There are well-understood methods, going back to Givental and Hori--Vafa, that to a Fano toric complete intersection X associate a Laurent polynomial f that corresponds to X under mirror symmetry. We describe a technique for inverting this process, c
onstructing the toric complete intersection X directly from its Laurent polynomial mirror f. We use this technique to construct a new four-dimensional Fano manifold.
We find at least 527 new four-dimensional Fano manifolds, each of which is a complete intersection in a smooth toric Fano manifold.
