There exist polyhedra which can be plaited (braided). Examples include a plaited cube and plaited icosahedron illustrated above (Pargeter 1959, Wells 1991). In the above figures, heavy lines indicate cuts, thin lines indicate folds, and polygons labeled "" are placed over polygons labeled "."
Plaited Polyhedron
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References
Gorham, J. Plaited Crystal Models. 1888.Pargeter, A. R. "Plaited Polyhedra." Math. Gaz. 43, 88-101, 1959.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 160, 1991.Referenced on Wolfram|Alpha
Plaited PolyhedronCite this as:
Weisstein, Eric W. "Plaited Polyhedron." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PlaitedPolyhedron.html