The Tutte polynomial and toric Nakajima quiver varieties

For a quiver $Q$ with underlying graph $\Gamma$ , we take $ {\mathcal {M}}$ an associated toric Nakajima variety. In this article, give direct relation between specialization of the Tutte polynomial Kac and Poincaré . We do by giving cell decomposition indexed spanning trees ‘geometrizing’ deletion contraction operators on graphs. These relations have been previously established in Hausel–Sturmfels [6] Crawley-Boevey–Van den Bergh [3], however methods here are more hands-on.