Knot theory socks!

I'm a knot theorist.  (I've posted about this before.)  I defended my Ph.D. thesis last week, my semester ends next week (with me grading lots of finals ... ), and my graduation will be in mid-May.  I can barely believe I'm actually done, and I'm really excited to start my new job in August!  These knot theory socks are a present for my advisor, to thank her for all of the many, many things she has done for me.  I feel so lucky to have such a great advisor.

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.

Being a mathematician improves my knitting

I am a mathematician.  The fact that I am a mathematician shapes the way I think, and slips out when I tell my mom that she doesn't need to worry about my apartment flooding because it's at a local maximum, or how I have a special appreciation for the non-simply-connected geological features at Arches National Park, or in my knitting.  Lately I've been noticing how my mathematical ways of thinking are helping me knit this little cable cardigan (Trellis from Knitty) for my nephew.

Modular Arithmetic: This particular cardigan has an 18-row cable pattern that repeats several times beginning with row 9 of the sweater body, while at the same time you knit a buttonhole every tenth row beginning in row 5.  So I know that I need to put a buttonhole in row 5, 15, 25, 35, and 45.  I want to start counting my rows with row 1 of the CABLE PATTERN, so using the new numbering system for rows, I'll be putting buttonholes in row 7 (this is the second buttonhole), 17, 27, and 37.  Then using modular arithmetic (also known as clock arithmetic) I reduce those modulo 18 and work buttonholes in rows 7, 17, 9, and 1 of the CABLE PATTERN. For me, this is much easier than trying to keep track of one count for the cable pattern and another for the buttonholes - instead, I just track everything in terms of the cable pattern.

Symmetry:  If you look closely at my photos in this post and compare them to the photos in the pattern, you'll notice that I changed some of the cable crossings.  Many mathematicians care a lot about symmetry (non-mathematicians care about this too, of course, but we're trained to notice it wherever we can).  This is a case where I think the pattern-writer was wrong.  If you imagine a vertical line going down the middle of the back of the sweater, in the center seed stitch column, and think of reflecting one side of the sweater across that line, you would get the other side of the sweater.  This is called a reflectional symmetry, and it makes for a much more pleasing image than what is written in the original pattern, with all of the large fancy cables twisting to the "right" and all of the little cables twisting "left."  I fixed this so that the two large fancy cables on each of the front and back twist toward the center, and each of the little cables twists toward the large fancy cable it frames.  

Braids:   This one doesn't really improve my knitting as much as add to my enjoyment.  My research is in knot theory, which is closely related to the study of mathematical braids. Every knitted cable is a braid; in this sweater, each of the fancy cables is a two-strand braid, and each of the little cables is a four-strand braid.  Referring back to symmetry for a moment, the mirror image of a braid is its inverse, so in this sweater we see braids paired with their inverses.  It makes me so happy when my work shows up in other areas of my life!  I'm so glad I'm a mathematician - if I wasn't, I wouldn't be able to properly appreciate this little sweater!  

Two weeks of CMS/Colloquium knitting

This is Taylor's second sock at the end of colloquium last week. Here it was at the end of colloquium yesterday:
I am almost done with the leg! Next week I should be able to post a heel picture.

Yesterday's colloquium was super cool, because the speaker talked about work that one of my undergraduate professors did. Also about trees and flowers and forests and gardens.

Knitted Surfaces

Today I was procrastinating on the interwebs, and I came across a variety of super-cool mathematical knitting resources. Most of these involve the knitting of various orientable and non-orientable surfaces.

Möbius bands are fairly ubiquitous these days, with varying degrees of mathematical precision and authenticity (I would contend that a möbius band with intrinsic twist is superior to one that is knitted as a rectangle and then stitched together). I haven't made one yet, but I'm interested ...

Orientable surfaces are pretty easy to knit (at least in theory - I'm sure that if I actually try to do it I'll discover differently), but many non-orientable surfaces (f.eks. the Klein bottle and RP^2) do not live in R^3. This makes knitting them rather difficult, since "immersing" them into our world involves singularities. Maybe this will be the subject of next year's CMS talk ...

Here are a few resources I found:

1. Dr. Miles Reid, a British mathematician, put together this series of articles on knitted surfaces when he was a grad student (I think). It's pretty technical - not recommended for non-mathematicians.
2. The Home of Mathematical Knitting is a much more accessible list of resources maintained by Dr. Sarah-Marie Belcastro.