Link (knot theory)

Implicit in this definition is that there is a trivial reference link, usually called the unlink, but the word is also sometimes used in context where there is no notion of a trivial link.

The notion of a link can be generalized in a number of ways.

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 embedding is not isotopic to the 1st.

neither of which yields non-trivial embeddings since the open end means that they can be shrunk to a point), so a possibly disconnected compact 1-manifold is a collection of n intervals

The condition that the boundary of X lies in says that intervals either connect two lines or connect two points on one of the lines, but imposes no conditions on the circles.

In this context, a braid is defined as a tangle which is always going down – whose derivative always has a non-zero component in the vertical (I) direction.

In particular, it must consist solely of intervals, and not double back on itself; however, no specification is made on where on the line the ends lie.

A string link is a tangle consisting of only intervals, with the ends of each strand required to lie at (0, 0), (0, 1), (1, 0), (1, 1), (2, 0), (2, 1), ... – i.e., connecting the integers, and ending in the same order that they began (one may use any other fixed set of points); if this has ℓ components, we call it an "ℓ-component string link".

The key technical value of tangles and string links is that they have algebraic structure.

Isotopy classes of tangles form a tensor category, where for the category structure, one can compose two tangles if the bottom end of one equals the top end of the other (so the boundaries can be stitched together), by stacking them – they do not literally form a category (pointwise) because there is no identity, since even a trivial tangle takes up vertical space, but up to isotopy they do.

For a fixed ℓ, isotopy classes of ℓ-component string links form a monoid (one can compose all ℓ-component string links, and there is an identity), but not a group, as isotopy classes of string links need not have inverses.

However, concordance classes (and thus also homotopy classes) of string links do have inverses, where inverse is given by flipping the string link upside down, and thus form a group.