Link Invariants
Examples
Related concepts
CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
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:
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.
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 $C_1, \ldots, C_m$, 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 $lk(C_i,C_j)$.
The linking number of the diagram $D$ us then the sum of the linking numbers of all pairs of components:
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.
The writhe is not an isotopy invariant, as it can be changed but twisting a stand of the knot (or link).
The writhe is an invariant of regular isotopy.
The linking number is a link invariant.
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.