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For polygons of fixed side lengths embedded in R3, a natural question is whether the embeddings are simply knot types, or there exist distinct embeddings of the same knot. This question was answered by J. Cantarella and H. Johnston [J. Knot Theory Ramifications 7 (1998), no. 8, 1027– 1039; MR1671500(99m:57002) ]. They showed that for some particular assignment of side lengths, there are three connected components corresponding to the unknot in Pol6(l1, l2, . . . , l6) (this notation is used for the space of embedding classes of polygons of 6 sides, where l1, l2, . . . , l6 are the lengths of the sides). Later, G. T. Toussaint [Beitr äge Algebra Geom. 42 (2001), no. 2, 301–306; MR1865519(2002k:57020) ] showed that there are two more classes for a different polygonal unknot with 6 sides (a hexagonal unknot). In this article, the authors consider the maximum number of embedding classes for the unknot, and show that there exists a hexagonal unknot with at least nine embedding classes.

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