Continuity mistake: After Gollum says, "Clever Hobbits, to climb so high!" he jumps onto Frodo and Sam. Gollum then attacks Frodo and when he turns Frodo onto his back, Frodo's shirt is wide open at his neck/chest and the necklace and ring are gone. In the next close-up as Gollum chokes Frodo, and the following shots, the necklace is back.
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