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Let $Br_n$ be the braid group and consider words in its generators (not in the inverses). Two such words define the same "positive" braid if one can be obtained from the other by commuting and braid moves. Put another way, there is a graph whose vertices are words and whose edges are these two sorts of moves, and we're interested in one connected component of this graph.

What is a set of cycles in this graph that generates its $\pi_1$?

For example, $\tau_1 \tau_3 \tau_7 \tau_9 \sim \tau_3 \tau_1 \tau_7 \tau_9 \sim \tau_3 \tau_1 \tau_9 \tau_7 \sim \tau_1 \tau_3 \tau_9 \tau_7 \sim \tau_1 \tau_3 \tau_7 \tau_9$ is a 4-cycle.

I expect it's not hard to figure out by thinking about the stratification of the space of pictures of a braid, so what I'd most like is a reference.

Also I expect the answer shouldn't be too different for Artin braid groups outside type A, and I'm interested in those as well.

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  • $\begingroup$ Unless I'm getting confused about what you're asking, I think this is usually just called the "graph of reduced words" for an element of a Coxeter group (and you don't really need the language of braids to talk about it). EDIT: whoops, of course I am confused, because you are not necessarily considering reduced words. Sorry! I will leave this comment in case anyone makes the same mistake. $\endgroup$
    – Sam Hopkins
    Commented Oct 9, 2022 at 16:13
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    $\begingroup$ You are really considering the braid semigroup $BS_n$, and (more generally, for any semigroup) your graphs are called the Squier complexes of the semigroup. The fundamental group of such a complex is called a diagram group, see e.g. Guba-Sapir. $\endgroup$
    – Carl-Fredrik Nyberg Brodda
    Commented Oct 9, 2022 at 16:15
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    $\begingroup$ According to ams.org/journals/tran/2013-365-05/S0002-9947-2012-05719-9/…: "Tits [31] gave explicit generators for [the graph of reduced words's] fundamental group" and [31] is "J.A. Tits, A local approach to buildings. The geometric vein, pp. 519–547, Springer, New York–Berlin, 1981." This only answers your question in a special case but it might be a place to start for a reference. $\endgroup$
    – Sam Hopkins
    Commented Oct 9, 2022 at 16:17
  • $\begingroup$ Yeah, nonreducedness is important for me. I'm checking out the Guba-Sapir article math.vanderbilt.edu/sapirmv/ftp/pub/diagramgroups/dg.pdf $\endgroup$
    – Allen Knutson
    Commented Oct 9, 2022 at 16:46
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    $\begingroup$ I believe that I have proved it, using the extremely helpful reference @SamHopkins found. I wrote up the proof on my version of the question. $\endgroup$
    – David E Speyer
    Commented Feb 18 at 20:22

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