Regular unimodular Hilbert triangulations of thick te-interlaces

This is a companion directory containing the polymake code and figures describing explicitly the combinatorics of the regular unimodular Hilbert triangulations of thick te-interlaces of size $4$ and $6$ in each of the two different types of Hilbert basis. Namely, types

1. $\mathcal{H}(C) = \left{\frac{1}{2} (r^i + r^j)\right}{i,j\in{1,\ldots,n}}\cup\left{\frac{1}{4}\sum{j} r^j\right},$
2. $\mathcal{H}(C) = \left{\frac{1}{2} (r^i + r^j)\right}{i,j\in{1,\ldots,n}}\cup\left{\frac{3}{4}r^i+\frac{1}{4}\sum{j\neq i} r^j\right}_{i\in{1,\ldots,n}}.$

In the different folders ./n4_type1, ./n4_type2, ./n6_type1, and ./n6_type2, each figure corresponds to a set of $n$ Hilbert basis elements generating a unimodular Hilbert cone in one of the four regular triangulations, for each pair $(n,t)$ for $n=4,6$ the size of the thick te-interlace and $t=1,2$ the type of Hilbert basis.

In each figure, one can find $n$ vertices labelled from $1$ to $n$ and colored loops, edges and circles that correspond to Hilbert basis elements as follows.

In both types

• $r^i$ for $i=1,\ldots,n$ is represented by a blue loop at vertex $i$ in each figure,
• $m^{ij}:=\frac{1}{2} (r^i + r^j)$ for $i\neq j$ is represented by a blue edge $ij$ in each figure.

Type 1

In that case there is a single additional Hilbert basis element:

• $h:=\frac{1}{4}\sum_{j} r^j$ that is represented by a green circle in the center of each figure.

Type 2

In that case there is a third type of of Hilbert basis elements:

• $h^i:=\frac{3}{4}r^i+\frac{1}{4}\sum_{j\neq i} r^j$, for $i=1,\ldots,n$, that is represented by a red circle around vertex $i$ in each figure.