Visible crew/equipment: After the Goblins sing, "Slime and snails... Thunder or lightning," they all sing, "Jump, magic, jump..." and when the Goblin, with the red scarves at his waist (same one as earlier shot, also visible in later shot), lands back on the ground the wires attached to the back of his shoulders are visible. Then in the next wide shot, the wires attached to the back of the waist on the tall Goblin with the long hair, are visible. (00:25:40)
Visible crew/equipment: When Sarah and Hoggle slip off the ledge at the Bog of Eternal Stench, Ludo breaks their fall below. As they fall, in the wide shot, the lines holding Sarah and Hoggle are visible. (00:56:20)
Visible crew/equipment: As Hoggle and Sarah climb the ladder after being chased by the cleaners, Hoggle says to Sarah 'I'm a coward, and Jareth scares me.' As he says this you can see a wire attached to him. (00:35:55)
Visible crew/equipment: While falling through the well of hands that lead to the oubliette, wires are visible attached to Sarah. Mostly noticeably when one hand says "Would you like us to let go?" and as she is dropped, wires are seen attached to her arms. (00:27:50)
Visible crew/equipment: Sarah runs to the edge of one of the platforms in the Escher-style room, and sees Toby. Just before she jumps, as she runs to the edge, a safety wire can briefly be seen attached behind her. (01:26:30)
Visible crew/equipment: Just after Jareth kicks the furry little creature way up towards the camera, he sings, "Jump, magic, jump..." To his left, when the little guy 'leaps' high into the air, in front of Jareth's throne, the wires attached to both sides of his shoulders are visible. (00:23:55)
Visible crew/equipment: As "Chilly Down" ends, Sarah is pulled up the ledge by Hoggle. When it shows Sarah, and the Fireys flying behind her, above the first Firey that flies up, the wires are seen.
Visible crew/equipment: Just after falling into the Bog of Eternal Stench, the duo are moving along the ledge. As it breaks away, there is an overhead shot where you can see a white and blue piece of equipment at the top left corner of the shot. I don't know what it is but it's not in the next shot so it's not something natural to the Labyrinth. (00:55:45)
Answer: The necklace is a symbol of Jareth's power throughout the story. All the time that Sarah is in the Labyrinth, Jareth has more power than her, because she doesn't really know what she's doing, and isn't taking one particular path, just getting lucky at the right times. At the end, as we all know, Sarah triumphs, and Jareth's necklace changes to admit his defeat.