Narrow Results

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 Z^p+2. We describe a lattice version of the spinning construction which establishes the embeddability of such surfaces in Z^p+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.

We prove a pattern theorem for 2-spheres in tubes in Z⁴ and use this to prove that all except exponentially few 2-spheres in a tube in Z⁴ are knotted. We sketch how the argument can be applied to prove the same result for p-spheres in a tube in Z^p+2, p > 2.

We consider the number of embeddings of k p-spheres in Z^d, with p+2≤d≤2p+1, stratified by the p-dimensional volumes of the spheres. We show for p+2<d that the number of embeddings of a fixed link type of k equivolume p-spheres grows with increasing p-dimensional volume at an exponential rate which is independent of the link type. For d=p+2 we derive similar results both for links of unknotted p-spheres and for "augmented" links where each component p-sphere can have any knot type, and similar but weaker results when the spheres are of specified knot type.