The
Hopf link is a simple example of a link with two components that is not an unlink.
The
Borromean rings form a link with three components that is not an unlink; however, any two of the rings considered on their own do form a two-component unlink.
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abKanenobu, Taizo (1986), "Hyperbolic links with Brunnian properties", Journal of the Mathematical Society of Japan, 38 (2): 295–308,
doi:10.2969/jmsj/03820295,
MR0833204