Link concordance

In mathematics, two links $$L_0 \subset S^n$$ and $$L_1 \subset S^n$$ are concordant if there exists an embedding $$f : L_0 \times [0,1] \to S^n \times [0,1]$$ such that $$f(L_0 \times \{0\}) = L_0 \times \{0\}$$ and $$f(L_0 \times \{1\}) = L_1 \times \{1\}$$.

By its nature, link concordance is an equivalence relation. It is weaker than isotopy, and stronger than homotopy: isotopy implies concordance implies homotopy. A link is a slice link if it is concordant to the unlink.

Concordance invariants
A function of a link that is invariant under concordance is called a concordance invariant.

The linking number of any two components of a link is one of the most elementary concordance invariants. The signature of a knot is also a concordance invariant. A subtler concordance invariant are the Milnor invariants, and in fact all rational finite type concordance invariants are Milnor invariants and their products, though non-finite type concordance invariants exist.

Higher dimensions
One can analogously define concordance for any two submanifolds $$M_0, M_1 \subset N$$. In this case one considers two submanifolds concordant if there is a cobordism between them in $$N \times [0,1],$$ i.e., if there is a manifold with boundary $$W \subset N \times [0,1]$$ whose boundary consists of $$M_0 \times \{0\}$$ and $$M_1 \times \{1\}.$$

This higher-dimensional concordance is a relative form of cobordism – it requires two submanifolds to be not just abstractly cobordant, but "cobordant in N".