Table 2:

K₃ → K₄

q	q = 0	q = 1	q = 2
${\mathcal{B}}_{q+1}^{3+1}$	$\begin{array}{ll}\hfill & \begin{array}{ccccc}01& 12& 23& 03& 24\end{array}\hfill \\ \begin{array}{c}0\\ 1\\ 2\\ 3\\ 4\end{array}\hfill & \left[\begin{array}{ccccc}-1& 0& 0& -1& 0\\ 1& -1& 0& 0& 0\\ 0& 1& -1& 0& -1\\ 0& 0& 1& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]\hfill \end{array}$	/	/
${\mathcal{B}}_q^3$	$\begin{array}{c}\begin{array}{ccccc}0& 1& 2& 3& 4\end{array}\\ \left[\begin{array}{ccccc}0& 0& 0& 0& 0\end{array}\right]\end{array}$	$\begin{array}{cc}& \begin{array}{cccc}01& 12& 23& 03\end{array}\\ \begin{array}{c}0\\ 1\\ 2\\ 3\\ 4\end{array}& \left[\begin{array}{cccc}-1& 0& 0& -1\\ 1& -1& 0& 0\\ 0& 1& -1& 0\\ 0& 0& 1& 1\\ 0& 0& 0& 0\end{array}\right]\end{array}$	/
${\mathcal{L}}_q^{3+1}$	$\left[\begin{array}{ccccc}2& -1& 0& -1& 0\\ -1& 2& -1& 0& 0\\ 0& -1& 3& -1& -1\\ -1& 0& -1& 2& 0\\ 0& 0& -1& 0& 1\end{array}\right]$	$\left[\begin{array}{cccc}2& -1& 0& 1\\ -1& 2& -1& 0\\ 0& -1& 2& 1\\ 1& 0& 1& 2\end{array}\right]$	/
${\beta }_q^{3+1}$	1	1	/
$\text{dim}\left({\mathcal{L}}_q^{3+1}\right)$	5	4	/
$\text{rank}\left({\mathcal{L}}_q^{3+1}\right)$	4	3	/
$\text{nullity }\left({\mathcal{L}}_q^{3+1}\right)$	1	1	/
$\text{Spectra }\left({\mathcal{L}}_q^{3+1}\right)$	{0, 0.8299, 2, 2.6889, 4.4812}	{0, 2, 2, 4}	/