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Tied monoids

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Abstract

We construct certain monoids, called tied monoids. These monoids result to be semidirect products finitely presented and commonly built from braid groups and their relatives acting on monoids of set partitions. The nature of our monoids indicate that they should give origin to new knot algebras; indeed, our tied monoids include the tied braid monoid and the tied singular braid monoid, which were used, respectively, to construct new polynomial invariants for classical links and singular links. Consequently, we provide a mechanism to attach an algebra to each tied monoid; this mechanism not only captures known generalizations of the bt-algebra, but also produces possible new knot algebras. To build the tied monoids it is necessary to have presentations of set partition monoids of types A, B and D, among others. For type A we use a presentation due to FitzGerald and for the other type it was necessary to built them.

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Acknowledgements

We would like to thank Prof. F. Aicardi for her important observations and many useful comments to improve our manuscript. We also thank Prof. J. East for his kind comments about the first version of the manuscript. Finally, we like to thank to the referee for many useful comments and carefully reading. The first author is part of the research group GEMA Res.180/2019 VRIP–UA and was supported, in part, by the Grant Fondo Apoyo a la Investigación DIUA179-2020. The second author was supported, in part, by the Grant FONDECYT Regular Nro.1180036 and FONDECYT Regular Nro.1210011.

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Authors and Affiliations

  1. Facultad de Ciencias de la Salud, Universidad Autónoma de Chile – Sede Talca, 5 Poniente 1670, 3460000, Talca, Chile

    Diego Arcis

  2. IMUV, Universidad de Valparaíso, Gran Bretaña 1111, Playa Ancha, 2340000, Valparaíso, Chile

    Jesús Juyumaya

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  1. Diego Arcis
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  2. Jesús Juyumaya
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Correspondence to Diego Arcis.

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Communicated by Benjamin Steinberg.

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Arcis, D., Juyumaya, J. Tied monoids. Semigroup Forum 103, 356–394 (2021). https://doi.org/10.1007/s00233-021-10212-y

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