Tied Monoids

TL;DR

This paper introduces tied monoids, a class of semidirect product monoids built from braid groups and set partition monoids, which can lead to new knot invariants and algebraic structures.

Contribution

It constructs tied monoids from braid and set partition monoids, providing a framework for new knot algebras and invariants, including presentations for various types.

Findings

Tied monoids include tied braid and tied singular braid monoids.

Provided presentations for set partition monoids of types A, B, and D.

Established a mechanism to attach algebras to 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 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.