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?



Are there piecewise-linear unknots that are not metrically unknottable?
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](http://arxiv.org/abs/cs.CG/9910009), figure 6.


