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 why is handedness defined in this way? Is there a practical reason or is it just because people got used to it?
For reference, here is the link: https://homepages.warwick.ac.uk/~masgar/Teach/2008_MA3F2/lecture1.pdf. Thanks
The right/left handed conventions for crossings is the "right" one given the way we say whether a given basis for 3D has a positive or negative orientation. This has to do with the right-hand rule in that if $v$ and $w$ are two vectors, then when $v\times w$ is nonzero you have that $v,w,v\times w$ is a positively oriented basis. Let's get into how I've convinced myself right/left-handed crossings are defined in a reasonable way.
Suppose you have an oriented disk and a line passing through it. There is a well-defined notion of the orientation of an intersection between the line and the disk. A way to calculate it is to take three linearly independent vectors: two from the disk itself, where the second vector is 90 degrees counterclockwise from the first with respect to the orientation of the disk, and the third is from the line. Take the determinant of the matrix whose columns are these three vectors in this order, and if the result is positive the intersection is "positive" and if it's negative the intersection is "negative." One can equivalently take a dot product of the surface normal for the disk with a tangent vector for the line rather that computing a determinant.
In the following illustration, the first is a positive intersection and the second is a negative intersection. I'm using the orientation of the boundary to indicate the orientation of the disk (it points in the direction you'd rotate the disk to get its positive "x-axis" to point in the direction of its positive "y-axis").
Now, notice that a line and the boundary of a given disk form a knot diagram. In the disk with a positive intersection, both crossings are positive ("right"). In the disk with a negative intersection, both crossings are negative ("left").
There's a mnemonic: if you grasp the line with your thumb pointing in the direction of the line, then if you did it with your right hand (resp. your left hand) then your four fingers point in the direction of the boundary of the right-handed/positive (resp. left-handed/negative) disk.
I think it's better to speak of crossings as being positive or negative rather than right- or left-handed. Handedness depends on a convention that positive-orientation is "right" and negative-orientation is "left." (At least the right-hand rule for orientations is well-established in math and physics. I imagine it's at least 150 hundred years old, but a cursory search didn't turn up the history.)