Knotty Knits are Tangles in Tori

Shashank G Markande and Elisabetta Matsumoto
Proceedings of Bridges 2020: Mathematics, Art, Music, Architecture, Education, Culture
Pages 103–112 Regular Papers


In this paper we outline a topological framework for constructing 2-periodic knitted stitches and an algebra for joining stitches together to form more complicated textiles. Our topological framework can be constructed from certain topological “moves" which correspond to “operations" that knitters make when they create a stitch. In knitting, unlike Jacquard weaves, a set of n loops may be combined in topologically nontrivial ways to create n 
stitches. We define a swatch as a mathematical construction that captures the topological manipulations a hand knitter makes. Swatches can capture the topology of all possible 2-periodic knitted motifs: standard patterns such as garter and ribbing, cables in which stitches connect one row of loops to a permutation of those same loops on the next row much like operators of a braid group, and lace or pieces with shaping which use increases and decreases to disrupt the underlying square lattice of stitches.