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 consider two self-avoiding polygons (2SAPs) each of which spans a tubular sublattice of ℤ³. A pattern theorem is proved for 2SAPs, that is any proper pattern (a local configuration in the middle of a 2SAP) occurs in all but exponentially few sufficiently large 2SAPs. This pattern theorem is then used to prove that all but exponentially few sufficiently large 2SAPs are topologically linked. Moreover, we also use it to prove that the linking number Lk of an n edge 2SAP G_n satisfies lim_n→∞ℙ(|Lk(G_n)| ≥ f(n))=1 for any function . Hence the probability of a non zero linking number for a 2SAP approaches one as the size of the 2SAP goes to infinity. It is also established that, due to the tube constraint, the linking number of an n edge 2SAP grows at most linearly in n.

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.