The term slave bracelet refers to a style of jewelry that has a chain or other attachment from the bracelet to one or more finger rings. Commonly there is a central piece (island or hand connector) between the ring and bracelet that rests on the back of the hand.
This fashion originated in Africa and is associated with India – harems, belly dancers and the like. The design was likely adopted by Native Americans for tourists after the bracelets became popular in the 1920s American flapper culture.
In spite of the negative connotation of the word slave and its various meanings, the style continues to be popular and has no other name that I am aware of.