Abstract
In this paper we give an alternative basis, \(\mathscr {B}_\mathrm{ST}\), for the Kauffman bracket skein module of the solid torus, \(\mathrm{KBSM}\left( \mathrm{ST}\right) \). The basis \(\mathscr {B}_\mathrm{ST}\) is obtained with the use of the Tempereley–Lieb algebra of type B and it is appropriate for computing the Kauffman bracket skein module of the lens spaces L(p, q) via braids.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
I. Diamantis, The Kauffman bracket skein module of the lens spaces \(L(p,q)\) via braids, in preparation
I. Diamantis, S. Lambropoulou, Braid equivalences in 3-manifolds with rational surgery description. Topol. Its Appl. 194, 269–295 (2015)
I. Diamantis, S. Lambropoulou, A new basis for the HOMFLYPT skein module of the solid torus. J. Pure Appl. Algebra 220(2), 577–605 (2016)
I. Diamantis, S. Lambropoulou, The braid approach to the HOMFLYPT skein module of the lens spaces \(L(p, 1)\), Algebraic Modeling of Topological and Computational Structures and Application. Springer Proceedings in Mathematics and Statistics (PROMS) (2017), arXiv:1702.06290v1 [math.GT]
I. Diamantis, S. Lambropoulou, An important step for the computation of the HOMFLYPT skein module of the lens spaces \(L(p,1)\) via braids, arXiv:1802.09376v1 [math.GT], to appear
I. Diamantis, S. Lambropoulou, The HOMFLYPT skein module of the lens spaces \(L(p,1)\) via braids, in preparation
I. Diamantis, S. Lambropoulou, J.H. Przytycki, Topological steps on the HOMFLYPT skein module of the lens spaces \(L(p,1)\) via braids, J. Knot Theory and Ramifications, 25, 14, (2016)
M. Flores, D. Goundaroulis, The Framization of a Temperley-Lieb algebra of type B, arXiv:1708.02014 [math.GT]
B. Gabrovšek, M. Mroczkowski, Link diagrams and applications to skein modules, Algebraic Modeling of Topological and Computational Structures and Applications. Springer Proceedings in Mathematics and Statistics (2017)
V.F.R. Jones, A polynomial invariant for links via Neumann algebras. Bull. Am. Math. Soc. 129, 103–112 (1985)
S. Lambropoulou, Solid torus links and hecke algebras of B-type, Quantum Topology, ed. by D.N. Yetter, World Scientiffic Press (1994) pp. 224–245
S. Lambropoulou, Knot theory related to generalized and cyclotomic Hecke algebras of type B. J. Knot Theory Its Ramif. 8(5), 621–658 (1999)
S. Lambropoulou, C.P. Rourke, Markov’s theorem in \(3\)-manifolds. Topology and its Applications 78, 95–122 (1997)
S. Lambropoulou, C.P. Rourke, Algebraic Markov equivalence for links in \(3\)-manifolds. Compos. Math. 142, 1039–1062 (2006)
J. Przytycki, Skein modules of 3-manifolds. Bull. Pol. Acad. Sci. Math. 39(1–2), 91–100 (1991)
V.G. Turaev, The Conway and Kauffman modules of the solid torus. Zap. Nauchn. Sem. Lomi 167, 79–89 (1988). English translation: J. Sov. Math. 2799–2805 (1990)
Acknowledgements
The author would like to acknowledge several discussions with Professor Sofia Lambropoulou. Moreover, financial support by the China Agricultural University, International College Beijing is gratefully acknowledged.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Diamantis, I. (2019). An Alternative Basis for the Kauffman Bracket Skein Module of the Solid Torus via Braids. In: Adams, C., et al. Knots, Low-Dimensional Topology and Applications. KNOTS16 2016. Springer Proceedings in Mathematics & Statistics, vol 284. Springer, Cham. https://doi.org/10.1007/978-3-030-16031-9_16
Download citation
DOI: https://doi.org/10.1007/978-3-030-16031-9_16
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-16030-2
Online ISBN: 978-3-030-16031-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)