A string is just a line segment

Yes, but the ends could be extended to infinity in arbitrary directions, so we can tie a knot in a line, not just a segment. You can't do that to the edges of a blanket knot (two sides run into the knot).

A mathematical knot is like a loop of string. There are no ends. Of course, you could cut any mathematical knot wherever you like and make the two ends go to infinity if you'd like.

Thank you, I appreciate the correct definition.

I understand a bit better after that, thanks.

But a knot in a plane segment doesn't need to fold the ends in - I can grab a blanket in the middle, and tie a knot there, then extend edges infinitely still?

I can imagine an infinitely long line with a know in the middle in 3d ... I guess the interesting question is "can you have an infinite plane with a knot in the middle in 5 dimensions?6 dimensions?"

if you selected a single point on a plane and pulled it upwards you could then knot it very similarly to a line couldn't you?

that's like tieing a knot in a loop of string, not like a granny knot ... I think the sort of knot I have in mind is a harder problem

A knot is an embedding of S1 in R3, i.e. a closed loop and not a line segment.