Kalpana Kalpana (Editor)

Link concordance

Updated on
Edit
Like
Comment
Share on FacebookTweet on TwitterShare on LinkedInShare on Reddit

In mathematics two links L 0 ⊂ S n and L 1 ⊂ S n are concordant if there is an embedding f : L 0 × [ 0 , 1 ] → S n × [ 0 , 1 ] such that f ( L 0 × { 0 } ) = L 0 × { 0 } and f ( L 0 × { 1 } ) = L 1 × { 1 } .

Contents

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 invariants.

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 ⊂ N . In this case one considers two submanifolds concordant if there is a cobordism between them in N × [ 0 , 1 ] , i.e., if there is a manifold with boundary W ⊂ N × [ 0 , 1 ] whose boundary consists of M 0 × { 0 } and M 1 × { 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".

References

Link concordance Wikipedia
(Text) CC BY-SA