Abstract
In this survey we collect all results regarding the construction of the Framization of the Temperley–Lieb algebra of type A as a quotient algebra of the Yokonuma–Hecke algebra of type A. More precisely, we present all three possible quotient algebras the emerged during this construction and we discuss their dimension, linear bases, representation theory and the necessary and sufficient conditions for the unique Markov trace of the Yokonuma–Hecke algebra to factor through to each one of them. Further, we present the link invariants that are derived from each quotient algebra and we point out which quotient algebra provides the most natural definition for a framization of the Temperley–Lieb algebra. From the Framization of the Temperley–Lieb algebra we obtain new one-variable invariants for oriented classical links that, when compared to the Jones polynomial, they are not topologically equivalent since they distinguish more pairs of non isotopic oriented links. Finally, we discuss the generalization of the newly obtained invariants to a new two-variable invariant for oriented classical links that is stronger than the Jones polynomial.
References
Aicardi, F., Juyumaya, J.: An algebra involving braids and ties. Preprint ICTP IC/2000/179, Trieste (2000)
Aicardi, F., Juyumaya, J.: Markov trace on the algebra of braids and ties. Moscow Math. J. 16, 397–431 (2016)
Aicardi, F., Juyumaya, J.: Tied links. J. Knot Theory Ramific. 25(9), 1641001 (2016)
Chlouveraki, M., Poulain d’Andecy, L.: Representation theory of the Yokonuma–Hecke. Adv. Math. 259, 134–172 (2014)
Chlouveraki, M., Juyumaya, J., Karvounis, K., Lambropoulou, S.: Identifying the invariants for classical knots and links from the Yokonuma–Hecke algebras, submitted for publication (2015). arXiv:1505.06666
Chlouveraki, M., Lambropoulou, S.: The Yokonuma–Hecke algebras and the Homflypt polynomial. J. Knot Theory and Its Ramif. 22 (2013)
Chlouveraki, M., Pouchin, G.: Determination of the representations and a basis for the Yokonuma-Temperley-Lieb algebra, Algebras Represent. Theory 18 (2015)
Chlouveraki, M., Pouchin, G.: Representation theory and an isomorphism theorem for the Framisation of the Temperley-Lieb algebra. Math. Z. 285(3), 1357–1380 (2017)
Espinoza, J., Ryom-Hansen, S.: Cell structures for the Yokonuma–Hecke algebra and the algebra of braids and ties, submitted for publication (2016). arXiv:1506.00715
Freyd, P., Yetter, D., Hoste, J., Lickorish, W., Millett, K., Ocneanu, A.: A new polynomial invariant of knots and links. Bull. AMS 12, 239–246 (1985)
Goundaroulis, D.: Framization of the Temperley-Lieb algebra and related link invariants, Ph.D. thesis, Department of Mathematics, National Technical University of Athens, 1 (2014)
Goundaroulis, D., Juyumaya, J., Kontogeorgis, A., Lambropoulou, S.: The Yokonuma-Temperley-Lieb algebra. Banach Center Pub. 103, 73–95 (2014)
Goundaroulis, D., Juyumaya, J., Kontogeorgis, A., Lambropoulou, S.: Framization of the Temperley-Lieb algebra. Math. Res. Letters 24(7), 299–345 (2017)
Goundaroulis, D., Lambropoulou, S.: Classical link invariants from the framizations of the Iwahori-Hecke algebra and the Temperley-Lieb algebra of type A. J. Knot Theory Ramific. 26(9), 1743005 (2017)
Goundaroulis, D., Lambropoulou, S.: A new two-variable generalization of the Jones polynomial. Submitted for publication (2016). arXiv:1608.01812 [math.GT]
Jones, V.: Hecke algebra representations of braid groups and link polynomials. Ann. Math. 126, 335–388 (1987)
Juyumaya, J.: Sur les nouveaux générateurs de l’algèbre de Hecke \({\cal{h}}(g, u,1)\). J. Algebra 204, 40–68 (1998)
Juyumaya, J.: Markov trace on the Yokonuma–Hecke algebra. J. Knot Theory Ramif. 13, 25–39 (2004)
Juyumaya, J., Kannan, S.: Braid relations in the Yokonuma–Hecke algebra. J. Algebra 239, 272–297 (2001)
Juyumaya, J., Lambropoulou, S.: \(p\)-adic framed braids. Topol. Appl. 154, 1804–1826 (2007)
Juyumaya, J., Lambropoulou, S.: An adelic extension of the Jones polynomial. In: Banagl, M., Vogel, D. (eds.) The Mathematics of Knots. Contributions in the Mathematical and Computational Sciences, vol. 1, pp. 825–840. Springer (2009)
Juyumaya, J., Lambropoulou, S.: An invariant for singular knots. J. Knot Theory Ramif. 18, 825–840 (2009)
Juyumaya, J., Lambropoulou, S.: Modular framization of the BMW algebra (2013). arXiv:1007.0092v1 [math.GT]
Juyumaya, J., Lambropoulou, S.: p-adic framed braids II. Adv. Math. 234, 149–191 (2013)
Juyumaya, J., Lambropoulou, S.: On the framization of knot algebras. In: Kauffman, L., Manturov, V. (eds.) New Ideas in Low-dimensional Topology. Series on Knots and everything. World Scientific (2014)
Kauffman, L.H., Lambropoulou, S.: New invariants of links and their state sum models (submitted for publication) (2016). arXiv: 1703.03655
Ko, K., Smolinsky, L.: The framed braid group and \(3\)-manifolds. Proc. AMS 115, 541–551 (1992)
Marin, I.: Artin groups and the Yokonuma–Hecke algebra. Int. Math. Res. Notices, rnx007 (2017). https://doi.org/10.1093/imrn/rnx007
Przytycki, J.H., Traczyk, P.: Invariants of links of Conway type. Kobe J. Math. 4, 115–139 (1987)
Yokonuma, T.: Sur la structure des anneux de Hecke d’un group de Chevalley fin. C.R. Acad. Sc. Paris 264, 344–347 (1967)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Goundaroulis, D. (2017). A Survey on Temperley–Lieb-type Quotients from the Yokonuma–Hecke Algebras. In: Lambropoulou, S., Theodorou, D., Stefaneas, P., Kauffman, L. (eds) Algebraic Modeling of Topological and Computational Structures and Applications. AlModTopCom 2015. Springer Proceedings in Mathematics & Statistics, vol 219. Springer, Cham. https://doi.org/10.1007/978-3-319-68103-0_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-68103-0_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-68102-3
Online ISBN: 978-3-319-68103-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)