= Eight-tile Neumann transplantation across orientability
{title2=$C=I+R^4+2(R+R^{-1})$}
Take eight congruent hexagonal tiles with three pairwise nonadjacent distinguished faces. On tile indices modulo eight let $\sigma(j)=1-j$, $t(j)=-j$, and use $u_1(j)=3j$ for one assembly and $u_2(j)=3j+4$ for the other. Pair faces by reflection and leave fixed faces, and the three remaining faces, with <Neumann boundary conditions>. If $Re_j=e_{j+1}$, the displayed matrix intertwines all three face actions: its circulant coefficients satisfy $c_r=c_{-r}=c_{3r+4}$. Its <eigenvalues> are $1+(-1)^k+4\cos(\pi k/4)$, which are nonzero for $0\leq k\leq7$. The <transplantation theorem> gives equal spectra. The second gluing graph is <bipartite>, with parts $\{0,5,6,7\}$ and $\{1,2,3,4\}$, so it is <orientable>. The first has the odd cycle $1,3,6,2,7,1$ and is nonorientable. Both graphs are connected with eight vertices and ten edges, and their bordered surfaces retract up to homotopy to the graphs. Both have <Euler characteristic> $-2$ and <fundamental groups> free of rank three. Neither is <simply connected>.
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