The path homology introduced by Grigor’yan, Lin, Muranov and Yau plays a central role in digraph topology and the emerging field of digraph homotopy theory more generally. Unfortunately, the computation of the path homology of a digraph is a two-step process, and until now no complete description of even the underlying chain complex has appeared in the literature. In particular, our understanding of the path chains is the primary obstruction to the development of fast path homology algorithms, which in turn would enable the practicality of a wide range of applications to directed networks.

I will introduce an inductive method of constructing elements of the path homology chain modules from elements in the proceeding two dimensions. When the coefficient ring has prime characteristic the inductive elements generate the path chains. Moreover, in low dimensions the inductive elements coincide with naturally occurring generating sets up to sign, making them excellent candidates to reduce to a basis.

Inductive elements provide a new concrete structure on the path chain complex that can be directly applied to understand path homology, under no restriction on the digraph. During the talk I will demonstrate how inductive elements yield the explicit structure of the dimension 3 path chains and enable the construction of a sequence of digraphs whose path Euler characteristic can differ arbitrarily depending on the choice of coefficients.