A Strange Family of Calabi-Yau 3-folds

We study the predictions of mirror symmetry for the 1-parameter family of Calabi-Yau 3-folds \(\tilde{X}\) with hodge numbers \(h^{11}=31,h^{21}=1\) constructed in \cite{BN}. We calculate the Picard-Fuchs differential equation associated to this family, and use it to predict the instanton numbers on the hypothetical mirror. These exhibit a strange vanishing in odd degrees. We also calculate the monodromy action on \(H^3(\tilde{X},\QQ)\) and find that it strangely predicts a positive Euler characteristic for its mirror. From a degenerate fiber of our family we construct a new rigid Calabi-Yau 3-fold. In an appendix we prove the expansion of the conifold period conjectured in \cite{ES} to hold for all 1-parameter families.