  We introduce families of four-regular graphs consisting of chains of
hourglasses which are attached to a finite kernel. We prove a formula for the
$c_2$ invariant of these hourglass chains which only depends on the kernel. For
different kernels these hourglass chains typically give rise to different $c_2$
invariants. An exhaustive search for these $c_2$ invariants for all kernels
with a maximum of ten vertices provides Calabi-Yau manifolds with point-counts
which match the Fourier coefficients of modular forms whose levels and weights
are [3,36], [4,8], [4,16], [6,4], [9,4]. We also confirm the conjecture that
curves (weight two modular forms) are absent in $c_2$ invariants up to level
69.
