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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asked Dec 8, 2009 at 0:29
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$\begingroup$ 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 $\endgroup$– Andrew StaceyCommented Dec 8, 2009 at 13:23
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2 Answers
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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.
answered Dec 8, 2009 at 0:41
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3$\begingroup$ There were examples given before these by Cantarella and Johnston. $\endgroup$ Commented Apr 14, 2010 at 6:57
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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.
answered Dec 8, 2009 at 0:49