### Context

#### Topology

topology

algebraic topology

## 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 $w\left(D\right)$.

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

This is a variant of the writhe that is more adapted for use with links.

Suppose we have an oriented link diagram $D$ with components ${C}_{1},\dots ,{C}_{m}$, 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 $\mathrm{lk}\left({C}_{i},{C}_{j}\right)$.

The linking number of the diagram $D$ us then the sum of the linking numbers of all pairs of components:

$\mathrm{Lk}\left(D\right)=\sum _{1\le iLk(D) = \sum_{1\le i\lt j\le m}lk(C_i,C_j).

## Examples

The writhe of the standard trefoil is 3, of the Hopf link (both components clockwise oriented) is +2, but that of the Borromean rings is 0 although it is a non-trivial link.

## Invariance?

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

###### Proposition

The writhe is an invariant of regular isotopy.