nLab linking number

Definitions

Signs of crossings

In an oriented link diagram, we can see there are two types of possible crossing. They are allocated a sign, + or -.

One method of remembering the sign convention is to imagine an approach to the crossing along the underpass in the direction of the orientation:

• if the overpass passes from left to right the crossing is counted as being positive;
• if it passes from right to left it counts as negative.

Writhe

The writhe of an oriented knot or link diagram is the sum of the signs of all its crossings. If $D$ is the diagram, we denote its writhe by $w(D)$.

The writhe is used in the definition of some of the knot invariants.

Linking number

The linking number is a variant of the writhe that is more adapted for use with links.

Consider an oriented link diagram $D$ with components $C_1, \ldots, C_m$.

Definition

The linking number of $C_i$ with $C_j$ where $C_i$ and $C_j$ are distinct components of $D$, is to be one half of the sum of the signs of the crossings of $C_i$ with $C_j$; it will be denoted $lk(C_i,C_j)$.

(e.g. Ohtsuki 2001 pp 7)

Definition

The total linking number of the diagram $D$ is then the sum of the linking numbers of all pairs of distinct components:

$$Lk(D) \;\coloneqq\; \sum_{1\le i\lt j\le m}lk(C_i,C_j) \,.$$

Definition

The linking matrix of a framed link is the square matrix indexed by the connected components of the link whose off-diagonal entries are the linking numbers and whose diagonal entries are the framing numbers.

(e.g. Ohtsuki 2001 p 227)

Notice that the linking matrix is a symmetric matrix.

Hence for a framed link the total linking number is equivalently half the sum of all non-diagonal entries of the linking matrix.

Examples

The writhe

• of the trefoil knot is 3,

• of the Hopf link (both components clockwise oriented) is +2,

• of the Borromean rings is 0 although it is a non-trivial link.

Properties

Invariance?

The writhe is not an isotopy invariant, as it can be changed by twisting a strand of the knot (or link).

Proposition

The writhe is an invariant of regular isotopy.

Proposition

The linking number is a link invariant.

Proof

We use Reidemeister moves so have to check that they do not change the linking number of a diagram. Any Reidemeister move that involves at least two components of the link (i.e. which must be an R2 or R3) leaves all linking numbers between components unchanged. An R2 move removes or introduces two crossings of opposite sign, whilst an R3 leaves the number of crossings and their signs unaltered.

We can conclude that the Hopf link is not isotopic to the two component unlink, (which is reassuring) as any assignment of orientations to the Hopf link leads to a non-zero linking number.

Linking number is an integer

Let $n_{1}$ (resp. $n_{2}$) be the sum of the signs of crossings between a pair of components $L_{1}$ and $L_{2}$ of a link in which the over arc belongs to $L_{1}$ (resp. $L_{2}$). The linking number is then clearly equal to $\frac{1}{2}(n_{1} + n_{2})$.

Now, take a crossing between $L_{1}$ and $L_{2}$ such that the over arc belongs to $L_{1}$. Suppose that we switch the crossing type, so that now the under arc belongs to $L_{1}$. Then, no matter what the sign of the crossing, it can be verified that the sum $n_{1} - n_{2}$ remains unchanged by this switch.

We may also obviously make crossing switches to the self-crossings of a component, or to crossings of the two components we are looking at with other components, without affecting the sum $n_{1} - n_{2}$.

In addition, it is straightforward to check that the Reidemeister moves do not change the sum $n_{1} - n_{2}$.

Now, it is obvious that any pair of components of a link can be unlinked from the link after making appropriate crossing switches. In the case that $L_{1}$ and $L_{2}$ are circles disjoint from each other and the rest of the link, we have that $n_{1} - n_{2} = 0$.

We deduce that $n_{1} - n_{2} = 0$ in all cases. Hence $n_{1} = n_{2}$, and the linking number is equal to $\frac{1}{2}(2n_{1}) = n_{1} = n_{2}$.

In particular, the linking number is an integer.

An immediate consequence of the fact that the sum of the signs of the crossings between $L_{1}$ and $L_{2}$ is even is that there must in fact be an even number of such crossings.

Related concepts

• framing number

• writhe

• crossing number

References

• Tomotada Ohtsuki, pp 7 in: Quantum Invariants – A Study of Knots, 3-Manifolds, and Their Sets, World Scientific (2001) [doi:10.1142/4746]

• Renzo L. Ricca, Bernardo Nipoti: Gauss’ Linking Number Revisited, Journal of Knot Theory and Its Ramifications 20 10 (2011) 1325-1343 [doi:10.1142/S0218216511009261, pdf]

• Wikipedia, Linking number