Linking of Random p-Spheres in Z^d

Abstract

We consider the number of embeddings of k p-spheres in Z , 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