Borromean Scarves

The Borromean Rings is a collection of 3 knots that satisfy the peculiar property where any 2 of the knots are not linked, but all 3 of them are linked together:

Removing any of the rings in this image would result in the other two rings being disconnected.

The Borromean scarves are a crochet project that I embarked on in March 2023. These 3 scarves are a physical realisation of the Borromean Rings.

I used the Caron Colorama Halo O'Go Yarn, although I did not use enough of the yarn for the gradient effect of the yarn to show.

A natural question to ask is: How many ways are there to wear these scarves?

Here are 2 possible ways to wear the scarf that I find comfortable.

Assuming that the scarf must be worn on the neck, so that no portion of the scarf is above your head or below your feet. Then one can infinitely extend the head and the feet of the body to meet at the point at infinity of the 3-sphere form a closed curve. From this one can see that there is a bijection between the ways that the scarf can be worn, and elements of the link group of the Borromean rings. i.e. the fundamental group of the link-complement.

Here are some other ways one might wear this scarf.