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How Many Links Fit in a Box?

Michael H. Freedman

TL;DR

The paper establishes a geometric packing bound for embedding multiple nontrivial links or knots inside the unit cube with a fixed separation between components. It extends prior results by providing a purely geometric argument, based on a bounded-geometry triangulation of $I^3$, a dual-cell $2$-coloring scheme, and neighborhoods $B_i$ for each component, coupled with a pigeonhole principle to force coincidences. Through ambient isotopy and compressions that reduce a Gromov–Thurston–like norm, it derives an exponential upper bound $N < e^{γ ε^{-3}}$ (and, for knots, $N < e^{δ ε^{-3}}$) that constrains packing density. These results, together with complementary lower bounds, delineate the limits of embedding nontrivial links and knots in three dimensions under locality constraints, with implications for topological packing in constrained volumes.

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.

How Many Links Fit in a Box?

TL;DR

The paper establishes a geometric packing bound for embedding multiple nontrivial links or knots inside the unit cube with a fixed separation between components. It extends prior results by providing a purely geometric argument, based on a bounded-geometry triangulation of , a dual-cell -coloring scheme, and neighborhoods for each component, coupled with a pigeonhole principle to force coincidences. Through ambient isotopy and compressions that reduce a Gromov–Thurston–like norm, it derives an exponential upper bound (and, for knots, ) that constrains packing density. These results, together with complementary lower bounds, delineate the limits of embedding nontrivial links and knots in three dimensions under locality constraints, with implications for topological packing in constrained volumes.

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 as increases.

Paper Structure

This paper contains 2 sections, 2 theorems.

Table of Contents

  1. Embedding Links
  2. Embedding Knots

Key Result

Theorem 1

Suppose $L$ is the disjoint union $L = \sqcup_{i=1}^N L_i$, where each $L_i$ is a true link, and that $L$ embeds in the unit cube $I^3$ so that the embedding restricted to each $L_i$ is an $\epsilon$-imbedding, then there is a constant $\gamma > 0$ (independent of $L$) so that $N < e^{\gamma \epsilo

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Theorem 2
  • proof