Triply invertible scarf sewing adventures (and instructions)

Abstract

We provide relevant math and detailed sewing instructions for constructing a toroidal scarf that reverses three ways and whose design uses the unique inversion properties of a particular torus geometry and particular 3-component link. We explain how the scarf’s sewing instructions are guided by the mathematics underlying its construction.

GRAPHICAL ABSTRACT

KEYWORDS:

• Torus
• topology
• knot theory
• links
• torus knots
• inversion
• toroidal scarf
• Möbius scarf
• fibre arts
• sewing

Acknowledgments

We would like to thank the editors and anonymous reviewers for their many helpful suggestions. The torus knot images in Figures 9 and 20 were produced using Robert Scharein’s KnotPlot (1998–2023), which is a marvellous tool. Thank you to Kent Christman for helping diagnose the misstep the first time Ellie tried to sew a $\left(1,1\right)$ torus knot seam on a rhombus layout, and for his excellent active listening in general.

Disclosure statement

No potential conflict of interest was reported by the author(s).