Are there piecewise-linear unknots that are not metrically unknottable? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T16:28:42Z http://mathoverflow.net/feeds/question/8145 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/8145/are-there-piecewise-linear-unknots-that-are-not-metrically-unknottable Are there piecewise-linear unknots that are not metrically unknottable? Scott Morrison 2009-12-08T00:29:39Z 2009-12-08T12:12:23Z <p>A stick knot is a just a piecewise linear knot. We could define "stick isotopy" as isotopy that preserves the length of each linear piece.</p> <p>Are there stick knots which are topologically trival, but not trivial via a stick isotopy?</p> http://mathoverflow.net/questions/8145/are-there-piecewise-linear-unknots-that-are-not-metrically-unknottable/8146#8146 Answer by David Eppstein for Are there piecewise-linear unknots that are not metrically unknottable? David Eppstein 2009-12-08T00:41:32Z 2009-12-08T00:41:32Z <p>Yes, there are. See "Locked and Unlocked Polygonal Chains in 3D", T. Biedl, E. Demaine, M. Demaine, S. Lazard, A. Lubiw, J. O'Rourke, M. Overmars, S. Robbins, I. Streinu, G. Toussaint, S. Whitesides, <a href="http://arxiv.org/abs/cs.CG/9910009" rel="nofollow">arXiv:cs.CG/9910009</a>, figure 6.</p> http://mathoverflow.net/questions/8145/are-there-piecewise-linear-unknots-that-are-not-metrically-unknottable/8148#8148 Answer by Greg Kuperberg for Are there piecewise-linear unknots that are not metrically unknottable? Greg Kuperberg 2009-12-08T00:49:52Z 2009-12-08T00:49:52Z <p>There is a <a href="http://people.csail.mit.edu/edemaine/papers/CRCLinkage/paper.pdf" rel="nofollow">survey paper</a> on this general topic by Robert Connelly and Erik Demaine. As David Eppstein just posted, the answer is yes in 3D. However, it is a <a href="http://www-cgi.cs.cmu.edu/afs/cs.cmu.edu/Web/People/fcady/papers/convexifying.pdf" rel="nofollow">famous result</a> of those two authors and Gunter Rote that the answer is no in 2D: all polygonal planar cyclic linkages can be made convex. This second problem was a long-standing conjecture called the "Carpenter's Rule Problem". There are at least two beautiful proofs of the conjecture: the original, and a second one using pseudo-triangulations due to Ileana Streinu.</p>