Laurent; March\'e, Julien, Multicurves and regular functions on the representation variety of a surface in SU(2), Comment. Math. Helv., 87, 2, 409-431 (2012) · Zbl 1246.57022
[12] J. G. van der Corput, \"Uber Systeme von linear-homogenen Gleichungen und Ungleichungen, Proceedings Koninklijke Akademie van Wetenschappen te Amsterdam 34 (1931), 368-371. · JFM 57.0232.01
[13] Dehn, Max, Papers on group theory and topology\upshape, translated from the German and with introductions and an appendix by John Stillwell; with an appendix by Otto Schreier, viii+396 pp. (1987), Springer-Verlag, New York · Zbl 1264.01046
[14] Fathi, A.; Laudenbach, F.; Po\'enaru, V., Travaux de Thurston sur les surfaces, S\'eminaire Orsay; with an English summary, Ast\'erisque 66, 284 pp. (1979), Soci\'et\'e Math\'ematique de France, Paris
[15] Frohman, Charles; Gelca, R\u azvan, Skein modules and the noncommutative torus, Trans. Amer. Math. Soc., 352, 10, 4877-4888 (2000) · Zbl 0951.57007
[16] Frohman, Charles; Gelca, R\u azvan; Lofaro, Walter, The A-polynomial from the noncommutative viewpoint, Trans. Amer. Math. Soc., 354, 2, 735-747 (2002) · Zbl 0980.57002
[17] Frohman, Charles; Kania-Bartoszynska, Joanna, The structure of the Kauffman bracket skein algebra at roots of unity, Math. Z., 289, 3-4, 889-920 (2018) · Zbl 1417.57013 · doi:10.1007/s00209-017-1980-2
[18] Garoufalidis, Stavros, On the characteristic and deformation varieties of a knot. Proceedings of the Casson Fest, Geom. Topol. Monogr. 7, 291-309 (2004), Geom. Topol. Publ., Coventry · Zbl 1080.57014
[19] Garoufalidis, Stavros; L\^e, Thang T. Q., The colored Jones function is \(q\)-holonomic, Geom. Topol., 9, 1253-1293 (2005) · Zbl 1078.57012
[20] Goodearl, K. R.; Warfield, R. B., Jr., An introduction to noncommutative Noetherian rings, London Mathematical Society Student Texts 61, xxiv+344 pp. (2004), Cambridge University Press, Cambridge · Zbl 1101.16001
[21] Hoste, Jim; Przytycki, J\'ozef H., A survey of skein modules of \(3\)-manifolds. Knots 90, Osaka, 1990, 363-379 (1992), de Gruyter, Berlin · Zbl 0772.57022
[22] Kashaev, R. M., Quantization of Teichm\"uller spaces and the quantum dilogarithm, Lett. Math. Phys., 43, 2, 105-115 (1998) · Zbl 0897.57014
[23] Khovanov, Mikhail, Crossingless matchings and the cohomology of \((n,n)\) Springer varieties, Commun. Contemp. Math., 6, 4, 561-577 (2004) · Zbl 1079.57009
[24] Lawton, Sean; Manon, Christopher, Character varieties of free groups are Gorenstein but not always factorial, J. Algebra, 456, 278-293 (2016) · Zbl 1354.14021
[25] L\^e, Thang T. Q., On Kauffman bracket skein modules at roots of unity, Algebr. Geom. Topol., 15, 2, 1093-1117 (2015) · Zbl 1315.57027
[26] T. T. Q. L\^e, Quantum Teichm\"uller spaces and quantum trace map, arXiv:1511.06054. · Zbl 1419.57036
[27] L\^e, Thang T. Q., Triangular decomposition of skein algebras, Quantum Topol., 9, 3, 591-632 (2018) · Zbl 1427.57011 · doi:10.4171/QT/115
[28] Luo, Feng; Stong, Richard, Dehn-Thurston coordinates for curves on surfaces, Comm. Anal. Geom., 12, 1-2, 1-41 (2004) · Zbl 1072.57012
[29] March\'e, Julien, The Kauffman skein algebra of a surface at \(\sqrt{-1} \), Math. Ann., 351, 2, 347-364 (2011) · Zbl 1228.57014
[30] McConnell, J. C.; Robson, J. C., Noncommutative Noetherian rings\upshape, with the cooperation of L. W. Small, Graduate Studies in Mathematics 30, xx+636 pp. (2001), American Mathematical Society, Providence, RI · Zbl 0980.16019
[31] Muller, Greg, Skein and cluster algebras of marked surfaces, Quantum Topol., 7, 3, 435-503 (2016) · Zbl 1375.13038
[32] Penner, R. C.; Harer, J. L., Combinatorics of train tracks, Annals of Mathematics Studies 125, xii+216 pp. (1992), Princeton University Press, Princeton, NJ · Zbl 0765.57001
[33] Papadopoulos, A.; Penner, R. C., Hyperbolic metrics, measured foliations and pants decompositions for non-orientable surfaces, Asian J. Math., 20, 1, 157-182 (2016) · Zbl 1343.30039
[34] Przytycki, J\'ozef H., Skein modules of \(3\)-manifolds, Bull. Polish Acad. Sci. Math., 39, 1-2, 91-100 (1991) · Zbl 0762.57013
[35] Przytycki, J\'ozef H., Fundamentals of Kauffman bracket skein modules, Kobe J. Math., 16, 1, 45-66 (1999) · Zbl 0947.57017
[36] J.H. Przytycki, The Kauffman bracket skein algebra of a surface times the interval has no zero divisors, Proceedings of the 46th Japan Topology Symposium at Hokkaido University (July 26-29, 1999), 52-61.
[37] Przytycki, J\'ozef H.; Sikora, Adam S., On skein algebras and \({\rm Sl}_2({\bf C})\)-character varieties, Topology, 39, 1, 115-148 (2000) · Zbl 0958.57011 · doi:10.1016/S0040-9383(98)00062-7
[38] Rapinchuk, A. S.; Benyash-Krivetz, V. V.; Chernousov, V. I., Representation varieties of the fundamental groups of compact orientable surfaces, Israel J. Math., 93, 29-71 (1996) · Zbl 0857.14012
[39] J. Roberts, Unpublished notes.
[40] Sikora, Adam S., Skein modules at the 4th roots of unity, J. Knot Theory Ramifications, 13, 5, 571-585 (2004) · Zbl 1079.57012
[41] Sikora, Adam S., Skein modules and TQFT. Knots in Hellas ’98 (Delphi), Ser. Knots Everything 24, 436-439 (2000), World Sci. Publ., River Edge, NJ · Zbl 0978.57017
[42] Sikora, Adam S., Character varieties, Trans. Amer. Math. Soc., 364, 10, 5173-5208 (2012) · Zbl 1291.14022
[43] Sikora, Adam S.; Westbury, Bruce W., Confluence theory for graphs, Algebr. Geom. Topol., 7, 439-478 (2007) · Zbl 1202.57004
[44] Turaev, V. G., The Conway and Kauffman modules of a solid torus, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI). J. Soviet Math., 167 52, 1, 2799-2805 (1990) · Zbl 0706.57004
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