Session K63: Fabrics, Knits, and Knots

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 BCE. 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.

At the 1D level, knits are composed of an interlocking series of slip knots. At the most basic level there is only one manipulation that creates a knitted stitch – pulling a loop of yarn through another loop. However, there exist hundreds of books with thousands of patterns of stitches with seemingly unbounded complexity.

The topology of knitted stitches has a profound impact on the geometry and elasticity of the resulting fabric. This puts a new spin on additive manufacturing – not only can stitch pattern control the local and global geometry of a textile, but the creation process encodes mechanical properties within the material itself. Unlike standard additive manufacturing techniques, the innate properties of the yarn and the stitch microstructure has a direct effect on the global geometric and mechanical outcome of knitted fabrics.   

A general geometric framework for knitted fabric elasticity

Knitting is a process in which yarn, an essentially filament-like material, is shaped in space to form a fabric, an essentially sheet-like material, via stitching together a lattice of slip-knots. Due to fabric-level dependence on the stitch pattern, a single yarn can be used to create a large variety of fabric geometries and material responses. Moreover, the elasticity of knits remains poorly understood, as evidenced by the lackluster performance of spring-lattice models. We seek a continuum elastic model that predicts the three-dimensional shape of knitted fabric. This model should have the flexibility to be adapted to describe a wide range of stitch patterns and elasticity models. To this end, we have developed a geometric framework for relating the yarn path to the emergent surface geometry of the fabric. The generality of our approach allows for a systematic coarse-graining of yarn degrees of freedom, without a priori specification of a model of yarn elasticity. Thus, we are able to arrive at a stitch pattern-dependent, continuum elastic model of knits by assuming a simple phenomenological model of yarn, whilst allowing for the possibility of including more realistic yarn mechanics and experimental comparison.   

A topological perspective on knitted fabrics

A knitted textile structure can be thought of as a series of slip knots stacked next to each other in multiple rows. In a given row, each loop is held in place by a loop in the preceding row. As a first step towards building a topological theory for knitted textile structures, we take advantage of the doubly periodic structure coming from the ordering of stitches into rows and columns. A two-periodic planar structure has two generators of translational symmetry. We get a minimal unit cell that tiles the original structure by modding out by these elements of symmetry. The resulting embedding of a curve inside the unit cell is equivalent to a knot sitting in the thickened two-torus. To study this class of knots, we aim to construct a knot invariant or link invariant based on the process of knitting -- using two needles to form slip knots in yarn -- to make an arbitrary two-periodic knitted textile structure. Such a knot invariant inherits an algebraic structure that reflects how and which elementary operations are used to make a given knitted textile structure and, as a result, tells us whether a given doubly periodic structure can be realized by knitting.   

Top Down Modeling of Complex Knit Structures: Beyond Jersey Knits

Modeling of knit textiles is popular in fields such as mechanical engineering and physics, where researchers are working to understand effects of yarn and loop geometries on fabric properties. Often, this can be computationally expensive, allowing only for modeling of planar fabrics such as jersey, made from all knit stitches. We see potential however, in complex self-folding structures, made with knit and purl stitches, as a means of engineering metamaterial textile properties. To aid in modeling of these structures, we have developed a topological framework for the knit structure using families of bicontinous surfaces which allow us to geometrically understand the physics of the characteristic boundary condition curling of jersey knits. We then consider complex self-folding as a result of competition between these boundaries, affected by contributing magnitudes of forces from course and wale directions. By characterizing these forces, we are developing a system of predicting this folding, through understanding of generalized behaviors that repeat regardless of material or machine. By studying interactions between segments of knit and purl stitches, rather than loops or yarns, we can more quickly understand this behavior, to engineer novel textile properties.   

Stable elastic knots with no self-contact

Knots are widespread, universal physical structures, from shoelaces to Celtic decoration to the many variants familiar to sailors. They are often simple to construct and aesthetically appealing, yet remain topologically and mechanically quite complex. Knots are also common in biopolymers such as DNA and proteins, with numerous and significant biological implications.

While self-contact is an inevitable feature of tight knots, here we go the other direction and ask whether a knotted filament with zero points of self-contact may be realized physically. Our focus is on the simple hand-held experiment of an elastic rod bent into a trefoil knot, with the ends held clamped. The question we consider is whether there exist stable configurations for which there are no points of self-contact. This idea can be fairly easily replicated with a thin strip of paper, but is more difficult or even impossible with a flexible wire. We search for such configurations within the space of three tuning parameters related to the degrees of freedom in the simple experiment. Mathematically, we show, both within standard Kirchhoff theory as well within an elastic strip theory, that stable and contact-free knotted configurations can be found, and we classify the corresponding parametric regions. Numerical results are complemented with an asymptotic analysis that demonstrates the presence of knots near the doubly-covered ring. In the case of the strip model, quantitative experiments of the region of good knots are also provided to validate the theory.
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Untangling the mechanics of elastic knots

In surgery, knots are used as ligatures to bind surgical thread during suturing. Even if breakage or unraveling of knotted configurations can be disastrous, suturing guidelines are mostly empirical. Knot theory, a well-established field of mathematics, tends to focus on idealized, non-elastic knots. Moreover, analytical models based on Kirchhoff’s theory for elastic rods are limited to simple knots in loose configurations. However, functioning knots are in general tight and involve elastic deformation of the thread, self-contact and nontrivial frictional interactions. We tackle this problem by performing high precision experiments to acquire unprecedented experimental data on the geometry and deformation of simple open-knots. We make use of X-ray micro computed tomography to acquire volumetric information of knotted configurations on homogeneous elastomeric rods. Emphasis is placed on systematically exploring how the mechanical properties of the rod, friction, and the externally applied loads, all conspire to dictate the mechanical performance of knotted structures. We hope that the physical insight gained from this experimental characterization will form the bases for future predictive models for physical knots.   

Not-knots as the building block of elastic knots

Knots are key for a wide variety of applications such as mooring ships to docks, ensuring the safety of a falling climber or fastening surgical suture threads. Even if knots have been used in hundreds of configurations for millennia, the understanding of their mechanical behavior remains mainly empirical. Past fundamental studies on knots include ideal knots, purely based on geometry and one-dimensional reduced elasticity models. For tight knots, intricate three-dimensional geometries, large deformations, and friction between rod strands are all present in setting a highly nonlinear and coupled behavior. To gain better insight into this complex class of problems, we study the ‘not-knot’; a simpler model system composed of a clasp of two bent elastic rods brought together in mechanical contact. We use X-ray computed tomography to probe the geometry of not-knots, as well as precise force-displacement measurements to quantify the interplay of bending curvature, elasticity, and friction. We believe that regarding complex tight knots as assemblies of simple not-knots will provide a solid foundation to develop much needed predictive models for knotted structures.   

From knots to weaved baskets: Unravelling the mechanics of a clasp between two contacting filaments through numerical experiments

Tight elastic knots and weaved structures tend to exhibit intricate modes of deformation with nontrivial regions of contact that call for a fully three-dimensional description. Given this complexity, there is a striking lack of predictive models for knotted or weaved structures, the design of which tends to rely mostly on accumulated experience and empirical craftsmanship. Here, we study the elementary, yet rich and informative, canonical case of ‘a clasp’ formed by two elastic rods in crossing contact. We first start with the case of rods that have an originally circular cross-section. We tackle this problem by performing finite element simulations of this clasp configuration and contrast our results with experimental data obtained using X-ray tomography of the corresponding physical structures. We compare our results to a well-established description for ideal claps of geometrically rigid strings (that exclude elasticity), finding that the latter acts as an underlying ‘backbone’ for the full elastic solution. Finally, we extend our framework to a clasp formed by strips of non-circular cross-sections, which is the building block in basket weaving.   

Detangling hair

Tangled hair is difficult to manage. We investigate the everyday problem of taming a tangle of hair with a comb by focusing on a minimal model of this system: a pair of chiral, elastic helical rods entangled to form a braid. As a single, stiff tine combs through the double-helix, it leaves two untangled filaments in its wake. We use experiments, theory and computation to characterize this problem with both mechanics and topology.   

Born in the Wrong Geometry: Longitudinal and transverse frustration in non-parallel filament bundles

Assemblies of one-dimensional, filamentous materials are commonplace in physical systems, from microscopic materials, such as columnar liquid crystals and biopolymer bundles, to familiar, macroscopic materials like wires, cables, and ropes. The geometry of continuus filament bundles with cross sectional ordering in three dimensions is highly constrained, and we show that only two families of such filament bundles permit equidistant configurations of their constituent filaments: those with bend, but no twist, and those with twist, but no bend. The elastic response of bent and twisted bundles with no such ordered ground state, such as those formed by DNA plasmids under confinement, is then doubly geometrically frustrated: the presence of twist frustrates crystalline order in the cross-section, and the presence of bend couples this compromise structure to the filament tangents. In filaments, as in many frustrated systems, the global response to deformations can not be adequately described by a linear energy. We present a fully nonlinear theory for the elasticity of bundles with one-dimensional components which can slide freely along their tangents, and discuss the response of Euclidean filament packings to the imposition of non-equidistant geometries.