How Many Links Fit in a Box?

TL;DR

This paper establishes a geometric upper bound on the number of non-trivial links that can be embedded in a unit cube with fixed separation, extending previous work on Hopf links and related invariants.

Contribution

It provides a new, purely geometric proof of an upper bound for all non-trivial links in a bounded region, generalizing earlier results based on linking number properties.

Findings

Derived a universal upper bound for embedding non-trivial links in a cube.

Extended previous bounds from Hopf links to all non-trivial links.

Connected bounds with recent lower bound results for bounded link types.

Abstract

In an earlier note [arXiv:2301.00295] it was shown that there is an upper bound to the number of disjoint Hopf links (and certain related links) that can be embedded in the unit cube where there is a fixed separation required between the components within each copy of the Hopf link. The arguments relied on multi-linear properties of linking number and certain other link invariants. Here we produce a very similar upper bound for all non-trivial links by a more-general, entirely geometric, argument (but one which, unlike the original, has no analog in higher dimensions). Shortly after the initial paper, [arXiv:2308.08064] proved lower bounds which still provide a converse to our Theorem 1 in the case that only a bounded number of link types appear among the set $\{L_i\}$ as $N$ increases.