nLab
link diagram

Contents

Idea

A link diagram is, roughly speaking, the combinatorial object obtained by projecting a link in ‘general position’ to a plane. Reidemeister's theorem establishes that one does not lose any essential information by passing from a link to its diagram, and thus it is possible to study knot theory in a way which takes link diagrams as primary: this is sometimes known as diagrammatic knot theory?.

Formal definition

The formal definition of a link diagram is in itself independent of the notion of a link.

Definition

A connected link diagram is a connected (undirected) 4-valent plane graph GG equipped with the following data for every vertex vv of GG.

  1. A choice of division of the four edges incident to vv into two pairs, say (e 0,e 1)(e_{0}, e_{1}) and (e 2,e 3)(e_{2}, e_{3}). We refer to the two edges in the first pair as over-edges, and to the two edges (e 1,e 2)(e_{1}, e_{2}) in the second pair as under-edges.

  2. A cyclic ordering of the four edges of GG incident to vv so that the over-edges and under-edges alternate.

Definition

A link diagram is a planar graph such that each connected component satisfies either 1. or 2. below.

  1. It is (up to homeomorphism) a circle, disjoint from the rest of the link diagram.

  2. It is a connected link diagram in the sense of Definition .

Remark

It is important to note that a plane graph consists, by definition, of an abstract graph together with a chosen embedding into the plane. There exist non-equivalent link diagrams which have the same underlying abstract graph, but for which the embedding in the plane is different.

Terminology

A component of a link diagram which satisfies 1. in Definition is typically referred to as an unknot, or an unknotted component.

Terminology

The vertices vv of a link diagram LL which do not belong to an unknotted component, together with the data of 1. and 2. in Definition for vv, are referred to as the crossings of LL.

A link diagram can be obtained from a link by choosing a plane in 3\mathbb{R}^{3}, and projecting the link onto this plane. A small change in the direction of projection will ensure that it is one-to-one except at the double points, which will become the crossings of the link diagram in the sense of Terminology , where the image curve of the knot crosses itself once transversely. Which strand of the two intersecting at the double point is the ‘over strand’ and which is the ‘under strand’ is recorded as the data needed for 1. and 2. in Definition .

Consider for example the parallel projection

p: 3 2p: \mathbb{R}^3 \to \mathbb{R}^2

defined by p(x,y,z)=(x,y,0)p(x,y,z) = (x,y,0).

(If you prefer your knots to be in S 3S^3, of course, you can remove a single point from the complement of KK and then project down to R 3\mathbf{R}^3. It does not matter which point you use.)

A point x\mathbf{x} in the image pKpK is called a multiple point if p 1(x)p^{-1}(\mathbf{x}) contains more than one point of KK. A double point occurs when there are exactly two points of KK in this, and similarly for a triple point, etc. Multiple points of infinite order could occur.

A knot is in regular position or general position with respect to pp if there are only double points and these are genuine crossings (i.e. no tangential touching occurs in the projected curve).

Any smooth or PL link LL is equivalent under an arbitrarily small rotation of R 3\mathbf{R}^3 to one in regular position with respect to pp.

A proof can be found in Crowell and Fox (page 7).

Terminology

A knot diagram is a link diagram which arises from a projection of a knot.

category: knot theory

Last revised on August 26, 2018 at 16:05:41. See the history of this page for a list of all contributions to it.