4 problems

Higher path-homology vanishing conjecture. The path homology of $\vec{C}_n^S$ vanishes in every degree at least three:

Let $G$ and $H$ be directed graphs, and let $G\squarediv H$ denote their strong product. A Künneth theorem for path homology would describe the path homology of $G\squarediv H$ in…

Let $p_{i,j}$ be the undirected circuits constructed in the preceding result, representing the relevant homology classes in the weighted digraph filtration. Simplicity conjecture.…

Torsion subgroup conjecture. The torsion subgroup of $H_1$ contains a subgroup isomorphic to $\mathbb{Z}/n\mathbb{Z}$.