Population dynamics of fisheries
In mathematics two links and are concordant if there is an embedding such that and .
![{\displaystyle f:L_{0}\times [0,1]\to S^{n}\times [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/382f5a09d722bbaca7ea6751d7b2b0e85ecaa39e)
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 . In this case one considers two submanifolds concordant if there is a cobordism between them in i.e., if there is a manifold with boundary whose boundary consists of and
![{\displaystyle N\times [0,1],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4404912db1fcc8e291daf48a99847ede15605e25)
![{\displaystyle W\subset N\times [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79813d4b179452edaa443db9ce5936d984b2650a)
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