On Freedman's link packings

TL;DR

This paper demonstrates that Freedman's previously large upper bounds on link packings are actually tight by constructing packings that match these bounds exponentially, and also provides improved bounds.

Contribution

It constructs explicit packings matching Freedman's bounds and offers improved, generalized upper bounds for link packings in bounded regions.

Findings

Constructed exponentially large link packings matching Freedman's bounds.

Proved Freedman's bounds are sharp for any link.

Provided improved upper bounds for link packings.

Abstract

Recently, Freedman [arXiv:2301.00295] introduced the idea of packing a maximal number of links into a bounded region subject to geometric constraints, and produced upper bounds on the packing number in some cases, while commenting that these bounds seemed far too large. We show that the smallest of these "extravagantly large" bounds is in fact sharp by constructing, for any link, a packing of exponentially many copies as a function of the available volume. We also produce improved and generalized upper bounds.