Topology of total cut complexes and cut complexes of grid graphs

Inspired by the work of Fr{ö}berg (1990) and Eagon and Reiner (1998), Bayer et al. recently introduced two new graph complexes: total cut complexes and cut complexes. In this article, we investigate these complexes specifically for (rectangular) grid graphs, focusing on $2\times n$ and $3\times n$ cases. We extend and refine the work of Bayer et al., proving and strengthening several of their conjectures, thereby enhancing the understanding of these graph complexes’ topological and combinatorial properties.