Any such knot could be topologically deformed into a knot made of arbitrarily thin plane segment, which would end up being the same as a line knot. So knots made out of finite plane segments are uninteresting.

I should have been more clear though, I was imagining an infinite 'blanket', which I grab in the 'middle' (not an end) and tie a knot there.

Or does that not count as (mathematical) knotting?

A sheet of paper or a blanket would only be a plane segment.

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.

For it to be a mathematical "knot", the edges have to be sewn together into form a knotted sphere. The trivial way to knot a blanket doesn't yield a trivial way to connect all the edges without untying the knot.

A sheet of paper or a blanket is equivalent to D2, the disc, which is not a plane.