On Freedman’s Link Packings

Freedman recently posed a new question in quantitative topology about link packings. Given a link L, define the $\epsilon$-diagonal packing number $n_{L(\epsilon)}$ to be the number of copies of L that can be simultaneously embedded in $[0,1]^3$ so that (1) Each copy of $L$ is contained in a ball which is disjoint from the other copies. (2) Within each copy, the components are separated by a distance of at least $\epsilon$. We’ll discuss a new construction for obtaining a lower bound on $n_{L(\epsilon)}$ and expand on Freedman’s ideas to obtain an upper bound on $n_{L(\epsilon)}$ when $L$ has a non-trivial Milnor Invariant. At the end we’ll mention several related open problems about link packings. This is joint work with Fedya Manin.