Diagrammatic representations of 3-periodic entanglements

TL;DR

The paper develops a diagrammatic framework for triply periodic tangles (TP tangles) by representing unit cells of TP tangles as links in the 3-torus $\mathbb{T}^3$ and introducing a three-projection (tridiagram) approach. It defines a robust equivalence scheme for TP tangles, including ambient isotopy, torus twists, lattice changes, and covering maps, unifying these into a single $U$-equivalence through unit-cell connections. A comprehensive diagrammatic theory is built with diagrams and tridiagrams projected along three non-coplanar axes, augmented by nine Reidemeister-type moves ($R_1$–$R_3$ plus $R_4$–$R_9$) and additional torus-related equivalences, culminating in a generalised Reidemeister theorem that links TP-tangle isotopy to tridiagram equivalence. Finally, a crossing-number invariant $c(K)$ is defined by minimising a triplet of crossings over unit cells and diagram moves, enabling practical classification with concrete examples such as the $\Pi^{+}$ and $\Sigma^{+}$ rod packings. The framework has potential applications in chemistry, materials science, and DNA origami, where triply periodic entanglements arise and systematic classification aids understanding of their topological and geometric structure.

Abstract

Diagrams enable the use of various algebraic and geometric tools for analysing and classifying knots. In this paper we introduce a new diagrammatic representation of triply periodic entangled structures (TP tangles), which are embeddings of simple curves in $\mathbb{R}^3$ that are invariant under translations along three non-coplanar axes. As such, these entanglements can be seen as preimages of links embedded in the 3-torus $\mathbb{T}^3 = \mathbb{S}^1 \times \mathbb{S}^1 \times \mathbb{S}^1$ in its universal cover $\mathbb{R}^3$, where two non-isotopic links in $\mathbb{T}^3$ may possess the same TP tangle preimage. We consider the equivalence of TP tangles in $\mathbb{R}^3$ through the use of diagrams representing links in $\mathbb{T}^3$. These diagrams require additional moves beyond the classical Reidemeister moves, which we define and show that they preserve ambient isotopies of links in $\mathbb{T}^3$. The final definition of a tridiagram of a link in $\mathbb{T}^3$ allows us to then consider additional notions of equivalence relating non-isotopic links in $\mathbb{T}^3$ that possess the same TP tangle preimage.

Figures (24)

• Figure 1: Diagrams of knotted structures: On the left, a diagram of the cinquefoil knot, obtained from a planar projection of an embedding of the knot. On the right, a motif diagram of a doubly periodic alternating weave, obtained from a projection of one of its motifs onto a 2-torus, represented as a square with identified edges.
• Figure 2: Simple examples of 3-periodic tangles are given by periodic packings of straight cylinders. These two structures are well known in structural chemistry, referred to as the $\Pi^{+}$ cylinder packing on the left and the $\Sigma^{+}$ cylinder packing on the right okeeffe2001. The cylinders are coloured by those that are parallel.
• Figure 3: Two unit cells of two TP tangles whose parallelepiped differ by a shearing and a uniform scaling of space. They are associated to the same link embedded in the $3$-torus.
• Figure 4: Two different lattices (represented by red points and blue points, the red ones belonging to the blue lattice as well) that preserve the periodicity of the TP tangle. Different unit cells can be taken out of those lattices, like the blue cube and the blue parallelepiped both belonging to the blue lattice or the red parallelepiped which is a unit cell of volume 2 with respect to the blue lattice. Those three unit cells are non-isotopic links in the 3-torus and highlight the difference between such links and 3-periodic tangles.
• Figure 5: Two unit cells that differ by the action of the matrix $M_1$.

Theorems & Definitions (78)

• Definition 2.1
• Remark 2.2
• Definition 2.3
• Definition 2.4
• Remark 2.5
• Definition 2.6
• Remark 2.7
• Remark 2.8
• Remark 2.9
• Definition 2.10