Abstract
In this paper we study the kernel of the homomorphism \(B_{g,n} \rightarrow B_n\) of the braid group \(B_{g,n}\) in the handlebody \(\mathscr {H}_g\) to the braid group \(B_n\). We prove that this kernel is semi-direct product of free groups. Also, we introduce an algebra \(H_{g,n}(q)\), which is some analog of the Hecke algebra \(H_n(q)\), constructed by the braid group \(B_n\).
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Acknowledgements
The author gratefully acknowledges Prof. Lambropoulou and her students and colleagues: Neslihan Gügümcü, Stathis Antoniou, Dimos Goundaroulis, Ioannis Diamantis, Dimitrios Kodokostas for the kind invitation to the Athens, where this paper was written, for conversation and interesting discussions.
This work was supported by the Russian Foundation for Basic Research (project 16-01-00414).
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Bardakov, V.G. (2017). Braid Groups in Handlebodies and Corresponding 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_9
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DOI: https://doi.org/10.1007/978-3-319-68103-0_9
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