  Bell et al. (1997) reported the first detection of the cyanopolyyne
HC$_{11}$N toward the cold dark cloud TMC-1; no subsequent detections have been
reported toward any source. Additional observations of cyanopolyynes and other
carbon-chain molecules toward TMC-1 have shown a log-linear trend between
molecule size and column density, and in an effort to further explore the
underlying chemical processes driving this trend, we have analyzed GBT
observations of HC$_9$N and HC$_{11}$N toward TMC-1. Although we find an
HC$_9$N column density consistent with previous values, HC$_{11}$N is not
detected and we derive an upper limit column density significantly below that
reported in Bell et al. Using a state-of-the-art chemical model, we have
investigated possible explanations of non-linearity in the column density
trend. Despite updating the chemical model to better account for ion-dipole
interactions, we are not able to explain the non-detection of HC$_{11}$N, and
we interpret this as evidence of previously unknown carbon-chain chemistry. We
propose that cyclization reactions may be responsible for the depleted
HC$_{11}$N abundance, and that products of these cyclization reactions should
be investigated as candidate interstellar molecules.
