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?
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? 


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. 


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 longstanding conjecture called the "Carpenter's Rule Problem". There are at least two beautiful proofs of the conjecture: the original, and a second one using pseudotriangulations due to Ileana Streinu. 

