The moduli problem of Lobb and Zentner and the colored \(\mathfrak{sl}(N)\) graph invariant (Q2859567)

A MOY graph \(\Gamma\) is an \(N\)-coloured, oriented, trivalent, plane graph. The MOY polynomial \(\langle \Gamma \rangle_N\) is a polynomial invariant of MOY graphs. These two concepts were introduced in [\textit{H. Murakami}, \textit{T. Ohtsuki}, and \textit{S. Yamada}, Enseign. Math., II. Sér. 44, No. 3--4, 325--360 (1998; Zbl 0958.57014)] to give a graphical calculus for the \(\mathfrak{sl}(N)\) quantum invariant (or the HOMFLY-PT polynomial). For a MOY graph \(\Gamma\), \textit{A. Lobb} and \textit{R. Zentner}, in [``The quantum \(\mathrm{sl}(N)\) graph invariant and a moduli space'', \url{arXiv:1204.5372}], introduced a moduli space \(\mathcal{M}(\Gamma)\) of colourings of \(\Gamma\) by associating an element of the complex Grassmannian \(\mathbb{G}(i,N)\) to an \(i\)-coloured edge of \(\Gamma\). It was shown in that paper that if \(\Gamma\) is \(2\)-coloured, then its MOY polynomial evaluated at 1 is the Euler characteristic of the moduli space, \(\chi(\mathcal{M}(\Gamma))=\langle \Gamma \rangle_N(1)\); and it was shown how \(\mathcal{M}(\Gamma)\) can be thought of as a representation variety. The paper under review extends these two results from \(2\)-coloured MOY graphs to \(N\)-coloured MOY graphs.