tangles – putting one tangle to the right of the other. For a fixed ℓ, isotopy classes of ℓ-component string links form a monoid (one can compose all ℓ-component string links, and there is an identity), but not a group, as isotopy classes of string links need not have inverses. However, concordance classes (and thus also homotopy classes) of string links do have inverses, where inverse is given by flipping the string link upside down, and thus form a group. Every link can be cut apart to form a string link, though this is not unique, and invariants of links can sometimes be understood as invariants of string links – this is the case for Milnor's invariants, for instance. Compare with closed braids. See also Hyperbolic link Unlink Link group References Manifolds