Brauer And Jones Tied Monoids

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 arising from knot theory and are interesting in themselves. 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 study 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.