Confusion About Definition of Handedness in Knot Theory

Question

I read that to determine the handedness of a crossing, you look at a small, almost-straight segment of rope that has the overcrossing. You then check to see if that segment has an overall slope that is negative or positive. If it is positive, then it is right-handed. Otherwise, it is left-handed.

But what if the slope is $0$?

For reference, here is the link: https://homepages.warwick.ac.uk/~masgar/Teach/2008_MA3F2/lecture1.pdf. Thanks

Answer 1

The definition of a knot diagram in the plane requires a number of properties:

1. The projection needs to be an immersion; this means it has to be smooth with no cusps.
2. Whenever the projection intersects itself, the two intersecting curves cannot be tangent to one another. (Equivalently, the tangent vectors at the crossing point have to span $\mathbb{R}^2$.)
3. The only self-intersections are double points (so there cannot be three different points projected to the same point)>

The second property, from the perspective of the notes, is exactly that the slope is not $0$.

In short, the reason the slope isn't $0$ is because by definition knot diagrams aren't allowed to have slope-$0$ crossings. It's possible to prove that every knot has a diagram that satisfies all the properties, using something like Sard's theorem.