The positive braid monoid on $n$ strands is the monoid with generators $s_1$, $s_2$, ..., $s_{n-1}$ and relations$$s_i s_{i+1} s_i = s_{i+1} s_i s_{i+1} \qquad s_i s_j = s_j s_i \text{for}\ |i-j| \geq 2.\qquad (\ast)$$For any element $\beta$ in the positive braid monoid, let $\Gamma(\beta)$ be the graph whose vertices are words for $\beta$, and whose edges correspond to using a single braid relation from the list $(\ast)$. For example, $\Gamma(s_1 s_1 s_2 s_1)$ looks like$$s_1 s_1 s_2 s_1 \longleftrightarrow s_1 s_2 s_1 s_2 \longleftrightarrow s_2 s_1 s_2 s_2.$$
This graph is a tree, but it is easy to find $\beta$ where this graph has cycles.
Main Question: What is a list of cycles which generates $\pi_1(\Gamma(\beta))$?
Here is the list of cycles which I conjecture works.
(1) If a word $w$ for $\beta$ can have two braid relations applied to it in non-overlapping positions, than we can apply the relations in either order, making a $4$-cycle in $\Gamma(\beta)$.
(2) If a word for $\beta$ contains a substring $s_p$, $s_q$, $s_r$, where $s_p$, $s_q$ and $s_r$ all commute, then we can glue in a hexagon, as in the figure below:
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(3) If a word for $\beta$ contains a substring $s_i s_j s_{j+1} s_j$ where $s_i$ commutes with $s_i$, $s_j$, then we can glue in an octagon, as in the figure below:
Image may be NSFW.
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(4) If a word for $\beta$ contains a reduced word for the longest element in $S_4$, then we can glue in a $14$-gon, as shown in the figure below.
Image may be NSFW.
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Question: Does this list of cycles generate? If not, can we always generate $\pi_1(\Gamma(\beta))$ with a list of cycles like this, coming from subwords of some bounded length?
