A presentation for the braid group is:
$$B_n = \{ s_1,...,s_{n-1} | s_is_{i+1}s_i=s_{i+1}s_is_{i+1},\ \text{ } s_is_j = s_js_i \text{ for } |i-j| \geq 2\}$$
As a set, the positive braid semigroup $B_n^+$ is then all the words in $B_n$ such that no negative exponents appear on any of the generators, and the binary operation is the restriction of the operation in $B_n$.
Elements $b \in B_n^+$ are equivalence classes, with $b \cong c$ if the braid word that represents $b$ differs from the braid word representing $c$ by finite many applications of the relations.
I am trying to understand, given an element $b \in B_n^+$, what is its cardinality as an equivalence class as an element of $B_n^+$?
For example, let $b=(1,2,3,1,2,3)$. Then I'm pretty sure, by brute force, I've calculated that there are $7$ different braid words that can represent $b$:
$(1,2,3,1,2,3)$
$(1,2,1,3,2,3)$
$(2,1,2,3,2,3)$
$(2,1,3,2,3,3)$
$(2,3,1,2,3,3)$
$(1,2,1,2,3,2)$
$(2,1,2,2,3,2)$
Does anyone know how I could figure this out in a more sophisitcated way? I know word problems such as this have been well studied. Also, is there a systemic way to determine when two braid words are equivalent? Thank you.