  In this work, we construct a new data type for hadronic jets in which the
traditional point-cloud representation is transformed into a simplicial complex
consisting of vertices, or 0-simplexes. An angular resolution scale, $r$, is
then drawn about each vertex, forming balls about hadrons. As $r$ grows, the
overlap of balls form 2- and 3-point connections, thereby appending 1- and
2-simplexes to the complex. We thus associate a jet with an
angular-resolution-dependent characterization of its substructure -- we dub
this data type the simplicial substructure complex $K_{\rm sub}(r)$. This data
type gives rise to two interesting representations. First, the subset of 0-and
1-simplexes lends itself naturally to a graph representation of a given jet's
substructure and we provide examples of valuable graph-theoretic calculations
such a representation affords. Second, the subset of 1- and 2-simplexes gives
rise to what is known as a Face-Counting-Vector, in topological combinatorics
parlance. We explore information-theoretic aspects of the components of this
vector, various metric properties which follow, as well as how this vector can
be used to define new jet-shape observables. The utility of these
representations is demonstrated in the context of the discrimination of jets
initiated by light quarks and gluons from those initiated by tops.
