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LINKING OF RANDOM p-SPHERES IN Zd

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