A study of 2-periodic weft-knitted textiles using the theory of knots and links

TL;DR

This work builds a rigorous bridge between two-periodic weft-knitted textiles and low-dimensional topology by modeling knitting motifs as links in the thickened torus $T^2\times I$ and translating them into classical $S^3$-links via Dehn filling along a Hopf link. It introduces swatches—a controlled, band-surgery–based construction that mirrors the act of knitting—and proves that textile links give rise to ribbon links in $S^3$, enabling obstruction via slice/ribbon invariants. The authors develop an algebra of swatches using meridional and longitudinal annulus sums, partition swatches into irreducible components, and analyze a range of invariants (hyperbolic volume, cusp shapes, trace fields, MVA, Jones polynomial, det) to classify and compare two-periodic motifs. They propose conjectures linking swatch composition to hyperbolic structure and multivariable Alexander polynomials, and they discuss implications for the topology-mechanics of knitted fabrics as well as potential universal knitting grammars grounded in 3-manifold operations.

Abstract

In this study, we use a correspondence between two-periodic weft-knitted textiles and links in the thickened torus to study the former using link invariants. We establish a criterion to identify the set of links whose elements are realized through techniques of weft-knitting leading to new, unconventional types of weft-knitting stitch patterns. A crucial topological underpinning of these links is shown to be their correspondence with ribbon knots and links in Euclidean three-space and equivalently in the three-sphere. Using the mechanics of weft-knitting, we propose a protocol for constructing and enumerating links in the thickened torus that can be knitted as a motif of a weft-knitted textile, and we call such links \emph{swatches}. Based on our analysis of link invariants of swatches, we propose conjectures on hyperbolic structure of the link complements of swatches and their multivariable Alexander polynomials.

Figures (31)

• Figure 1: Construction of a basic knit stitch. The top images show 3D manipulations of the knitting needles and yarn. In planar knot diagrams (bottom images), the needles act like punctures in $\mathbb{R}^2$ that the yarn cannot pass through. When the pointed ends of the needles are facing into the page, they are marked with an (X) and when they are facing out of the page, they are marked with an (O).
• Figure 2: (a) The bulk of stockinette fabric. (b) A space curve representation of a finite patch of the two-periodic knit projected orthogonally on to the $xy$-plane. The block in red is the fundamental translational unit and the symbol k denotes the repeating motifs known as knits. (c) The planar diagram of the corresponding textile knot in $T^2\times[0,1]$.
• Figure 3: Images of some weft-knitted fabrics made from knit and purl stitches and the corresponding two-periodic versions.
• Figure 4: The Hopf link, $H\subset\mathbb{R}^3$ along with multiple two-dimensional open sliced tori embedded in $\mathbb{R}^3\setminus H$. The complement of the Hopf link is $T^2 \times I$.
• Figure 5: Textile knots naturally live in $T^2\times I$. However, we can construct textile knots as three component links in $S^3$ as illustrated in the above figures.

Theorems & Definitions (3)

• proof
• proof : Proof of Theorem \ref{['theorem:ribbon']}
• proof