Accumulations of infinite links

We consider smooth links in S3 which consist of a countably infinite collection of tamely embedded circles having the property of being locally straight. We show that in general there are no restrictions on the types of accumulations that may appear in terms of the knot types: e.g., trefoils may accumulate on figure-eights in a smooth manner. Our techniques rely on properties of “universal templates”, a class of branched 2-manifolds. Our results apply to closed orbits of flows on S3.