Knots, Coloring and Applications
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1 Knots, Coloring and Applications Ben Webster University of Virginia March 10, 2015 Ben Webster (UVA) Knots, Coloring and Applications March 10, / 14
2 This talk is online at Ben Webster (UVA) Knots, Coloring and Applications March 10, / 14
3 What s a knot? What I want to tell you about is the mathematical theory of knots. Knots are just tangles of string; I ll always want them to be closed, like I ve taken a rope and burned the ends together. The secret of mathematics is forgetting things; that may sound a little strange, but point is to throw out a lot of details and see what you can conclude from the few things you still remember. Thus I m not going to worry about: the material that the knot is made out of, how thick it is, how long the knot is. I ll imagine that the knot is made out of infinitely thin and infinitely strong material. In particular, I m not allowed to cut the material. how the knot is positioned in space. I ll draw knots flat on the page (what s called a projection) but exactly how it s arranged is not what I m worried about. Ben Webster (UVA) Knots, Coloring and Applications March 10, / 14
4 Examples of knots Ben Webster (UVA) Knots, Coloring and Applications March 10, / 14
5 Why should we care? I think there s something to be said for studying these things for their own sake, but let me try to make some arguments for the usefulness of this approach. The original motivation for studying knots was the (ultimately very wrong) theorem that atoms where vortices in the aether like little knotted strings, and the shape of the knot determined the chemical propeties. More recently, people have applied knot theory to quantum computing. Very loosely, you can do computations by setting up the answer to be the probability that particles in a certain system will trace out a particular knot and observe this probability directly. Ben Webster (UVA) Knots, Coloring and Applications March 10, / 14
6 Quantum computing These applications are based on the theory of anyons. These are physical states that behave like they are trapped in a 2-dimensional medium. In a 2-dimensional medium, when you look at, say, the probability that two particles will switch places, there are two different ways that they might have switched. These sorts of physical systems can be realized by taking two crystals with slightly different compositions, and applying a magnetic field to excite things. Ben Webster (UVA) Knots, Coloring and Applications March 10, / 14
7 Quantum computing So now our program consists of a link, and running the program involves cooking up one of these systems physically and observing the probability that the particles trace out our chosen link. This approach allows us to do any computation we can do with a usual computer, and lots of hard programs (prime factorization, discrete logs, etc.) can be done in polynomial time. At the moment, it s difficult and expensive to get the physical system going and do the observations, but progress is being made. I ve heard this described as the second least insane approach to quantum computing. Ben Webster (UVA) Knots, Coloring and Applications March 10, / 14
8 DNA Maybe the most exciting application to appear in recent years is to biology in the structure of DNA and proteins. It might not seem immediately to matter too much exactly how DNA is tangled up (which it inevitably becomes). But remember how bacteria reproduce: they take their genome, split it through the middle of helix, and turn each half into a new copy of the genome. That is not going to work very well if the genome is tied up into a complicated knot. Enzymes can fix this, but how long it takes is important. Ben Webster (UVA) Knots, Coloring and Applications March 10, / 14
9 Telling knots apart Let s step back into the abstract. I showed you a slide with all the different knots which can be draw with 7 or more crossings. But how do I know that they are really different? If I hand you two knots, and you think they are the same, you can just check it. If you think they are different, how do you check that? If I hand you a rubber band, how can you be sure that someone must have slit it open and messed with it? You would need to try infinitely many different ways of rearrange things. Who has the time for that? The knots below are called the unknot and the trefoil. I hope you believe they are different. But how can you be sure? Ben Webster (UVA) Knots, Coloring and Applications March 10, / 14
10 Colorings Instead, you have to extract some piece of information that s easier to compare, like a number. Let me tell you my favorite way of getting such a number. Definition A coloring of a knot diagram is a coloring of each arc of the picture red, blue or green such that at any crossing (where three arcs meet) there are either a single color or all three different colors. OK bad OK So, the thing I want to consider is the number of colorings of a knot. Ben Webster (UVA) Knots, Coloring and Applications March 10, / 14
11 Colorings You ll note: I pulled a bait and switch there. I defined a coloring of a knot diagram, and hopefully you believe I can count those. But what I want is to tell knots apart, so if this number might change when I lay the knot down on the page differently, it s not going to be helpful. Luckily, this is not actually a problem. The number of colorings is the same for any diagram of a given knot. The unknot has 3 colorings, where the trefoil has 9, so they are definitely not the same knot. Ben Webster (UVA) Knots, Coloring and Applications March 10, / 14
12 Colorings You ll note: I pulled a bait and switch there. I defined a coloring of a knot diagram, and hopefully you believe I can count those. But what I want is to tell knots apart, so if this number might change when I lay the knot down on the page differently, it s not going to be helpful. Luckily, this is not actually a problem. The number of colorings is the same for any diagram of a given knot. The unknot has 3 colorings, where the trefoil has 9, so they are definitely not the same knot. Ben Webster (UVA) Knots, Coloring and Applications March 10, / 14
13 Coloring is the n = 3 case. A coloring with n = 5 shows that the figure eight is non-trivial. Ben Webster (UVA) Knots, Coloring and Applications March 10, / 14 Invariants But it s not time to declare victory yet. The next most complicated knot is the figure eight, and like the unknot, it only has three colorings. Thus, this particular invariant doesn t show that the figure eight is non-trivial. Luckily, there is a much more general notion of coloring. For example, you can color a knot with the integers modulo n with the rule that you must have: 2a b a b
14 Invariants Luckily, there are a lot more invariants than that. You can use a lot more techniques and different areas of mathematics. Knot theorist isn t even the hat I wear most of the time; most I work with other tools that you can then bring to bear on questions about knots. That s part of what s great about mathematics and science more generally: you never know when a tool or approach that started one place can do something remarkable in another. Ben Webster (UVA) Knots, Coloring and Applications March 10, / 14
15 Thanks. Ben Webster (UVA) Knots, Coloring and Applications March 10, / 14
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