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Let $\rho:B_n \rightarrow H_2(\overline{C_2 P_n})$ denote the Lawrence–Krammer representation of the braid group on $n$ symbols. The group $H_2(\overline{C_2 P_n})$ is a free $\mathbb{Z}[q,t]$-module of rank $\binom{n}{2}$. The braid group admits a surjective map to the symmetric group on $n$ symbols via “shrinking” or ignoring the braids, the “time” direction. There is an “unlabeling” action of $\Sigma_n$ on a complete (weighted) graph, or network, of dimension $\binom{n}{2}$.

Q: Is there a sense in which this L–K representation quantizes the unlabeling action on the moduli space of networks in the limit as $q \rightarrow 1$ and $t \rightarrow 1$?

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    $\begingroup$ I suppose one could check for starters if the characters agree in that limit? $\endgroup$
    – Qiaochu Yuan
    Commented Oct 19, 2022 at 22:32
  • $\begingroup$ Good idea. IIRC, the characters of the unlabeling action are characters of the symmetric group, Specht modules. In this case, it is the second symmetric power of the standard representation. In Young module notation, $(2,n-2)$, and is irreducible. $\endgroup$
    – Jackson Walters
    Commented Oct 21, 2022 at 20:44

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