Abstract
The mixed braid groups \(B_{2,n}, \ n \in \mathbb {N}\), with two fixed strands and n moving ones, are known to be related to the knot theory of certain families of 3-manifolds. In this paper, we define the mixed Hecke algebra \(\mathrm {H}_{2,n}(q)\) as the quotient of the group algebra \({\mathbb Z}\, [q^{\pm 1}] \, B_{2,n}\) over the quadratic relations of the classical Iwahori–Hecke algebra for the braiding generators. We further provide a potential basis \(\Lambda _n\) for \(\mathrm {H}_{2,n}(q)\), which we prove is a spanning set for the \(\mathbb {Z}[q^{\pm 1}]\)-additive structure of this algebra. The sets \(\Lambda _n,\ n \in \mathbb {Z}\) appear to be good candidates for an inductive basis suitable for the construction of Homflypt-type invariants for knots and links in the above 3-manifolds.
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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. A Spanning Set and Potential Basis of the Mixed Hecke Algebra on Two Fixed Strands. Mediterr. J. Math. 15, 192 (2018). https://doi.org/10.1007/s00009-018-1240-7
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DOI: https://doi.org/10.1007/s00009-018-1240-7