Link concordance
In mathematics, two links [math]\displaystyle{ L_0 \subset S^n }[/math] and [math]\displaystyle{ L_1 \subset S^n }[/math] are concordant if there exists an embedding [math]\displaystyle{ f : L_0 \times [0,1] \to S^n \times [0,1] }[/math] such that [math]\displaystyle{ f(L_0 \times \{0\}) = L_0 \times \{0\} }[/math] and [math]\displaystyle{ 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,[1] though non-finite type concordance invariants exist.
Higher dimensions
One can analogously define concordance for any two submanifolds [math]\displaystyle{ M_0, M_1 \subset N }[/math]. In this case one considers two submanifolds concordant if there is a cobordism between them in [math]\displaystyle{ N \times [0,1], }[/math] i.e., if there is a manifold with boundary [math]\displaystyle{ W \subset N \times [0,1] }[/math] whose boundary consists of [math]\displaystyle{ M_0 \times \{0\} }[/math] and [math]\displaystyle{ 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
- ↑ 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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