  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 the $c_2$ invariants of hourglass chains
with kernels that have a maximum of ten vertices provides Calabi-Yau manifolds
with point-counts which match the Fourier coefficients of modular forms whose
weights and levels are [4,8], [4,16], [6,4], and [9,4]. Assuming the completion
conjecture, we show that no modular form of weight 2 and level $\leq1000$
corresponds to the $c_2$ of such hourglass chains. This provides further
evidence in favour of the conjecture that curves are absent in $c_2$ invariants
of $\phi^4$ quantum field theory.
