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The two curves of this (2, 8)-torus link have linking number four. In mathematics, the linking number is a numerical invariant that describes the linking of two closed curves in three-dimensional space. Intuitively, the linking number represents the number of times that each curve winds around the other.
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 ...
The first (2-fold) Milnor invariant is simply the linking number (just as the 2-fold Massey product is the cup product, which is dual to intersection), while the 3-fold Milnor invariant measures whether 3 pairwise unlinked circles are Borromean rings, and if so, in some sense, how many times (that is to say, the Borromean rings have a Milnor 3 ...
Frequently the word link is used to describe any submanifold of the sphere diffeomorphic to a disjoint union of a finite number of spheres, .. In full generality, the word link is essentially the same as the word knot – the context is that one has a submanifold M of a manifold N (considered to be trivially embedded) and a non-trivial embedding of M in N, non-trivial in the sense that the 2nd ...
1 link/Hopf link - the simplest nontrivial link with more than one component; it consists of two circles linked together exactly once (L2a1) 4 2 1 link/Solomon's knot (a two component "link" rather than a one component "knot") - a traditional decorative motif used since ancient times (L4a1) 5 2
The number of colorings meeting these conditions is a knot invariant, independent of the diagram chosen for the link. A trivial link with three components has n 3 − n {\displaystyle n^{3}-n} colorings, obtained from its standard diagram by choosing a color independently for each component and discarding the n {\displaystyle n} colorings that ...