Anyone with necklaces or lace-up shoes has some first-hand experience with knots, but believe it or not (knot?), there is an entire mathematical discipline dedicated to studying knots and some closely related concepts. A mathematical knot is almost like a real-world knot, but it can't have any ends. So if you're thinking of a shoelace, imagine tying it in a normal knot and then gluing the aglets together. (OK, I just wanted to show off the fact that I know the word "aglet." It's the little plastic bit on the end of your shoelace.) I don't study knot theory, but I have found myself involved with mathematical knots twice at the Joint Math Meetings, which is two more times than most weeks.
My first knotty experience was on Thursday afternoon during the joint prize session. This is a time for the mathematical community to get together and pat each other on the back. Prizes are awarded for research (including the Oswald Veblen Prize in Geometry to Ian Agol, some of whose work is described in this beautiful article by Erica Klarreich, and Daniel Wise), teaching, service to the mathematical community, and public communication and outreach. Scientific American readers may be interested to learn that John Allen Paulos, author of several popular books about mathematics, including Innumeracy and A Mathematician Reads the Newspaper, and many articles in Scientific American and other magazines, received the Communications Award from the Joint Policy Board for Mathematics.
But as a young(-ish) female mathematician, I was most excited about the Alice T. Schafer Prize for Excellence in Mathematics by an Undergraduate Woman, which is awarded annually by the Association for Women in Mathematics. This year's winner is MurphyKate Montee, a senior at the University of Notre Dame.
Montee's research has been primarily in the area of knot theory. Mathematicians generally work with "projections," or diagrams, of knots, ways of drawing them two-dimensionally. Sometimes two drawings of a knot represent the same knot in three-dimensional space, just seen from a different angle or with the "rope" of the knot jiggled around. Any amount of stretching and twisting is allowed, but the knot can't be cut. For example, you can see that the two diagrams below represent the same knot. (It is just a boring circle, also called the unknot.)
One of the problems knot theorists tackle is how to tell knot diagrams apart. It isn't too hard to see that the two pictures above are equivalent, but sometimes it can be trickier. For example, below is another picture of the unknot. So it would be possible to unfold and untwist this sucker to get a regular circle, but I would certainly have trouble doing it.
In one of the papers Montee has coauthored [pdf], she and her colleauges developed a new way of creating knot projections that she describes as "daisy-shaped." Instead of separating all of the crossings where one strand passes over another, this type of projection pushes all the crossings to a single point, with loops, or "petals" surrounding it.
They use this projection to develop a new knot invariant, which can be used to distinguish two different knots. Invariants are kind of like filters or sieves. It generally assigns a number or polynomial expression to a knot diagram. Any two different ways of drawing the same knot need to yield the same number or expression, or else it's pretty useless as an invariant, but some invariants are more sensitive than others, in some sense a finer sieve. Montee's research shows that the new knot invariant is as sensitive as possible: it can distinguish knots uniquely, which is unusual. But that sensitivity comes at a cost. It can be difficult to compute for a complicated diagram, and if you can't compute the invariant, you can't use its powerful sensitivity property.
Montee says she and her colleagues have some ideas of how to continue this project, and she is looking forward to exploring math wherever she goes to graduate school. I only got to talk to her for a few minutes, but she definitely makes me optimistic about up-and-coming young mathematicians!
My second knot-ical experience was being part of "Flash Knots" on Friday at noon. Initially conceived to be a flash mob reenactment of a portion of Diana Davis's award-winning "Dance Your Ph.D." submission, it was decided that a human knot would be easier to implement and could involve more people. So at noon, a bunch of us stood outside of the exhibition hall and grabbed hands to become a giant human knot. We tried to untangle ourselves as much as possible, but the knot dissolved before we were able to determine whether it was the unknot or not. Afterward, some of us assembled for a picture to celebrate the 125th birthday of the American Mathematical Society.
I had fun playing with knots at the Joint Math Meetings, but I think I'm knot cut out for it in the long term.