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.
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