Framization and Deframization

TL;DR

This paper introduces the concepts of framization and deframization for braid-related monoids and algebras, providing a geometric foundation and detailed procedures, with implications for knot theory.

Contribution

It defines and studies framization and deframization of several monoids and algebras, expanding the algebraic framework related to knot theory.

Findings

Defined framization and deframization procedures

Applied these procedures to specific monoids and algebras

Provided detailed examples and geometric interpretations

Abstract

Starting from the geometric construction of the framed braid group, we define and study the framization of several Brauer-type monoids and also the set partition monoid, all of which appear in knot theory. We introduce the concept of deframization, which is a procedure to obtain a tied monoid from a given framed monoid. Furthermore, we show in detail how this procedure works on the monoids mentioned above. We also discuss the framization and deframization of some algebras, which are deformations, respectively, of the framized and deframized monoids discussed here.