Notes for AKT-140108/0:08:12

18S-AKT Question: Why is this sum divisible by 2? Why the $\frac{1}{2}$?

Attempt 1 The factor $\frac{1}{2}$ I think is as a result of the projection of the link unto the plane making each sign appear twice.

Jordan Curve Theorem. If $C$ is a simple closed curve in $\mathbb{R}^2$, then the complement ${\mathbb R}^2\setminus C$ has two components, the interior and the exterior, with $C$ the boundary of each.

Attempt 2The Jordan curve theorem implies that two distinct components in a diagram for a link $L$ intersect an even number of times. Hence we add up an even number of $\pm 1$'s in the computation of $lk(L)$, which yields an even number. It is always an integer. This is why we have a factor of $\frac12$.

Attempt 3 Improved The Jordan curve theorem requires that we have simple closed curves and in general the components of a link are not simple closed curves. However, in the definition of the linking number, only the crossings between the components count. So we can smooth out crossings that are not between the components and this will not affect the linking number; we will then get components that are simple closed curves. From this argument, we can then apply the Jordan curve theorem to get the desired result.