Robin Endelman

Associate Professor of Mathematics
Department of Mathematics & Statistics, University of the Fraser Valley
Abbotsford, BC, Canada

When not doing mathematics (abstract algebra, representation theory), I am an excessive knitter. The design for this piece was born in 2003, while I was a post-doc, and the Yang Baxter equation was fresh in my mind. I am excited that now (only 10 years later!) I have been able to create it.

The colours chosen were inspired by the 2013 thematic year of the Mathematics of Planet Earth (MPE2013), to represent water, sun, and life on earth. The yarn company (Manos del Uruguay) was chosen for its commitment to sustainable development, local farmers and artisans, and fair trade. The fibre (Silk Blend) was chosen for its softness and comfort, as the scarf is meant to be worn!

The Yang-Baxter Scarf
The Yang-Baxter Scarf
Manos del Uruguay's Silk Blend (hand-dyed merino/silk)

The scarf depicts the Yang Baxter equation of statistical mechanics, a variation of which is the braid equation in algebra, or the 3rd Reidemeister move in knot theory. Assigning the numbers 1,2,3 to the colors blue, green, gold, respectively, the Yang-Baxter equation reads: R_12 R_13 R_23 = R_23 R_13 R_12, where R_ij denotes strand i crossing strand j. The two sides of the equation are depicted in the ends of the scarf, with equality represented by the middle portion.

For the equation to read correctly, the fabric needed to be double-sided. To create the two-sided fabric and traveling cables simultaneously, I introduced a hybrid of double-knitting and tubular knitting. As a result, the colored strands are produced as embedded I-cords.