Applications of knot theory - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T10:56:42Z http://mathoverflow.net/feeds/question/48222 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/48222/applications-of-knot-theory Applications of knot theory Sam Nead 2010-12-03T22:23:17Z 2013-02-02T14:35:59Z <p>An <a href="http://mathoverflow.net/questions/48191/how-mathematicians-knowledge-is-organized-closed" rel="nofollow">answer</a> of André Henriques' inspired the following closely related CW question. Parts of the following is extracted from his answer and my comments. </p> <blockquote> <p>I regularly teach a knot theory class. Every time, students ask about applications. What should I say?</p> </blockquote> <p>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 <a href="http://en.wikipedia.org/wiki/Topoisomerase" rel="nofollow">DNA</a>, mixing <a href="http://www.math.wisc.edu/~jeanluc/talks/columbia2010a.pdf" rel="nofollow">liquids</a>, and the structure of the Sun's <a href="http://www.suntrek.org/magnetic-sun/tangled-sun/tangled-sun.shtml" rel="nofollow">corona</a>. I'm interested hearing about other applications. I am <em>also</em> interested in hearing your take on the pedagogical issues involved. Thank you! </p> http://mathoverflow.net/questions/48222/applications-of-knot-theory/48225#48225 Answer by Qiaochu Yuan for Applications of knot theory Qiaochu Yuan 2010-12-03T22:46:56Z 2010-12-03T23:49:50Z <p>Colin Adams' <a href="http://books.google.com/books?id=M-B8XedeL9sC&amp;printsec=frontcover&amp;dq=colin+adams+knot+book" rel="nofollow">The knot book</a> discusses the following applications:</p> <ul> <li>Knotting in DNA,</li> <li><a href="http://en.wikipedia.org/wiki/Molecular_knot" rel="nofollow">Molecular knots</a>, </li> <li>Statistical mechanics (e.g. the <a href="http://en.wikipedia.org/wiki/Potts_model" rel="nofollow">Potts model</a>).</li> </ul> <p>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 <a href="http://books.google.com/books?id=T0z882H1928C&amp;printsec=frontcover&amp;dq=kauffman+knots+and+physics" rel="nofollow">Knots and physics</a>. The related notion of a <a href="http://en.wikipedia.org/wiki/Tangle_(mathematics)#Rational_and_algebraic_tangles" rel="nofollow">tangle</a> 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 <a href="http://www.ams.org/samplings/feature-column/fcarc-lorenz" rel="nofollow">Lorenz attractor</a>.</p> http://mathoverflow.net/questions/48222/applications-of-knot-theory/48226#48226 Answer by Ryan Budney for Applications of knot theory Ryan Budney 2010-12-03T23:00:30Z 2012-04-05T08:29:38Z <p>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 $exp_3 S^1$. </p> <p>$$exp_3 S^1 \simeq S^3$$</p> <p>$SO_2$ 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. </p> <p>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:</p> <p>$$SO_2 \times \mathbb C^2 \to \mathbb C^2$$</p> <p>$$(\alpha, (z_1,z_2)) \longmapsto (\alpha^3z_1,\alpha^2z_2)$$</p> <p>here I'm thinking of $SO_2$ as the unit complex numbers. </p> <p>Anyhow, the "high point" of this cycle of observations is that the free orbits of this action on $S^3$ are trefoil knots. <a href="http://www.msp.warwick.ac.uk/agt/2002/02/p043.xhtml" rel="nofollow">http://www.msp.warwick.ac.uk/agt/2002/02/p043.xhtml</a></p> <p>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.</p> <p>(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)</p> http://mathoverflow.net/questions/48222/applications-of-knot-theory/48227#48227 Answer by Ryan Budney for Applications of knot theory Ryan Budney 2010-12-03T23:11:52Z 2010-12-03T23:24:37Z <p>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</p> <p>$$z_1^{p_1}+z_2^{p_2}+\cdots+z_n^{p_n} = 0$$</p> <p>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. </p> <p>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).</p> <p>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". </p> http://mathoverflow.net/questions/48222/applications-of-knot-theory/48233#48233 Answer by Joseph O'Rourke for Applications of knot theory Joseph O'Rourke 2010-12-03T23:55:00Z 2010-12-03T23:55:00Z <p>If I may steal some thunder from Peter Shor, his paper, <a href="http://arxiv.org/abs/1004.5127" rel="nofollow">Quantum money from knots</a> (with Edward Farhi, David Gosset, Avinatan Hassidim, and Andrew Lutomirski) relies for the security of its "quantum money scheme" on</p> <blockquote> <p>the assumption that given two different looking but equivalent knots, it is difficult to explicitly find a transformation that takes one to the other.</p> </blockquote> <p>The Alexander polynomial plays a prominent role in the paper.</p> http://mathoverflow.net/questions/48222/applications-of-knot-theory/49897#49897 Answer by Agol for Applications of knot theory Agol 2010-12-19T18:32:49Z 2010-12-19T18:32:49Z <p><a href="http://www.ams.org/mathscinet-getitem?mr=1114611" rel="nofollow">Freedman et al.</a> 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. </p> http://mathoverflow.net/questions/48222/applications-of-knot-theory/49902#49902 Answer by jc for Applications of knot theory jc 2010-12-19T19:34:40Z 2010-12-19T19:34:40Z <p>Along the lines of Kelvin's idea, discussed in the comments, <a href="http://arxiv.org/abs/hep-ph/0209339" rel="nofollow">Buniy and Kephart wrote a speculative paper in 2002</a> that the mass spectrum of <a href="http://en.wikipedia.org/wiki/Glueball" rel="nofollow">glueballs</a> 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:</p> <p><img src="http://i583.photobucket.com/albums/ss275/jaspercrowne/buniykephart.png" alt="Buniy and Kephart"></p> <p>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 <a href="http://en.wikipedia.org/wiki/Ropelength" rel="nofollow">ropelength</a> of knots.</p> <p>I'm not an expert on QCD so take all of the following with a grain of salt.</p> <p>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 <strong>QCD is not understood well enough to confirm that these measured masses are indeed all glueballs</strong>, e.g. the state at 1270 MeV also has a quark model interpretation.</p> http://mathoverflow.net/questions/48222/applications-of-knot-theory/49945#49945 Answer by Bruno Martelli for Applications of knot theory Bruno Martelli 2010-12-20T09:46:14Z 2010-12-20T10:00:10Z <p>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. </p> <p>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.</p> http://mathoverflow.net/questions/48222/applications-of-knot-theory/120588#120588 Answer by sabit for Applications of knot theory sabit 2013-02-02T13:26:48Z 2013-02-02T13:26:48Z <p>can we apply knot theory in resources modelling and allocation in computer </p> http://mathoverflow.net/questions/48222/applications-of-knot-theory/120591#120591 Answer by Ronnie Brown for Applications of knot theory Ronnie Brown 2013-02-02T14:35:59Z 2013-02-02T14:35:59Z <p>See some some knots and links as sculptures <a href="http://wwww.popmath.org.uk" rel="nofollow">here1</a> and <a href="http://wwww.bradshawfoundation.com/jr" rel="nofollow">here2</a>. Students should also learn, indeed should be taught, to see mathematics in the <a href="http://pages.bangor.ac.uk/~mas010/context.html" rel="nofollow">context</a> of human development, and knots must surely be the oldest applied geometry, used for making baskets, fishing nets, rafts, clothing and housing, etc. </p> <p>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. </p>