c2 Invariants of Hourglass Chains via Quadratic Denominator Reduction

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 c2 invariant of these hourglass chains which only depends on the kernel. For different kernels these hourglass chains typically give rise to different c2 invariants. An exhaustive search for the c2 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 <= 1000 corresponds to the c2 of such hourglass chains. This provides further evidence in favour of the conjecture that curves are absent in c2 invariants of phi(4) quantum field theory.