Link concordance explained

L₀\subsetSⁿ

and

L₁\subsetSⁿ

are concordant if there exists an embedding

f:L₀ x [0,1]\toSⁿ x [0,1]

such that

f(L₀ x \{0\})=L₀ x \{0\}

and

f(L₀ x \{1\})=L₁ x \{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,M₁\subsetN

. In this case one considers two submanifolds concordant if there is a cobordism between them in

N x [0,1],

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

W\subsetN x [0,1]

whose boundary consists of

M₀ x \{0\}

and

M₁ x \{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".

Link concordance explained

Concordance invariants

Higher dimensions

See also

Further reading