Vrije Universiteit AmsterdamWebpage
Weaving with periodicity in one dimension: braids
Weavings in one dimension can be described as braids. This talk presents and compares the results of different generative models of classes of braids that might have some relevance for entanglements of 1-dimensional coordination polymers.
University of PotsdamWebpage
Using periodic surfaces as a scaffold is a convenient route to making periodic entanglements. I will present a systematic way of building new tangled periodic structures, using low-dimensional topology and combinatorics, posing the question of how to characterise the structures more completely.
A mathematical model for bobbin lace grounds
Randall D. Kamien
University of PennsylvaniaWebpage
Scaffolds for Knitting
Doubly and triply periodic surfaces can be used as scaffolds to capture the topology of knit fabric. We discuss the construction of these scaffolds from more basic units and show how we can optimize over their geometry to inform the design of three-dimensional, self-folding knit patterns.
University of LiverpoolWebpage
Introduction to Periodic Geometry and Topology
Periodic textiles were traditionally studied up to continuous deformations in a fixed thickened torus or with a fixed lattice basis. The more natural equivalence of textiles is a periodic isotopy whose invariants were first introduced by Morton and Grishanov. However, this branch of Periodic Topology has many more open questions than answers. The talk will focus on the related area of Periodic Geometry where the similar ambiguity of changing a lattice basis plays a key role. The key object is a periodic point set obtained by lattice translations from a finite motif of points in a unit cell. The most practical equivalence of periodic point sets is rigid motion or isometry, which preserves all interpoint distances. Real crystal structures can be reliably distinguished only by isometry invariants that should be also continuous under perturbations of points. Such recent invariants are density functions (Proceedings SoCG 2021, arXiv:2104.11046), average minimum distances (AMD, arXiv:2009.02488) and complete invariant isosets (Proceedings DGMM 2021, arXiv:2103.02749). The ultra-fast AMD enabled invariant-based visualizations of huge crystal datasets within a few hours on a modest desktop. The above work is based on joint papers with several colleagues at the Materials Innovation Factory, Liverpool, UK.
Introduction to define, construct and classify weaves
From innovative woven artificial muscles to garments made from traditional woven fabrics, weavings are historically well-known structures, and still represent a very active research topic in materials science. The study of weaves as new mathematical objects is very interesting in itself but also as an interdisciplinary subject, with the aim of better understanding their geometric and topological structure, often associated with physical properties. This talk attempts to introduce weaves from a formal mathematical point of view. First, we will define Euclidean and hyperbolic weaves and state a new construction methodology. Next, an idea of classification will be discussed.
The Tait First Conjecture for Alternating Weaving Diagrams, Sonia Mahmoudi, preprint. Link
Georgia Institute of TechnologyWebpage
Twisted topological tangles or: the knot theory of knitting
Imagine a 1D curve, then use it to fill a 2D manifold that covers an arbitrary 3D object – this computationally intensive materials challenge has been realized in the ancient technology known as knitting. This process for making functional materials 2D materials from 1D portable cloth dates back to prehistory, with the oldest known examples dating from the 11th century CE. Knitted textiles are ubiquitous as they are easy and cheap to create, lightweight, portable, flexible and stretchy. As with many functional materials, the key to knitting’s extraordinary properties lies in its microstructure.
University of LiverpoolWebpage
Capturing the topology of periodic textile patterns
I will discuss the use of diagrams on a torus in giving a compact description of a range of woven and knitted textile patterns. Following work with S. Grishanov I will show how the multi-variable Alexander polynomial can be used to calculate and display invariants of a pattern. Features of the polynomial mean that it can handle changes of scale, and it can also retrieve underlying geometric information. I will comment on some of these advantages, and also on some shortcomings.
Doubly periodic textile patterns, H. R. Morton, S. Grishanov, J. Knot Theory Ramif. 18 (2009), 1597-1622.
Vuong M. Ngo & Mohand Tahar Kechadi
Trinity College Dublin & University College DublinEmail ; Webpage
Structural textile pattern recognition and processing based on hypergraphs
To facilitate the clustering and search textiles at the level of thread structure, we introduce an approach for recognising similar weaving patterns based on their structures for textile archives. We first represent textile structures using hypergraphs and extract multisets of k-neighbourhoods describing weaving patterns from these graphs. Then, for retrieval, the resulting multisets are compared and ranked by various distance measures. While, for clustering, the multisets are clustered using various distance measures and various clustering algorithms.
Structural textile pattern recognition and processing based on hypergraphs, V. M. Ngo et al., Information Retrieval Journal, 24 (2021), 137–173.
University of TennesseeWebpage
Effects of entanglement in mechanical properties of material
Material composed by entangled filamentous structures (like textiles, polymers and metal wires) exhibit complex mechanical properties. Modeling of such systems usually includes Periodic Boundary Conditions (PBC). To measure entanglement in PBC, we use the periodic linking number. We show that the periodic linking number correlates with the mechanical response of these systems. To account for multi-chain entanglement, we introduce the Jones polynomial in PBC and discuss its properties. A problem with applying the Jones polynomial to physical systems is that its calculation is computationally expensive. We discuss how Vassiliev measures and Vassiliev invariants can provide efficient alternatives.
Jessica S. Purcell
Geometry of biperiodic alternating links
A biperiodic alternating link is a link in the plane that has a quotient that is an alternating link in the thickened torus; the infinite weave is an example of such a link. We consider invariants of these links using hyperbolic geometry. We present a family of such links, called semi-regular links, that have their hyperbolic geometry completely determined by a Euclidean tiling. In this talk, we will describe the links and their underlying hyperbolic geometry. This is joint work with Abhijit Champanerkar and Ilya Kofman.
3-dimensional topology and polycontinuous pattern
We study polycontinuous patterns that appear as microphase separation in block copolymers using the 3-dimensional topology. A triply periodic pattern gives a handlebody decomposition of the 3-dimensional torus. We will discuss a characterization of handlebody decompositions of the 3-dimensional torus.