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 giv ...

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 In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...

. 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.

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 In knot theory, an area of mathematics, the link group of a link is an analog of the knot group of a knot. They were described by John Milnor in his Ph.D. thesis, . Notably, the link group is not in general the fundamental group of the link complem ...

, 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 In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary (French '' bord'', giving ''cobordism'') of a manifold. Two manifolds of the same ...

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