A new class of stuck unknots in Pol6
Godfried Toussaint · 2001
We consider embedding classes of hexagonal unknots with edges of fixed length. Cantarella and Johnston [3] recently showed that there exist "stuck" hexagonal unknots which cannot be reconfigured to convex hexagons for suitable choices of edge lengths. Here we uncover a new class of stuck unknotted hexagons, thereby proving that there exist at least five classes of nontrivial embeddings of the unknot. 1 Introduction A closed chain of n line segments with lengths l 1 ; :::; l n embedded in R 3 forms a space polygon. The space of such polygons is denoted (using the notation of Cantarella and Johnston [3]) by P ol n (l 1 ; :::; l n ). We are concerned here with simple space polygons or unknots (also trivial knots). Recently Cantarella and Johnston [3] and independently, Biedl, et al. [1] studied the embedding classes of such objects and discovered that there exist stuck (or locked) simple polygons. The polygon of Biedl, et al. [1] contains 10 edges whereas the example of Cantarella and ...