Measuring the entanglement complexity of 3-periodic networks through the untangling number

Periodic networks serve as models for the structural organisation of biological and chemical crystalline systems. Single or multiple networks can have different configurations in space, where entanglement may arise due to the way the (possibly curvilinear) edges weave around each other. This entanglement influences the functional, physical, and chemical properties of the materials modelled by the networks, which highlights the need to quantify its complexity. In this paper, we define the least tangled embeddings of 3-periodic networks that we call ground states, through the use of knot-theoretic crossing diagrams. The concept of a ground state permits the definition of a measure of entanglement complexity called the untangling number that quantifies the distance between a given 3-periodic structure and its least tangled version.

💡 Research Summary

The paper addresses a fundamental gap in the quantitative description of entanglement in three‑dimensional periodic networks, which are central models for a wide range of crystalline materials such as metal‑organic frameworks, covalent organic frameworks, and liquid‑crystal polymers. While knot and link theory provide powerful invariants for finite closed curves, extending these ideas to infinite, multi‑component, and possibly curvilinear graph embeddings has remained challenging.

To overcome this, the authors introduce a rigorous framework that defines the “least tangled” embedding of a 3‑periodic network, termed a ground state, and a corresponding quantitative measure of how far any given embedding lies from this ground state, called the untangling number. The construction proceeds in several steps.

1. Unit Cell and Tridiagram Representation – A periodic embedding is decomposed into a fundamental parallelepiped (unit cell). By projecting the three mutually orthogonal faces (front, top, right) onto a square and recording over‑/under‑crossing information, a set of three planar diagrams—collectively a tridiagram—is obtained. Thick dots and open circles encode intersections with the front and back faces, respectively, ensuring that the full three‑dimensional topology is captured in two dimensions.

2. Crossing Number for Periodic Networks – For each diagram D₁, D₂, D₃ the numbers of crossings a, b, c are recorded. The authors define a scalar crossing measure c(T)=a²+b²+c² for a tridiagram T. The crossing number of a particular embedding with respect to a chosen unit cell, c(K,U), is the minimum of c(T) over all tridiagrams representing the same ambient isotopy class of that cell. The global crossing number of the network is then the minimum of c(K,U) over all admissible unit cells. This invariant generalises the classical crossing number of links to the periodic setting.

3. Families Connected by Crossing Changes – Analogous to the unknotting number for links, the authors consider the equivalence relation generated by crossing changes (flipping an over‑crossing to an under‑crossing) applied to any of the three diagrams in a tridiagram. The set of all embeddings reachable from a given embedding K via a finite sequence of crossing changes is denoted U(K,U). Within this family, the subfamily G(K,U) consists of those embeddings whose crossing number attains the minimal possible value among all members of U(K,U).

4. Ground States – Elements of G(K,U) are defined as ground states. By construction, a ground state is an embedding that cannot be further simplified (in terms of crossing number) by any sequence of crossing changes that respects the periodicity of the network. The paper demonstrates that familiar high‑symmetry embeddings—such as the barycentric embeddings of the srs, dia, and pcu nets—are ground states, each achieving the triplet (0,0,0) and thus a crossing number of zero. Even networks that are not strictly three‑periodic (e.g., the square‑lattice sql net) become ground states when extended periodically in the third dimension.

5. Untangling Number – For any embedding K, the untangling number τ(K) is defined as the minimal number of crossing changes required to transform K into a ground state within its family U(K,U). This invariant captures the “distance” from the most ordered configuration and is a direct analogue of the unknotting number, but adapted to the richer setting of infinite periodic graphs.

6. Algorithmic and Computational Aspects – The authors outline how to compute crossing numbers and untangling numbers in practice. Ambient isotopies correspond to a set of planar moves (R‑moves) detailed in the Supplementary Information. Minimum tridiagrams can be obtained by systematic reduction of crossings using these moves, while the ropelength energy (total edge length divided by edge diameter) provides an alternative, physically motivated metric; the authors note that embeddings minimizing ropelength often coincide with ground states. Existing tools such as the Systre algorithm (for barycentric embeddings) and the PB‑SONO algorithm (for ropelength minimization) are employed to generate concrete examples.

7. Illustrative Examples and Physical Relevance – The paper presents several case studies. The srs net, a chiral three‑fold coordinated network, is shown in both its single‑handed and enantiomorphic double‑handed forms. The minimal tridiagram for the double‑handed embedding yields a crossing number of 48, whereas the single‑handed embedding attains zero. The authors also discuss a ravel embedding of the θ‑graph, which lacks any knotted or linked cycles yet remains tangled—a phenomenon that traditional cycle analysis would miss. By applying the untangling number, such subtle forms of entanglement become quantifiable.

8. Implications for Materials Science – Since entanglement influences mechanical stiffness, diffusion pathways, and optical activity, the ability to assign a numeric untangling number to a crystal framework opens avenues for systematic design. For instance, a low untangling number could correlate with higher porosity or easier guest‑molecule transport, while a high untangling number might indicate robust mechanical interlocking. The authors suggest that future work could explore statistical correlations between untangling numbers and experimentally measured properties across databases of MOFs, COFs, and liquid‑crystal polymers.

In conclusion, the paper delivers a mathematically rigorous, computationally tractable, and physically meaningful framework for assessing the entanglement complexity of 3‑periodic networks. By defining ground states and the untangling number, it extends classical knot‑theoretic concepts to infinite, multi‑component crystalline structures, providing a new lens through which the relationship between topology and material functionality can be explored.