ALMOST UNKNOTTED EMBEDDINGS OF GRAPHS IN Z3 AND HIGHER DIMENSIONAL ANALOGUES

We consider the number of almost unknotted embeddings of graphs in Z3. We show that the number of such embeddings is the same, to exponential order, as the number of unknotted embeddings. We also consider some higher dimensional analogues, ie almost unknotted embeddings of surfaces which are p-dimensional analogues of Θ-graphs in Zp+2. We describe a lattice version of the spinning construction which establishes the embeddability of such surfaces in Zp+2 and show that the number of embeddings is the same, to exponential order, as the number of unknotted embeddings. The proofs of our upper bounds feature a novel application of the classical Loomis–Whitney inequality.