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.