  The study of topological band structures have sparked prominent research
interest the past decade, culminating in the recent formulation of rather
prolific classification schemes that encapsulate a large fraction of phases and
features. Within this context we recently reported on a class of unexplored
topological structures that thrive on the concept of {\it sub-dimensional
topology}. Although such phases have trivial indicators and band
representations when evaluated over the complete Brillouin zone, they have
stable or fragile topologies within sub-dimensional spaces, such as planes or
lines. This perspective does not just refine classification pursuits, but can
result in observable features in the full dimensional sense. In three spatial
dimensions (3D), for example, sub-dimensional topologies can be characterized
by non-trivial planes, having general topological invariants, that are
compensated by Weyl nodes away from these planes. As a result, such phases have
3D stable characteristics such as Weyl nodes, Fermi arcs and edge states that
can be systematically predicted by sub-dimensional analysis. Within this work
we further elaborate on these concepts. We present refined representation
counting schemes and address distinctive bulk-boundary effects, that include
momentum depended (higher order) edge states that have a signature dependence
on the perpendicular momentum. As such, we hope that these insights might spur
on new activities to further deepen the understanding of these unexplored
phases.
