Existence of folded states which do not admit folding motions from the unfolded state

抄録

In 2004, Demaine et al. showed that if P is a simply connected polygonal region in the plane, then for any piecewise-C² folded state (f, λ) of P, there exists a folding motion from P to (f, λ). In this paper, we show that if P is not simply connected, then the conclusion of the theorem does not necessarily hold; in fact, there exists an annulus P in ℝ² such that there exist piecewise-C² folded states of P which do not admit folding motions. The theorem is proved using an argument from a branch of low-dimensional topology called knot theory.