Thursday, April 21, 2016
Knot theory socks!
So, about the socks: purple is the unofficial team color of our research group, so choosing purple yarn was obvious. I wanted a design that was knot-theory related, so this sock has a braid on the outside of the leg.
A bit about the math:
A knot is an embedding of a circle into 3-dimensional space. Intuitively, we can think of a length of string that has been tied up in a knot and then had its ends glued together. It is in some essential sense still a circle - if you were a tiny ant walking along the string, you would eventually get back to where you started, so all you would be able to tell would be that it is a circle (you wouldn't be able to gather any information about how it was knotted up). In simplest terms, the basic question of knot theory is how to tell different knots apart. More generally, knots and links (with more than one circle, but still knotted up somehow) are really important to how mathematicians understand 3- and 4-dimensional spaces.
Every knot and link can be represented as the closure of a braid. A mathematical braid is a set of strands, all oriented in one direction, that can cross over each other but not loop back on themselves. To take the braid closure, we glue the top and bottom ends together, with the rightmost top end glued to the rightmost bottom end, and so forth. If we take the braid closure of the braid on this sock, we get the figure eight knot, which is a really cool knot!
The figure eight knot is the second-simplest non-trivial knot (the simplest is the trefoil), and it has the property of being amphicheiral, which means that it can be stretched and rearranged into its mirror image. This is a very cool property!
The second sock (which I've already started) will have a braid whose closure is the Borromean Rings. I'll explain why they're so cool once I've finished knitting the braid.