Bayer et al.'s Betti-number conjecture for total cut complexes of grid graphs

Let $\mathcal{G}_{m\times n}$ denote the $m\times n$ grid graph, and let $\Delta_k^t(G)$ be the total $k$-cut complex of a graph $G$. For a simplicial complex $K$, write $b_i(K)$ for its $i$th Betti number. Bayer et al.'s Betti-number conjecture. The nonzero Betti numbers satisfy

$$b_{2n-2k}(\Delta_k^t(\mathcal{G}_{2\times n}))=\binom{n-1}{k-1},\qquad k\geq 2,$$

and

$$b_{3n-6}(\Delta_3^t(\mathcal{G}_{3\times n}))=\binom{2n-2}{2}.$$

The conjecture predicts the nonzero Betti numbers of total cut complexes for two families of grid graphs. It was subsequently confirmed by homotopy-equivalence results giving wedges of spheres with the indicated multiplicities and dimensions.