I was reading this Wikipedia article on writhe, and it said the following for the definition of writhe of a link diagram:
In knot theory, the writhe is a property of an oriented link diagram. The writhe is the total number of positive crossings minus the total number of negative crossings.
A direction is assigned to the link at a point in each component and this direction is followed all the way around each component. For each crossing one comes across while traveling in this direction, if the strand underneath goes from right to left, the crossing is positive; if the lower strand goes from left to right, the crossing is negative. One way of remembering this is to use a variation of the right-hand rule.
Accompanied by the following visual aid:
This definition immediately seemed odd to me. For one, what if the lower strand is straight up and down? It also seemed odd that positivity or negativity of a crossing would not be preserved by rotation. In fact for certain diagrams, the total writhe would change with rotation.
I searched a bit for another definition, and found the following on Wolfram MathWorld:
A knot property, also called the twist number, defined as the sum of crossings $p$ of a link $L$,
$$w(L)=\sum_{p \in C(L)}\epsilon(p), $$
where $\epsilon(p)$ defined to be $\pm1$ if the overpass slants from top left to bottom right or bottom left to top right and $C(L)$ is the set of crossings of an oriented link.
This seems the same as Wikipedia's definition, except it depends on the direction of the upper strand instead of the lower one. This makes things even more confusing, because these definitions aren't equivalent.
This diagram for example doesn't have a defined writhe by Wikipedia's definition, but by Wolfram's definition, its writhe would be +2 or -2, depending on the direction of the upper strand.
Even if we only treat diagrams where no crossings have a fully vertical strand, we still run into this issue.
This diagram, according to Wikipedia's definition has a total write of 0, where by Wolfram's definition, it has a writhe of -2. Note that the sign can vary with rotation.
In this figure, the -2 in Wolfram's definition becomes 2. Note that the 0 in Wikipedia's definition would become a 2 and Wolfram's would become a 0 if which strand was on top were switched. But, writhe is supposedly preserved by Reidermeister moves 2 and 3, which is why I started looking at this in the first place. But we can switch which strand is on top by applying Reidermeister move 2 twice.
So now according to Wikipedia's definition, the writhe of this diagram is 2, where as Wolfram would say it's 0. But both changed, so in neither case was the writhe preserved by Reidermeister 2 moves. In addition, they are both equivalent to the diagram with no crossings. By the definitions, it would seem a diagram with no crossings would have writhe 0. So it clearly shouldn't be able to become a diagram with write 2 with just a Reidermeister 2 move.
So why do these definitions seem wrong? Is there a better definition of writhe, or am I making a mistake here?