Michael Matthew John Treacy

Finite tangles are embeddings of graphs that include knots or links as subcomponents. We describe vertex-transitive tangles with 12 or fewer crossings, as well as tangles with just two crossings that feature the Hopf link. Additionally, we consider tangles formed by adding edges to a ravel. This article is part of the theme issue ‘Geometry, materials and the imagination’.

Vertex- and edge-transitive graphs that admit tilings (tessellate graphs) have been known for 20 years. In this paper, we identify additional families of vertex- and edge-transitive graphs with symmetric embeddings, for which the order of the point symmetry at a vertex equals the vertex's coordination number. Remarkably, symmetric tessellate graphs appear over 150 000 times in crystal structures, while there is only one reported occurrence of a symmetric decussate (untessellable) graph.

Three-dimensional atomic structure is routinely determined for periodic crystals. However, extending such analysis to amorphous materials remains a substantial challenge, despite the scientific and technological importance^1,2. In this context, a recent report describing the three-dimensional structure determination of an amorphous solid using atomic-resolution electron tomography (AET) is truly remarkable³. If validated, such an analysis would be groundbreaking. Here we address this issue and investigate whether and when AET can identify all or most atoms in an amorphous nanoparticle. By simulating AET, we reveal limitations on the structural and chemical information AET can determine from noisy electron images. For monoatomic nanoparticles, the structure can be determined with an atomic-position accuracy of tens of picometres under stringent fluence, sampling and projection requirements. For multi-element amorphous nanoparticles, chemical identification resolution is determined by noise and experimental sampling. Heavier atoms are more easily resolved than lighter ones, and large chemical analysis uncertainties emerge when atomic peak and background intensities overlap. Using these insights, we delineate nanoparticle size, composition, electron fluence and image sampling requirements for AET. The results serve as a benchmark for future experiment design and demonstrate a viable approach for amorphous structure determination validation using AET.

Tangles are defined as embeddings of connected graphs that contain knots and/or links as substructures. We enumerate and describe piecewise-linear embeddings of cubic and icosahedral tangles and linked polyhedra with transitivity [1 2]; that is, with one kind of vertex (all related by symmetry) and two kinds of edge. There are eight families of catenated polyhedra and ten families of tangles. For each family, we only describe the members with the largest girth, where girth is defined as the ratio of the shortest distance between edges with no shared vertices to the length of the longest edge.

Supersymmetric structures are defined as edge-transitive spatial graphs where the order of the site symmetry at each vertex is at least twice the coordination number of the vertex. Only 11 vertex- and edge-transitive connected periodic graphs possess this property, which we describe. We identify 32 examples of supersymmetric vertex 2-transitive, edge-transitive nets, also detailed here. Additionally, we list supersymmetric interpenetrating and interwoven structures, including weavings and polycatenanes.

Anomalously low values of the normalised variance in fluctuation electron microscopy (FEM) have been frequently reported. We present three experimental corrections for quantitative interpretation that significantly modify conventional approaches. FEM relies on measurements of intensity statistics in coherent nanodiffraction patterns. We demonstrate that sampling the nanodiffraction patterns with a pixelated detector removes high‐frequency signals and reduces statistical variance. The most significant impact is on the background normalised variance, which arises from random atomic alignments and is distinct from the normalised variance peaks associated with the correlated alignments of medium‐range order. Indeed, we show that if the peaks are background‐subtracted, their height is much less affected by the detector effect, provided the experimental conditions are optimised. We show that shot noise correction must also be adjusted to account for the camera Modulation Transfer Function (MTF) effects. Additionally, we demonstrate through experiment that the traditional method of thickness correction for a‐Si is inadequate and propose an alternative approach to address thickness variations. We speculate on the origin of the anomalous thickness effect in terms of displacement decoherence due to sample ‘fluttering’ under irradiation.

We provide a systematic account using symmetry and coordinates to explore symmetric (vertex- and edge-transitive) 3-periodic weavings of piecewise linear threads. The dia-w graph, derived from the diamond structure, generates an infinite family of supersymmetric 3-periodic thread weaves – a unique property shared only with the 2-periodic sql-w family of fabric weaves. Additionally, we describe a selection of symmetric 3-periodic, 3-coordinated weavings related to the srs graph and a supersymmetric Borromean weave of hcb graphs.

We investigate how data acquisition rate affects the decoherence of diffraction speckles in fluctuation electron microscopy (FEM) experiments on amorphous silicon at 80 kV. Surprisingly, reducing acquisition time from 256 ms to 1 ms does not significantly enhance the intensity variance peaks related to medium-range order. This suggests that decoherence processes operate at timescales faster than 1 ms. At the highest acquisition rates, noise complicates the variance background estimation. A significant source of non-Poisson noise is the spread of electron signals across adjacent detector pixels. We mostly restore the discrete pulse counting needed to mitigate Poisson noise by rounding pixel intensity to the nearest integer-electron value. However, a residual negative-variance offset grows as the acquisition rate increases. Efficient electron pulse counting in detectors is crucial for processing Poisson noise in FEM, especially with weak signals.

A systematic description of 1-periodic polycatenanes is given. The description uses piecewise-linear embeddings (straight edges) and is limited to structures with symmetry-related vertices (isogonal). Components linked are polygons, including knotted polygons and polyhedra. The structures described are generally those with the order of rotational symmetry up to 10. An account is given of transitivity and intransitivity in patterns of links.

We report symmetric (vertex- and arc-transitive) embeddings of catenated rings, polyhedra and rods. Linked triangles form infinite families of structures, and we limit this report to only structures with each ring linked to three or six others. For linked squares, hexagons, tetrahedra, octahedra, cubes and rods, only a small number of symmetric structures were found, and all are reported.