{{Short description|Link equivalence relation weaker than isotopy but stronger than homotopy}} In mathematics, two links <math>L_0 \subset S^n</math> and <math>L_1 \subset S^n</math> are '''concordant''' if there exists 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 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,<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 |doi-access=free }}</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 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 319–347, Elsevier, Amsterdam, 2005. {{MathSciNet | id = 2179265 }} {{isbn|0-444-51452-X}}
Category:Knot invariants Category:Manifolds