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Link concordance

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Link equivalence relation weaker than isotopy but stronger than homotopy

In mathematics, two links L 0 ⊂ S n {\displaystyle L_{0}\subset S^{n}} and L 1 ⊂ S n {\displaystyle L_{1}\subset S^{n}} are concordant if there exists an embedding f : L 0 × [ 0 , 1 ] → S n × [ 0 , 1 ] {\displaystyle f:L_{0}\times \to S^{n}\times } such that f ( L 0 × { 0 } ) = L 0 × { 0 } {\displaystyle f(L_{0}\times \{0\})=L_{0}\times \{0\}} and f ( L 0 × { 1 } ) = L 1 × { 1 } {\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 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 , M 1 ⊂ N {\displaystyle M_{0},M_{1}\subset N} . In this case one considers two submanifolds concordant if there is a cobordism between them in N × [ 0 , 1 ] , {\displaystyle N\times ,} i.e., if there is a manifold with boundary W ⊂ N × [ 0 , 1 ] {\displaystyle W\subset N\times } whose boundary consists of M 0 × { 0 } {\displaystyle M_{0}\times \{0\}} and M 1 × { 1 } . {\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

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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