Applications of knot theory

An answer of André Henriques' inspired the following closely related CW question. Parts of the following is extracted from his answer and my comments.

I regularly teach a knot theory class. Every time, students ask about applications. What should I say?

I have two off-the-cuff replies when students ask. The first is that knot theory is a treasure chest of examples for several different branches of topology, geometric group theory, and certain flavours of algebra. The second is a list of engineering and scientific applications: untangling DNA, mixing liquids, and the structure of the Sun's corona. I'm interested hearing about other applications. I am also interested in hearing your take on the pedagogical issues involved. Thank you!

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Lord Kelvin had hypothesized that atoms were made of knots. Moreover, if I remember correctly, I think the idea was that molecules are linked knots. I guess this is not really an application since it turned out to be untrue... It is still a nice story though. – Kevin H. Lin Dec 3 '10 at 22:30
@Kevin: And not just a nice story, but in fact both surprisingly modern and very fruitful. Fruitful in that it directly influenced Tait to start his enumeration of knots, which is perhaps the birth of knot theory. And modern in that it is perhaps one of the first instances of an idea which is periodically revived. Kelvin postulated that atoms were knots in the ether, but replacing "knots" by "topologically stable configurations" and "ether" by "field", then this is the idea behind the Skyrme model for hadrons, etc... – José Figueroa-O'Farrill Dec 3 '10 at 23:19
I have put together some of the source material on knots and their applications on my home page maths.ed.ac.uk/~aar/knots – Andrew Ranicki Dec 4 '10 at 6:28
@Andrew - you need to fix your hagfish link. Also, I am definitely mentioning that application of knot theory next time I teach the class. youtube.com/watch?v=RrPvMMkQkk0 – Sam Nead Dec 5 '10 at 16:28
@Kevin - I always thought that connect sums were supposed to form molecules? Is there a canonical reference? After looking at the Wikipedia page I found the following very romantic article: southalabama.edu/mathstat/personal_pages/silver/scottish.pdf – Sam Nead Dec 5 '10 at 17:01

If I may steal some thunder from Peter Shor, his paper, Quantum money from knots (with Edward Farhi, David Gosset, Avinatan Hassidim, and Andrew Lutomirski) relies for the security of its "quantum money scheme" on

the assumption that given two different looking but equivalent knots, it is difficult to explicitly find a transformation that takes one to the other.

The Alexander polynomial plays a prominent role in the paper.

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This sounds super cool. However, I am very confused by their remarks at the very bottom/top of page 17/18. True, there is no poly-time algorithm known to detect the unknot. BUT 1. the problem is in $\bf{NP} \cap \bf{coNP}$ and 2. there are several heuristic methods that work amazingly well. In fact, SnapPea is very very good at proving that knots have isotopic, a much harder problem than detecting the unknot. It would be nice to know what references they are relying on! – Sam Nead Dec 4 '10 at 1:07
Perhaps their reduction goes the other way? Ie, a successful attack solves the knot isotopy problem, while the knot isotopy problem does not necessarily give an attack? – Sam Nead Dec 4 '10 at 1:10
The problem you need to solve to break the cryptosystem (at least if you use the obvious attack) is not: given two knots, tell whether they are isotopic, but: given two isotopic knots, efficiently find a sequence of Reidemeister moves that takes one to the other. Any information about how hard this is would be very useful. Certainly it's at least as hard as the isotopy problem – Peter Shor Dec 19 '10 at 20:34
@Peter - Thank you. So my second comment was more along the correct lines. I believe that Alex Coward and Marc Lackenby have upper bounds on the number of R-moves required. But, perhaps, their current bounds are too large to be relevant... – Sam Nead Dec 20 '10 at 9:56
@Sam: The number of Reidemeister moves is really not the important thing here. What you would need to break the cryptosystem is something like an efficient algorithm for the problem: given two knots which are $k$ Reidemeister moves apart (where $k$ is polynomial in the size of the knot diagram), find a sequence of Reidemeister moves which is polynomial in $k$. There are some caveats I should give here. First, there might be a clever way to break the algorithm that doesn't use this attack. Second, this isn't quite enough to break the algorithm. – Peter Shor Dec 20 '10 at 13:56

Colin Adams' The knot book discusses the following applications:

Constructing invariants of knots is also related to constructing 3d topological quantum field theories. A good reference for these kinds of connections might be Kauffman's Knots and physics. The related notion of a tangle turns out to be related to continued fractions and $\text{SL}_2(\mathbb{Z})$. There is another connection (which may or may not be the same connection) involving the Lorenz attractor.

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@Qiaochu - +1 for DNA and molecular knots. Rational tangles are part of the course already. I googled a bit for Potts model and knots, but didn't find anything transparent. Do you have suggestions? In my mind, TQFTs and Lorenz knots (following Rob Ghrist, say?) are less "applications" and more like "advanced topics"... – Sam Nead Dec 5 '10 at 16:37

A nice softie "application" (?connection?) that I like is the observation that the space of subsets of $S^1$ containing at most three elements is homeomorphic to $S^3$. The space of subsets of $S^1$ with at most $3$ elements is usually denoted $\exp_3 S^1$.

$$\exp_3 S^1 \simeq S^3$$

$SO_2$ acts on $\exp_3 S^1$ by rotation. The action is fixed-point free. Consider its orbit stratification. There are the free orbits, the orbits with isotropy $\mathbb Z_2$ and orbits with isotropy $\mathbb Z_3$, and that's it.

This is enough information to deduce that $\exp_3 S^1$ as an $SO_2$ space is equivalent to $S^3 \subset \mathbb C^2$ as an $SO_2$-space with the action:

$$SO_2 \times \mathbb C^2 \to \mathbb C^2$$

$$(\alpha, (z_1,z_2)) \longmapsto (\alpha^3z_1,\alpha^2z_2)$$

here I'm thinking of $SO_2$ as the unit complex numbers.

Anyhow, the "high point" of this cycle of observations is that the free orbits of this action on $S^3$ are trefoil knots. http://www.msp.warwick.ac.uk/agt/2002/02/p043.xhtml

Depending on what path you take maybe you're not using any knot theory at all for this, but the fact that a non-trivial knot arises naturally IMO is pretty cool. I believe the trefoil comes up in a few other branches of mathematics quite naturally.

(edit: earlier I said symmetric product. Technically the symmetric product is $(S^1)^3/\Sigma_3 \simeq S^3$ is homeomorphic to $D^2 \times S^1$. There is a natural onto map $(S^1)^3 / \Sigma_3 \to \exp_3 S^1$ which collapses the boundary torus to a Moebius band)

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I think you missed a superscript $^3$ in your first displayed equation. – José Figueroa-O'Farrill Dec 3 '10 at 23:20
Thanks. Thanks is too short for a comment. :) – Ryan Budney Dec 3 '10 at 23:22
I don't know if I've run across the term "symmetric product" before, but it seems just about self-explanatory. I recall seeing a paper in the American Mathematical Monthly that said (in the language used in the answer above, but not in the paper I saw, as far as I recall) that the symmetric product of the circle with itself is the Möbius band. Is that well-known to topologists? Wikipedia's article en.wikipedia.org/wiki/Symmetric_product_of_an_algebraic_curve limits the concept to algebraic curves. Should it get edited to make it more general, or should there be a separate article? – Michael Hardy Dec 4 '10 at 17:41
If anyone cares, I've started a discussion on how to organize Wikipedia's material on symmetric products: en.wikipedia.org/wiki/… – Michael Hardy Dec 4 '10 at 18:12
Symmetric products are a common construction in algebraic topology, try a google search on "infinite symmetric product". The Moebius band example is a well-known one as there's a canonical embedding of $(S^1)^2/\Sigma_2$ in the space of straight lines in $\mathbb R^2$ which is a Moebius band (given a line in the plane intersect it with the unit circle) -- the space of straight lines in $\mathbb R^2$ is the canonical/classifying bundle over $\mathbb RP^1$, for example. This is in Milnor and Stasheff's book on characteristic classes. – Ryan Budney Dec 4 '10 at 18:34

I think historically one of the big formal motivations for knot theory were things like the Brieskorn varieties, i.e. looking at solutions to equations of the form

$$z_1^{p_1}+z_2^{p_2}+\cdots+z_n^{p_n} = 0$$

in $\mathbb C^n$, for various $p_k$. $0$ is generally a singular point and one way to study it is to intersect the variety with a small sphere centred about $0$. In the $n=2$ case you get knots in spheres, in many cases you get homology spheres (sometimes homotopy-spheres) knotted in spheres. The knot type informs on the singularity.

In the $n=3$ case you get the 3-dimensional Brieskorn varieties. By forgetting various coordinates this expresses the Brieskorn manifolds as branched covering spaces of $S^3$ branched over a knot. So again knots arrise. Looking at M.Epple's "Geometric aspects in the development of knot theory" apparently this perspective on branched coverings and knots goes back to Wirthinger (1895).

This perspective is well written-up in Milnor's "Singular points of complex hypersurfaces" and pursued further in Eisenbud and Neumann's "Three-dimensional link theory and invariants of plane curve singularities".

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Thank you for the references! – Sam Nead Dec 5 '10 at 16:39

Freedman et al. considered the energy of orbits of a divergence-free flow (such as a magnetic field), and obtain lower bounds on the energy in terms of knot invariants. However, I'm not sure this is strictly an application (even to another area of science), since I'm not sure if their estimates have been applied to a concrete physical experiment.

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Along the lines of Kelvin's idea, discussed in the comments, Buniy and Kephart wrote a speculative paper in 2002 that the mass spectrum of glueballs in QCD might be related to the spectrum of lengths of tightened knots. Here's a figure from their paper, comparing some experimental masses to lengths of some knots:

The model in their paper argues that the mass of a glueball (viewed as some kind of flux tube) should be proportional to the length of the tube, and hence the knot energy considered here is the ropelength of knots.

I'm not an expert on QCD so take all of the following with a grain of salt.

Before you get too excited, this model so far as I can tell is based on some phenomenological considerations, not QCD itself, and furthermore, their identifications of glueball states with knots is just the best fit of some experimental masses to corresponding entries in a table of knot lengths. As the final section of their paper says, there are some knots without measured glueball states, more importantly, it's my impression that QCD is not understood well enough to confirm that these measured masses are indeed all glueballs, e.g. the state at 1270 MeV also has a quark model interpretation.

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If you manage to convince your students that smooth manifolds are among the most beautiful and interesting objects in mathematics, expecially dimensions 3 and 4 that model our universe, then you can say that (among other things) knots form a fundamental ingredient in understanding and constructing such models.

For instance, you can tell that by removing a (well-chosen) knot from $S^3$ we can get the simplest possible universe with a hyperbolic geometry having finite volume. Or that every 3-manifold may be constructed by removing and "regluing" (finitely many) knots.

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See some some knots and links as sculptures here1 and here2. Students should also learn, indeed should be taught, to see mathematics in the context of human development, and knots must surely be the oldest applied geometry, used for making baskets, fishing nets, rafts, clothing and housing, etc.

The advantage I found in teaching knot theory, as against say homology theory, was that the basic problems could be stated at the beginning, and some methods were given, relating as said above to other nice mathematics, for some measure of solution. So students were able to see the point of the course.

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