Link concordance

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

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

This higher-dimensional concordance is a relative form of cobordism – it requires two submanifolds to be not just abstractly cobordant, but "cobordant in N".

References

^ 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

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