New Constructions Related to the Polynomial Sphere Recognition Problem

We construct a simply connected 2-complex C embeddable in 3-space such that for any embedding of C in 𝕊^3 , any edge contraction forms a minor of the 2-complex not embeddable in 3-space. Additionally, we construct the 2-complex C such that every edge of C forms a nontrivial knot in any of the embeddings of C in 𝕊^3 . This answers a question of Saul Schleimer.