# Are there piecewise-linear unknots that are not metrically unknottable?

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.

Are there stick knots which are topologically trival, but not trivial via a stick isotopy?

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At the risk of being overly facetious, can we do a s/(?:s)t(?:i)//g on this question? I'm also reminded of one of my children's favourite stories: amazon.co.uk/Stick-Man-Julia-Donaldson/dp/1407106171 –  Loop Space Dec 8 '09 at 13:23

## 2 Answers

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, arXiv:cs.CG/9910009, figure 6.

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There were examples given before these by Cantarella and Johnston. –  Dev Sinha Apr 14 '10 at 6:57

There is a survey paper 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 famous result 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.

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