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In [[mathematics]] two [[link (knot theory)|links]] <math>L_0 \subset S^n</math> and <math>L_1 \subset S^n</math> are '''concordant''' if there is an [[embedding]] <math>f : L_0 \times [0,1] \to S^n \times [0,1]</math> such that <math>f(L_0 \times \{0\}) = L_0 \times \{0\}</math> and <math>f(L_0 \times \{1\}) = L_1 \times \{1\}</math>.
 
By its nature, '''link concordance''' is an [[equivalence relation]]. It is weaker than [[Homotopy#Isotopy|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 invariant|finite type]] concordance invariants are Milnor invariants and their products,<ref name="kontsevich">{{citation
|title=The Kontsevich integral and Milnor's invariants
|journal=Topology
|first1=Nathan
|last1=Habegger
|first2=Gregor
|last2=Masbaum
|volume=39
|year=2000
|issue=6
|pages=1253–1289
|doi=10.1016/S0040-9383(99)00041-5 
|postscript=, [http://people.math.jussieu.fr/~masbaum/K39.ps.gz preprint].
}}</ref> though non-finite type concordance invariants exist.
 
== Higher dimensions ==
One can analogously define concordance for any two submanifolds <math>M_0, M_1 \subset N</math>. In this case one considers two submanifolds concordant if there is a [[cobordism]] between them in <math>N \times [0,1],</math> i.e., if there is a manifold with boundary <math>W \subset N \times [0,1]</math> whose boundary consists of <math>M_0 \times \{0\}</math> and <math>M_1 \times \{1\}.</math>
 
This higher dimensional concordance is a [[relative homotopy|relative]] form of cobordism – it requires two submanifolds to be not just abstractly cobordant, but "cobordant in ''N''".
 
==See also==
*[[Slice knot]]
 
==References==
<references/>
 
==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&ndash;347, [[Elsevier]], Amsterdam, 2005. {{MathSciNet | id = 2179265 }} ISBN 0-444-51452-X
 
[[Category:Knot invariants]]
[[Category:Manifolds]]

Revision as of 18:11, 27 January 2014

In mathematics two links L0⊂Sn and L1⊂Sn are concordant if there is 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.

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 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

References

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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. Template:MathSciNet ISBN 0-444-51452-X
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