Gordian Unlinks

TL;DR

The paper proves the existence of gordian unlinks—thick links that cannot be separated via thick isotopies—by developing a geometric framework based on coned surfaces and CAT(0) spaces and presenting explicit constructions. It introduces concrete 2-component gordian unlinks, notably a planar $\beta$ with length $8+4\pi$ through which a second component $\alpha$ threads, and shows these configurations cannot be disentangled without violating thickness or length, extending the construction to $L(m,n)$ for arbitrary nonzero $m,n$ and to $n$-component analogues for all $n\ge 2$. The core argument uses a four-point property, isoperimetric bounds, and transversality to force a planar cone angle $\theta=2\pi$ and to maintain nontrivial linking data throughout thick isotopies, thereby obstructing separation. The results imply that the thick-link configuration space is not homotopy equivalent to the space of configurations of unlinked circles, revealing notable differences between thick and classical knot theories and motivating further work on gordian unknots and related configurations.

Abstract

This paper gives the first examples of gordian unlinks. The components of these unlinks cannot be separated while maintaining constant length and thickness. We construct infinite families of 2-component gordian unlinks and also construct $n$-component gordian unlinks for each $n \geq 2$.

Figures (6)

• Figure 1: Two of the simplest examples in a family $L(m,n)$ of gordian unlinks. The unlink on the left is $L(1,1)$, and on the right is $L(1, -1)$.
• Figure 2: The convex hulls of $\cup p_i$ and $\cup D_i$ have boundaries $K$ in blue and $b$ in red, respectively.
• Figure 3: The family of gordian unlinks $L(m,n)$ and an example $L(3,-2)$.
• Figure 4: The planar curve $\beta$ contains four disjoint unit radius disks whose centers have distance two from $\beta$. The second component $\alpha$ passes through the centers of the disks in the indicated order. The curves $\gamma_1 = \alpha_1 \cup \delta_1$ and $\gamma_3 = \alpha_3 \cup \delta_3$ have linking number $m$, while the curves $\gamma_2 = \alpha_2 \cup \delta_2$ and $\gamma_ 4 = \alpha_4 \cup \delta_4$ have linking number $n$. In the figure at right, $m=-1$ and $n=1$.
• Figure 5: The link component $\beta$ is one of a family of planar convex curves, each of length $8 + 4\pi$.

Theorems & Definitions (14)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• Lemma 3.1
• proof
• Lemma 3.2