  Gyroscopic metamaterials --- mechanical structures composed of interacting
spinning tops --- have recently been found to support one-way topological edge
excitations. In these structures, the time reversal symmetry breaking that
enables their topological behavior emerges directly from the lattice geometry.
Here we show that variations in the lattice geometry can therefore give rise to
more complex band topology than has been previously described. A `spindle'
lattice (or truncated hexagonal tiling) of gyroscopes possesses both clockwise
and counterclockwise edge modes distributed across several band gaps. Tuning
the interaction strength or twisting the lattice structure along a Guest mode
opens and closes these gaps and yields bands with Chern numbers of $|C| > 1$
without introducing next-nearest-neighbor interactions or staggered potentials.
A deformable honeycomb structure provides a simple model for understanding the
role of lattice geometry in constraining the effects of time reversal symmetry
and inversion symmetry breaking. Lastly, we find that topological band
structure generically arises in gyroscopic networks, and a simple protocol
generates lattices with topological excitations.
