In mathematics, two links

L_0 \subset S^n

and

L_1 \subset S^n

are concordant if there exists an

embedding In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is g ...

f : L_0 \times,1 \to S^n \times,1 /math> such that f(L_0 \times \) = L_0 \times \ and f(L_0 \times \) = L_1 \times \ .

By its nature, link concordance is an equivalence relation . It is weaker than isotopy, and stronger than

homotopy In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a defor ...

: isotopy implies concordance implies homotopy. A link is a slice link if it is concordant to the

unlink In the mathematics, mathematical field of knot theory, an unlink is a Link (knot theory), link that is equivalent (under ambient isotopy) to finitely many disjoint circles in the plane. Properties * An ''n''-component link ''L'' ⊂&nbs ...

Concordance invariants

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

linking number In mathematics, the linking number is a numerical invariant that describes the linking of two closed curves in three-dimensional space. Intuitively, the linking number represents the number of times that each curve winds around the other. In E ...

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,1

i.e., if there is a manifold with boundary

W \subset N \times,1 /math> whose boundary consists of M_0 \times \ and M_1 \times \. This higher-dimensional concordance is a relative form of cobordism – it requires two submanifolds to be not just abstractly cobordant, but "cobordant in ''N''".

Concordance invariants

Higher dimensions

See also

References

Further reading