A Study of Two-Periodic Knitted Fabrics using Tangles

We have previously created a framework to classify and characterize two-periodic stitch patterns of knitted fabrics that is based on knot and link theory. Using this theory, we are able to associate a unique link in three dimensional sphere S³, to every two-periodic stitch pattern of a knitted fabric. In every such link, the subset of loops representing the yarn satisfy ribbonness or forms a ribbon link. A link is said to be ribbon if each of its component loops bound a 2D disk that pierces through itself making a finite number of slit like intersections lying in the interior of the collection of disks. Based on the ribbonness, and different types of yarn-needle moves catalogued in the knitting literature, we are able to describe knittable stitch patterns composed of -- knit and purl with twists, yarn overs, knit or purl two together, knit front back, slip stitch, cable etc. The description uses the notion of a tangle -- a collection of properly embedded arcs inside a euclidean three ball. This set up lets us define operator invariants for two-periodic knitted stitch patterns because tangles can be described using tensor products and composition by contraction of indices.