arXiv subjects

Myfanwy E. Evans

Publications and source records attributed to Myfanwy E. Evans.

14 recordsLinked to original sources

arXiv2026

The untangling number of 3-periodic tangles

The entanglement of curves within a 3-periodic box provides a model for complicated space-filling entangled structures occurring in biological, chemical and physical systems. Quantifying the complexity of the entanglement within these models enhances the characterisation of these structures. In this paper, we introduce a new measure of entanglement complexity through the untangling number, reminiscent of the unknotting number in knot theory. The untangling number quantifies the minimum distance between a given 3-periodic structure and its least tangled version, called ground state, through a sequence of operations in a diagrammatic representation of the structure. For entanglements that consist of only infinite open curves, we show that the generic ground states are crystallographic rod packings, well known in structural chemistry.

math.GT↗

arXiv2026

The untangling number as a measurement of entanglement complexity in 3-periodic networks

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.

math.GT↗

arXiv2026

Entanglement by design: Symmetry-guided periodic helical assemblies

In this paper, a selection of elegant, highly symmetric examples of three-periodic tangled nets and filaments are presented. They are constructed via familiar crystal nets using edges as geometric scaffolds for n-fold helical windings. Rather than providing a complete classification, this gallery of examples highlights recurring geometric motifs, offering insight into how periodic tangles are organised in crystalline, molecular, and biological systems.

cond-mat.soft↗

arXiv2025

Topological potentials guiding protein self-assembly

The simulated self-assembly of molecular building blocks into functional complexes is a key area of study in computational biology and materials science. Self-assembly simulations of proteins using physically-motivated potentials for non-polar interactions, can identify the biologically correct assembly as the energy-minimizing state. Short-range potentials, however, produce rugged energy landscapes, which lead to simulations becoming trapped in non-functional local minimizers. Successful self-assembly simulations depend on the physical realism of the driving potentials as well as their ability to efficiently explore the configuration space. We introduce a long-range topological potential, quantified via weighted total persistence, and combine it with the morphometric approach to solvation-free energy. This combination improves the assembly success rate in simulations of the tobacco mosaic virus dimer and other protein complexes by up to sixteen-fold compared with the morphometric model alone. It further enables successful simulation in systems that don't otherwise assemble during the examined timescales. Compared to previous topology-based work, which has been primarily descriptive, our approach uses topological measures as an active energetic bias that is independent of electrostatics or chemical specificity and depends only on atomic coordinates. Therefore, the method can, in principle, be applied to arbitrary systems where such coordinates are optimized. Integrating topological descriptions into an energy function offers a general strategy for overcoming kinetic barriers in molecular simulations, with potential applications in drug design, materials development, and the study of complex self-assembly processes.

cond-mat.soft↗

arXiv2025

Transforming tubular packings to bicontinuous surfaces

The link between bicontinuous architectures in biological membranes and triply-periodic minimal surfaces (TPMS) is a well established example of stunning geometric form in nature. The prolamellar body (PLB) in early plant plastid development is a classic example, forming the Diamond TPMS in a lipid-protein-pigment membrane. However, the early development of such spectacular geometric structures is poorly understood. Inspired by the presence of tubules in the micrographs of early plastid membrane formation, we explore here geometric modelling of transformations of packings of cylinders that coalesce together to form bicontinuous structures. Using computational modelling, we find that specific cylinder packings with cubic symmetry transform into highly symmetric TPMS, which now stand as a candidate set of surfaces for further investigation into the PLB, as well as other occurrences of bicontinuous membranes.

cond-mat.soft↗

arXiv2025

Can solvents tie knots? Helical folds of biopolymers in liquid environments

Helices are the quintessential geometric motif of the microscale, from alpha-helices in proteins to double helices in DNA. Assembly of the helical geometry of biopolymers is a foundational step in a hierarchy of structure that eventually leads to biological activity. By simulating folding in a simplified setting we probe the role of the solvent in the collaborative processes governing biomaterials. Using a simulation technique based on the morphometric approach to solvation, we performed computer experiments in which a short, flexible tube-modelling a biopolymer in an aqueous environment-folds solely based on the interaction of the tube with the solvent. Our findings reveal a variety of helical structures that assemble depending on solvent conditions, including overhand knots and symmetric double helices. By differentiating the role of solvation, our work illuminates the environment of all soluble biomolecules, demonstrating that the solvent can drive fundamental rearrangements, even up to tying a simple overhand knot.

cond-mat.soft↗

arXiv2025

Hierarchical woven fibrillar structures in developing single gyroids in butterflies

Nature offers a remarkable diversity of nanomaterials that have extraordinary functional and structural properties. Intrinsic to nature is the impressive ability to form complex ordered nanomaterials via self-organization. One particularly intriguing nanostructure is the Gyroid, a network-like structure exhibiting high symmetry and complex topology. Although its existence in cells and tissues across many biological kingdoms is well documented, how and why it forms remains elusive and uncovering these formation mechanisms will undoubtedly inform bioinspired designs. A beautiful example is the smooth single gyroid that is found in the wing scales of several butterflies, where it behaves as a photonic crystal generating a vibrant green colour. Here, we report that the gyroid structures of the Emerald-patched Cattleheart, Parides sesostris, develop as woven fibrillar structures, disputing the commonly held assumption that they form as smooth constructs. Ultramicroscopy of pupal tissue reveals that the gyroid geometry consists of helical weavings of fibres, akin to hyperbolic line patterns decorating the gyroid. Interestingly, despite their fibrillar nature, electron diffraction reveals the absence of crystalline order within this material. Similar fibrillar structures are also observed in the mature wing scales of P. sesostris specimens with surgically altered pupal development, leading to a blue colouration. Our findings not only introduce a fundamentally new variation of the gyroid in biology but also have significant implications for our understanding of its formation in nature.

cond-mat.soft↗

arXiv2025

Diagrammatic representations of 3-periodic entanglements

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.

math.GT↗

arXiv2023

Robust self-assembly of nonconvex shapes in 2D

We present fast simulation methods for the self-assembly of complex shapes in two dimensions. The shapes are modeled via a general boundary curve and interact via a standard volume term promoting overlap and an interpenetration penalty. To efficiently realize the Gibbs measure on the space of possible configurations we employ the hybrid Monte Carlo algorithm together with a careful use of signed distance functions for energy evaluation. Motivated by the self-assembly of identical coat proteins of the tobacco mosaic virus which assemble into a helical shell, we design a particular nonconvex 2D model shape and demonstrate its robust self-assembly into a unique final state. Our numerical experiments reveal two essential prerequisites for this self-assembly process: blocking and matching (i.e., local repulsion and attraction) of different parts of the boundary; and nonconvexity and handedness of the shape.

physics.comp-ph↗

arXiv2023

Robust geometric modeling of 3-periodic tensegrity frameworks using Riemannian optimization

Framework materials and their deformations provide a compelling relation between materials science and algebraic geometry. Physical distance constraints within the material transform into polynomial constraints, making algebraic geometry and associated numerical strategies feasible for finding equilibrium configurations and deformation pathways. In this paper, we build the necessary geometric formulations and numerical strategies to explore the mechanics of two examples of 3-periodic tensegrity frameworks through non-linear optimization, eventually showing that the structures are auxetic by multiple definitions.

math.AG↗

arXiv2021

Re-entrant tensegrity: A three-periodic, chiral, tensegrity structure that is auxetic

We present a three-periodic, chiral, tensegrity structure and demonstrate that it is auxetic. Our tensegrity structure is constructed using the chiral symmetry Π+ cylinder packing, transforming the cylinders themselves to the elastic elements and cylinder contacts to incompressible rods. The resulting structure displays local re-entrant geometry at its vertices, and is shown to be auxetic when modelled as an equilibrium configuration of spatial constraints subject to a quasi-static deformation. When the tensegrity structure is subsequently modelled as a lattice material with elastic elements, the auxetic behaviour is again confirmed through finite element modelling as well as demonstrated with a 3d printed example. The cubic symmetry of the structure means that the behaviour is independent of the chosen stretching direction and the auxetic behaviour is observed in both perpendicular directions. This structure could be the simplest three-dimensional analogue to the two-dimensional re-entrant honeycomb. This, alongside the chirality of the structure, make it an interesting design target for multifunctional materials.

cond-mat.mtrl-sci↗

arXiv2020

Enumerating Isotopy Classes of Tilings guided by the symmetry of Triply-Periodic Minimal Surfaces

We present a technique for the enumeration of all isotopically distinct ways of tiling a hyperbolic surface of finite genus, possibly nonorientable and with punctures and boundary. This provides a generalization of the enumeration of Delaney-Dress combinatorial tiling theory on the basis of isotopic tiling theory. To accomplish this, we derive representations of the mapping class group of the orbifold associated to the symmetry group in the group of outer automorphisms of the symmetry group of a tiling. We explicitly give descriptions of certain subgroups of mapping class groups and of tilings as decorations on orbifolds, namely those that are commensurate with the Primitive, Diamond and Gyroid triply-periodic minimal surfaces. We use this explicit description to give an array of examples of isotopically distinct tilings of the hyperbolic plane with symmetries generated by rotations, outlining how the approach yields an unambiguous enumeration.

math.GT↗

arXiv2020

Isotopic tiling theory for hyperbolic surfaces

In this paper, we develop the mathematical tools needed to explore isotopy classes of tilings on hyperbolic surfaces of finite genus, possibly nonorientable, with boundary, and punctured. More specifically, we generalize results on Delaney-Dress combinatorial tiling theory using an extension of mapping class groups to orbifolds, in turn using this to study tilings of covering spaces of orbifolds. Moreover, we study finite subgroups of these mapping class groups. Our results can be used to extend the Delaney-Dress combinatorial encoding of a tiling to yield a finite symbol encoding the complexity of an isotopy class of tilings. The results of this paper provide the basis for a complete and unambiguous enumeration of isotopically distinct tilings of hyperbolic surfaces.

math.GT↗

arXiv2013

Network-like propagation of cell-level stress in random foams

Quasistatic simple shearing flow of random monodisperse soap froth is investigated by analyzing Surface Evolver simulations of spatially periodic foams. Elastic-plastic behavior is caused by irreversible topological rearrangements (T1s) that occur when Plateau's laws are violated; the first T1s occur at the elastic limit and at large strains frequent cascades of T1s, composed of one or more individual T1s, sustain the yield-stress plateau. The stress and shape anisotropy of individual cells is quantified by $Q$, a scalar measure derived from the interface tensor that gauges each cell's contribution to the global stress. During each T1 cascade, the connected set of cells with decreasing $Q$, called the \textit{stress release domain}, is network-like and highly non-local. Geometrically, the network-like nature of the stress release domains is corroborated through morphological analysis using the Euler characteristic. The stress release domain is distinctly different from the set of cells that change topology during a T1 cascade. Our results highlight the unique rheological behavior of foams, where complex large-scale cooperative rearrangements of foam cells are observed as a consequence of distinctly local events.

cond-mat.soft↗