Link concordance

Short description: Link equivalence relation weaker than isotopy but stronger than homotopy

In mathematics, two links $L_0\subset S^n$ and $L_1\subset S^n$ are concordant if there exists an embedding $f:L_0\times [0,1]\to S^n\times [0,1]$ such that $f(L_0\times \{0\})=L_0\times \{0\}$ and $f(L_0\times \{1\})=L_1\times \{1\}$.

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 $M_0,M_1\subset N$. In this case one considers two submanifolds concordant if there is a cobordism between them in $N\times [0,1],$ i.e., if there is a manifold with boundary $W\subset N\times [0,1]$ whose boundary consists of $M_0\times \{0\}$ and $M_1\times \{1\}.$

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

• Slice knot

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