Toroidal Algorithms for Mesh Geometries of Root Orbits of the Dynkin Diagram $\mathbb{D}_4$

By applying symbolic and numerical computation and the spectral Coxeter analysis technique of matrix morsifications introduced in our previous paper [Fund. Inform. 124(2013)], we present a complete algorithmic classification of the rational morsifications and their mesh geometries of root orbits for the Dynkin diagram $\mathbb{D}_4$. The structure of the isotropy group $Gl(4, \mathbb{Z})_{\mathbb{D}_4}$ of $\mathbb{D}_4$ is also studied. As a byproduct of our technique we show that, given a connected loop-free positive edge-bipartite graph Δ, with n ≥ 4 vertices (in the sense of our paper [SIAM J. Discrete Math. 27(2013)]) and the positive definite Gram unit form $q_\Delta : \mathbb{Z}^n \rightarrow \mathbb{Z}$, any positive integer d ≥ 1 can be presented as d = q_Δ(v), with $v \in \mathbb{Z}^n$. In case n = 3, a positive integer d ≥ 1 can be presented as d = q_Δ(v), with $v \in \mathbb{Z}^n$, if and only if d is not of the form 4^a(16 · b + 14), where a and b are non-negative integers.