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On Knotted Necklaces of Pearls

A knotted necklace of pearls is a necklace whose string forms a nontrivial knot. We assume that all pearls are balls of the same size, and any two consecutive pearls in a knotted necklace are tangent to each other. How many pearls are necessary to make a knotted necklace, say a trefoil knot? We show that 15 pearls are sufficient, and that to make a knotted necklace which can be put in the shallowest-possible show-case (a box with a glass-lid), 16 pearls are necessary and sufficient.