Abstract
We construct some Hecke-type algebras, and most notably the quotient algebra \(\mathrm {H}_{2,n}(q)\) of the group-algebra \({\mathbb Z}\, [q^{\pm 1}] \, B_{2,n}\) of the mixed braid group \(B_{2,n}\) with two identity strands and n moving ones, over the quadratic relations of the classical Hecke algebra for the braiding generators. The groups \(B_{2,n}\) are known to be related to the knot theory of certain families of 3-manifolds, and the algebras \(\mathrm {H}_{2,n}(q)\) are aimed for the construction of invariants of oriented knots and links in these manifolds. To this end, one needs a suitable basis of \(\mathrm {H}_{2,n}(q)\), and we have singled out a subset \(\varLambda _n\) of this algebra for which we proved it is a spanning set, whereas ongoing research aims at proving it to be a basis.
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References
Ariki, S., Koike, K.: A Hecke algebra of \({\mathbb{Z}} / r{\mathbb{Z}} \wr S_n\) and construction of its irreducible representations. Adv. Math. 106, 216–243 (1994)
Bardakov, V: Braid groups in handlebodies and corresponding Hecke algebras,. In: Lambropoulou, S., Stefaneas, P., Theodorou, D., Kauffman, L.(eds) Algebraic Modeling of Topological and Computational Structures and Applications, Springer Proceedings in Mathematics and Statistics (PROMS) (2017)
Buck, D., Mauricio, M.: Connect sum of lens spaces surgeries: application to hin recombination. Math. Proc. Camb. Phil. Soc. 150(3), 505–525 (2011). https://doi.org/10.1017/S0305004111000090
Diamantis, I., Lambropoulou, S.: Braid equivalences in 3-manifolds with rational surgery description. Topol. Appl. 194(2), 269–295 (2015). https://doi.org/10.1016/j.topol.2015.08.009
Diamantis, I., Lambropoulou, S.: A new basis for the Homflypt skein module of the solid torus. J. Pure Appl. Algebra 220(2), 557–605 (2015). http://doi.org/10.1016/j.jpaa.2015.06.014
Diamantis, I., Lambropoulou, S., Przytycki, J.H.: Topological steps toward the HOMFLYPT skein module of the lens spaces L(p,1) via braids. J. Knot Theory Ramif. 25(14), 1650084 (2016)
Dipper, R., James, G.D.: Representations of Hecke algebras of type \(B_n\). J. Algebra 146, 454–481 (1992)
Geck, M., Lambropoulou, S.: Markov traces and knot invariants related to Iwahori-Hecke algebras of type \(B\). J. für die reine und angewandte Mathematik 482, 191–213 (1997)
Häring-Oldenburg, R., Lambropoulou, S.: Knot theory in handlebodies. J. Knot Theory Ramif. 11(6), 921–943 (2002)
Jones, V.F.R.: Hecke algebra representations of braid groups and link polynomials. Ann. Math. 126, 335–388 (1987)
D. Kodokostas, S. Lambropoulou A spanning set and potential basis of the mixed Hecke algebra on two fixed strands, (submitted for publication)
Lambropoulou, S.: Solid torus links and Hecke algebras of B-type. In: Yetter, D.N. (ed.) Quantum Topology, pp. 225–245. World Scientific Press, USA (1994)
Lambropoulou, S., Rourke, C.P.: Markov’s theorem in 3-manifolds. Topol. Appl. 78, 95–122 (1997)
Lambropoulou, S.: Knot theory related to generalized and cyclotomic Hecke algebras of type B. J. Knot Theory Ramif. 8(5), 621–658 (1999)
S. Lambropoulou, Braid structures in knot complements, handlebodies and 3–manifolds. In: Proceedings of Knots in Hellas ’98. Series of Knots and Everything, vol. 24, pp. 274–289. World Scientific Press, USA (2000)
Lambropoulou, S., Rourke, C.P.: Algebraic Markov equivalence for links in \(3\)-manifolds. Compos. Math. 142, 1039–1062 (2006)
Acknowledgements
This research has been co-financed by the European Union (European Social Fund - ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF) - Research Funding Program: THALES: Reinforcement of the interdisciplinary and/or inter-institutional research and innovation.
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Kodokostas, D., Lambropoulou, S. (2017). Some Hecke-Type Algebras Derived from the Braid Group with Two Fixed Strands. 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_8
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