Braids via term rewriting

Abstract

We present a brief introduction to braids, in particular simple positive braids, with a double emphasis: first, we focus on term rewriting techniques, in particular, reduction diagrams and decreasing diagrams. The second focus is our employment of the colored braid notation next to the more familiar Artin notation. Whereas the latter is a relative, position dependent, notation, the former is an absolute notation that seems more suitable for term rewriting techniques such as symbol tracing. Artin's equations translate in this notation to simple word inversions. With these points of departure we treat several basic properties of positive braids, in particular related to the word problem, confluence property, projection equivalence, and the congruence property. In our introduction the beautiful diamond known as the permutohedron plays a decisive role.

Confluence of the Chinese Monoid

“…The well-known braid presentation does also not admit for its 'canonical' tiles an application of decreasing diagrams. Nevertheless confluence by tiling does hold there for its canonical tiles, and it can be proven using decreasing diagrams employing a more complex labelling [9].…”

“…As a preparation for the main section of this note where we prove confluence for the Chinese monoid C n on n generators, we introduce this method, somewhat informally, guided by a few examples of monoids. More complete expositions of reduction diagram construction can be found in [1,9,22]. In the latter paper the exposition is for positive braid words.…”