mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 (mathematics), group that is a subgroup. When some object X is said to be embedded in another object Y ...

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. A simpler example is equ ...

. It is weaker than isotopy, and stronger than

homotopy In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. ...

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

unlink In the mathematical field of knot theory, an unlink is a link that is equivalent (under ambient isotopy) to finitely many disjoint circles in the plane. The two-component unlink, consisting of two non-interlinked unknots, is the simplest pos ...

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 The signature of a knot is a topological invariant in knot theory. It may be computed from the Seifert surface. Given a knot ''K'' in the 3-sphere, it has a Seifert surface ''S'' whose boundary is ''K''. The Seifert form of ''S'' is the pairin ...

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 (knot theory), link is an analog of the knot group of a Knot (mathematics), knot. They were described by John Milnor in his Ph.D. thesis, . Notably, the link group is not in general t ...

, 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 space, compact manifolds of the same dimension, set up using the concept of the boundary (topology), boundary (French ''wikt:bord#French, bord'', giving ''cob ...

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 Relative may refer to: General use *Kinship and family, the principle binding the most basic social units of society. If two people are connected by circumstances of birth, they are said to be ''relatives''. Philosophy *Relativism, the concept t ...

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