Link concordance

Short description: Link equivalence relation weaker than isotopy but stronger than homotopy

In mathematics, two links L0⊂Sn and L1⊂Sn are concordant if there exists an embedding f:L0×[0,1]->Sn×[0,1] such that f(L0×{0})=L0×{0} and f(L0×{1})=L1×{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.

## Contents

* 1 Concordance invariants
* 2 Higher dimensions
* 3 See also
* 4 References
* 5 Further reading

## 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,[1] though non-finite type concordance invariants exist.

## Higher dimensions

One can analogously define concordance for any two submanifolds M0,M1⊂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 M0×{0} and M1×{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".

## See also

* Slice knot

## References

1. ↑ Habegger, Nathan; Masbaum, Gregor (2000), "The Kontsevich integral and Milnor's invariants", Topology 39 (6): 1253–1289, doi:10.1016/S0040-9383(99)00041-5

## Further reading

* J. Hillman, Algebraic invariants of links. Series on Knots and everything. Vol 32. World Scientific.
* Livingston, Charles, A survey of classical knot concordance, in: Handbook of knot theory, pp 319-347, Elsevier, Amsterdam, 2005. MR2179265 ISBN 0-444-51452-X

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