Tied Monoids

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. To build the tied monoids it is necessary to find presentations of set partition monoids of types A, B and D, among others; these presentations are interesting in their own right and seem to be absent in the literature.