Link concordance

In mathematics, two links L 0 ⊂ S n {displaystyle L_{0}subset S^{n}} and L 1 ⊂ S n {displaystyle L_{1}subset S^{n}} are concordant if there exists an embedding f : L 0 × [ 0 , 1 ] → S n × [ 0 , 1 ] {displaystyle f:L_{0} imes o S^{n} imes } such that f ( L 0 × { 0 } ) = L 0 × { 0 } {displaystyle f(L_{0} imes {0})=L_{0} imes {0}} and f ( L 0 × { 1 } ) = L 1 × { 1 } {displaystyle f(L_{0} imes {1})=L_{1} imes {1}} . In mathematics, two links L 0 ⊂ S n {displaystyle L_{0}subset S^{n}} and L 1 ⊂ S n {displaystyle L_{1}subset S^{n}} are concordant if there exists an embedding f : L 0 × [ 0 , 1 ] → S n × [ 0 , 1 ] {displaystyle f:L_{0} imes o S^{n} imes } such that f ( L 0 × { 0 } ) = L 0 × { 0 } {displaystyle f(L_{0} imes {0})=L_{0} imes {0}} and f ( L 0 × { 1 } ) = L 1 × { 1 } {displaystyle f(L_{0} imes {1})=L_{1} imes {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. 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. One can analogously define concordance for any two submanifolds M 0 , M 1 ⊂ N {displaystyle M_{0},M_{1}subset N} . In this case one considers two submanifolds concordant if there is a cobordism between them in N × [ 0 , 1 ] , {displaystyle N imes ,} i.e., if there is a manifold with boundary W ⊂ N × [ 0 , 1 ] {displaystyle Wsubset N imes } whose boundary consists of M 0 × { 0 } {displaystyle M_{0} imes {0}} and M 1 × { 1 } . {displaystyle M_{1} imes {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'.