  In this article, we summarize combinatorial description of complete
intersection Calabi-Yau threefolds in Hibi toric varieties. Such Calabi-Yau
threefolds have at worst conifold singularities, and are often smoothable to
non-singular Calabi-Yau threefolds. We focus on such non-singular Calabi-Yau
threefolds of Picard number one, and illustrate the calculation of topological
invariants, using new motivating examples.
