Brauer and Jones tied monoids

TL;DR

This paper introduces ramified monoids related to Brauer-type monoids, including symmetric, Jones, and Brauer monoids, exploring their algebraic structures and connections to knot theory.

Contribution

It defines a new class of ramified monoids for Brauer-type monoids and provides presentations, also studying related tied-like monoids not classified as ramified.

Findings

Presented a presentation for the ramified monoid of the Brauer monoid.

Connected ramified monoids to knot theory and tied braid monoids.

Introduced and analyzed two new tied-like monoids.

Abstract

We introduce a ramified monoid, attached to each Brauer--type monoid, that is, to the symmetric group, to the Jones and Brauer monoids among others. Ramified monoids correspond to a class of tied monoids which arise from knot theory and are interesting in itself. The ramified monoid attached to the symmetric group is the Coxeter-like version of the so--called tied braid monoid. We give a presentation of the ramified monoid attached to the Brauer monoid. Also, we introduce and studied two tied-like monoids that cannot be described as ramified monoids. However, these monoids can also be regarded as tied versions of the Jones and Brauer monoids.