On the skein module of the product of a surface and a circle

Gilmer, Patrick; Masbaum, Gregor

doi:10.1090/proc/14553

On the skein module of the product of a surface and a circle
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by Patrick M. Gilmer and Gregor Masbaum PDF

Proc. Amer. Math. Soc. 147 (2019), 4091-4106 Request permission

Abstract:

Let $\Sigma$ be a closed oriented surface of genus $g$. We show that the Kauffman bracket skein module of $\Sigma \times S^1$ over the field of rational functions in $A$ has dimension at least $2^{2g+1}+2g-1.$

References

Lars V. Ahlfors, Complex analysis. An introduction to the theory of analytic functions of one complex variable, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953. MR 0054016
C. Blanchet, N. Habegger, G. Masbaum, and P. Vogel, Three-manifold invariants derived from the Kauffman bracket, Topology 31 (1992), no. 4, 685–699. MR 1191373, DOI 10.1016/0040-9383(92)90002-Y
C. Blanchet, N. Habegger, G. Masbaum, and P. Vogel, Topological quantum field theories derived from the Kauffman bracket, Topology 34 (1995), no. 4, 883–927. MR 1362791, DOI 10.1016/0040-9383(94)00051-4
Francis Bonahon and Helen Wong, Representations of the Kauffman bracket skein algebra I: invariants and miraculous cancellations, Invent. Math. 204 (2016), no. 1, 195–243. MR 3480556, DOI 10.1007/s00222-015-0611-y
Doug Bullock, On the Kauffman bracket skein module of surgery on a trefoil, Pacific J. Math. 178 (1997), no. 1, 37–51. MR 1447403, DOI 10.2140/pjm.1997.178.37
Alessio Carrega, Nine generators of the skein space of the 3-torus, Algebr. Geom. Topol. 17 (2017), no. 6, 3449–3460. MR 3709652, DOI 10.2140/agt.2017.17.3449
Benson Farb and Dan Margalit, A primer on mapping class groups, Princeton Mathematical Series, vol. 49, Princeton University Press, Princeton, NJ, 2012. MR 2850125
Charles Frohman and Răzvan Gelca, Skein modules and the noncommutative torus, Trans. Amer. Math. Soc. 352 (2000), no. 10, 4877–4888. MR 1675190, DOI 10.1090/S0002-9947-00-02512-5
Charles Frohman, Joanna Kania-Bartoszynska, and Thang Lê, Unicity for representations of the Kauffman bracket skein algebra, Invent. Math. 215 (2019), no. 2, 609–650. MR 3910071, DOI 10.1007/s00222-018-0833-x
Patrick M. Gilmer, On the Kauffman bracket skein module of the 3-torus, Indiana Univ. Math. J. 67 (2018), no. 3, 993–998. MR 3820238, DOI 10.1512/iumj.2018.67.7327
Patrick M. Gilmer and John M. Harris, On the Kauffman bracket skein module of the quaternionic manifold, J. Knot Theory Ramifications 16 (2007), no. 1, 103–125. MR 2300430, DOI 10.1142/S0218216507005208
Patrick M. Gilmer and Gregor Masbaum, Integral lattices in TQFT, Ann. Sci. École Norm. Sup. (4) 40 (2007), no. 5, 815–844 (English, with English and French summaries). MR 2382862, DOI 10.1016/j.ansens.2007.07.002
Patrick M. Gilmer and Gregor Masbaum, Irreducible factors of modular representations of mapping class groups arising in integral TQFT, Quantum Topol. 5 (2014), no. 2, 225–258. MR 3229042, DOI 10.4171/QT/51
John M. Harris, The Kauffman bracket skein module of surgery on a $(2,2b)$ torus link, Pacific J. Math. 245 (2010), no. 1, 119–140. MR 2602685, DOI 10.2140/pjm.2010.245.119
Jim Hoste and Józef H. Przytycki, The $(2,\infty )$-skein module of lens spaces; a generalization of the Jones polynomial, J. Knot Theory Ramifications 2 (1993), no. 3, 321–333. MR 1238877, DOI 10.1142/S0218216593000180
Jim Hoste and Józef H. Przytycki, The Kauffman bracket skein module of $S^1\times S^2$, Math. Z. 220 (1995), no. 1, 65–73. MR 1347158, DOI 10.1007/BF02572603
Kenneth Ireland and Michael Rosen, A classical introduction to modern number theory, 2nd ed., Graduate Texts in Mathematics, vol. 84, Springer-Verlag, New York, 1990. MR 1070716, DOI 10.1007/978-1-4757-2103-4
Louis H. Kauffman, State models and the Jones polynomial, Topology 26 (1987), no. 3, 395–407. MR 899057, DOI 10.1016/0040-9383(87)90009-7
Thang T. Q. Lê, The colored Jones polynomial and the $A$-polynomial of knots, Adv. Math. 207 (2006), no. 2, 782–804. MR 2271986, DOI 10.1016/j.aim.2006.01.006
Thang T. Q. Lê, Triangular decomposition of skein algebras, Quantum Topol. 9 (2018), no. 3, 591–632. MR 3827810, DOI 10.4171/QT/115
Thang T. Q. Le and Anh T. Tran, The Kauffman bracket skein module of two-bridge links, Proc. Amer. Math. Soc. 142 (2014), no. 3, 1045–1056. MR 3148538, DOI 10.1090/S0002-9939-2013-11835-6
W. B. R. Lickorish, The skein method for three-manifold invariants, J. Knot Theory Ramifications 2 (1993), no. 2, 171–194. MR 1227009, DOI 10.1142/S0218216593000118
Julien Marché, The skein module of torus knots, Quantum Topol. 1 (2010), no. 4, 413–421. MR 2733247, DOI 10.4171/QT/11
J. Marché and R. Santharoubane, Asymptotics of quantum representations of surface groups, arXiv:1607.00664, 2016
Maciej Mroczkowski, Kauffman bracket skein module of a family of prism manifolds, J. Knot Theory Ramifications 20 (2011), no. 1, 159–170. MR 2777022, DOI 10.1142/S0218216511008681
M. Mroczkowski and M. K. Dabkowski, KBSM of the product of a disk with two holes and $S^1$, Topology Appl. 156 (2009), no. 10, 1831–1849. MR 2519219, DOI 10.1016/j.topol.2009.03.045
Józef H. Przytycki, Skein modules of $3$-manifolds, Bull. Polish Acad. Sci. Math. 39 (1991), no. 1-2, 91–100. MR 1194712
Józef H. Przytycki, Fundamentals of Kauffman bracket skein modules, Kobe J. Math. 16 (1999), no. 1, 45–66. MR 1723531
Józef H. Przytycki, Kauffman bracket skein module of a connected sum of 3-manifolds, Manuscripta Math. 101 (2000), no. 2, 199–207. MR 1742248, DOI 10.1007/s002290050014
Justin Roberts, Skein theory and Turaev-Viro invariants, Topology 34 (1995), no. 4, 771–787. MR 1362787, DOI 10.1016/0040-9383(94)00053-0
Pierre Sallenave, Structure of the Kauffman bracket skein algebra of $T^2\times I$, J. Knot Theory Ramifications 8 (1999), no. 3, 367–372. MR 1691417, DOI 10.1142/S0218216599000237
V. G. Turaev, The Conway and Kauffman modules of a solid torus, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 167 (1988), no. Issled. Topol. 6, 79–89, 190 (Russian, with English summary); English transl., J. Soviet Math. 52 (1990), no. 1, 2799–2805. MR 964255, DOI 10.1007/BF01099241