Neighborly 2-Manifolds with 12 Vertices

We provide a complete list of 59 orientable neighborly 2-manifolds with 12 vertices of genus 6, and we study their possible flat embeddings in Euclidean 3-space. Whereas the question of embeddability remains open in its general form, we obtain several properties of the embedding (polyhedral realization) under the assumption that it does exist: 1. The order of the geometrical automorphism group of any polyhedral realization would not exceed 2. 2. The polyhedral realization would not be obtainable via the Schlegel diagram of any 4-polytope; moreover, none of our orientable neighborly 2-manifolds with 12 vertices can be found within of the 2-skeleton of any 4-polytope. 3. The polyhedral realization would not allow a tetrahedral subdivision without inserting new vertices. By using a weaker version of the manifold property, we obtain neighborly polyhedra with 2nvertices for everyn⩾3.