Link concordance

In mathematics two links ${\displaystyle L_{0}\subset S^{n}}$ and ${\displaystyle L_{1}\subset S^{n}}$ are concordant if there is an embedding ${\displaystyle f:L_{0}\times [0,1]\to S^{n}\times [0,1]}$ such that ${\displaystyle f(L_{0}\times \{0\})=L_{0}\times \{0\}}$ and ${\displaystyle f(L_{0}\times \{1\})=L_{1}\times \{1\}}$.

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

Higher dimensions

One can analogously define concordance for any two submanifolds ${\displaystyle M_{0},M_{1}\subset N}$. In this case one considers two submanifolds concordant if there is a cobordism between them in ${\displaystyle N\times [0,1],}$ i.e., if there is a manifold with boundary ${\displaystyle W\subset N\times [0,1]}$ whose boundary consists of ${\displaystyle M_{0}\times \{0\}}$ and ${\displaystyle M_{1}\times \{1\}.}$

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, preprint.

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. MR 2179265 ISBN 0-444-51452-X