  Inspired by work of Fr\"oberg (1990), and Eagon and Reiner (1998), we define
the \emph{total $k$-cut complex} of a graph $G$ to be the simplicial complex
whose facets are the complements of independent sets of size $k$ in $G$. We
study the homotopy types and combinatorial properties of total cut complexes
for various families of graphs, including chordal graphs, cycles, bipartite
graphs, the prism $K_n \times K_2$, and grid graphs, using techniques from
algebraic topology and discrete Morse theory.
