What is the difference between a knot and a plane curve? I am curious about geometry topics, and I could not understand the Wikipedia definition of knot because I don't have the pre-requisites (I guess). But looking at the Wikipedia picture below they seem to match what I understand as a plane curve. In special, the unknot looks like a circle. Sorry if this question is stupid.

I just want to know what differs a knot from a curve, or if knots are
special cases of curves (maybe?).

Thank you.

 A: I would start off by saying that your question skirts close to being philosophical in nature, but I will stick with the literal.  Now to the specific question at hand.
For me (a low-dimensional topologist) a plane curve is any 1-manifold in the plane.  That means a function $C:I \to \mathbb{R}^2$, where $I$ is an open or closed interval in $\mathbb{R}$.  So you should think of this as the parabola that you are used to, which we can write as
$$f:\mathbb{R} \to \mathbb{R}^2, \text{ where } f(x) = (x,x^2)$$
But it can be the circle you mentioned too, $$C:[0,1] \to  \mathbb{R}^2, \text{ where } 
C(x) = (\cos (2\pi x), \sin (2\pi x) )$$

I mention what I take as a plane curve, as the definitions provided on Wikipedia and Wolfram MathWorld seem to leave a bit out, but this might just be me.
Now, a knot is almost the same thing, but we usually put it in three dimensions and require that it start and end at the same point.  And that circle can be a knot by just adding a $z$ variable to our $(x,y)$ plane.
$$K_0:[0,1] \to  \mathbb{R}^3, \text{ where } 
K_0(x) = (\cos (2\pi x), \sin (2\pi x), 0 )$$

So you really should think of knots as living in 3-space, instead of on the plane, which make the unknot, $K_0$ a bad example.  So here is a trefoil, and the function which makes it look like what we usually envision.
$$K_3:[0,2\pi] \to  \mathbb{R}^3, \text{ where } 
K_3(t) = (\sin t + 2\sin 2t, \cos t - 2\cos 2t, -\sin 3t )$$

So you can probably see from the nice images on the Wikipedia page and this graph (which you can play with here: Math3d.org) that this thing does not live in the plane.
Now, what about those pictures you have in your question?  Well those are not actually knots, but diagrams of knots.  Which means, that we want to look at them on a page, since we can.  It is easier than having to envision these things in 3 dimensions all the time.  So to do this take any knot and project it onto your favorite plane.  This usually is going to be the $(x,y)$ plane again, but there is a catch, we want it to be clear what is "going over" and "under" each other piece.  So there are some rules about what diagrams look like, but just remember that only two lines can cross and they must be actually crossing at any one point.  And you should be thinking: "Doesn't this sound like a plane curve?" and you would be right!  We now have a plane curve.  But we often break the line so you can see which is the "overarc" and "underarc." So knot diagrams are plane curves with some more information in them.
To wrap up, your intuition that these were almost the same thing are kind of right.  But math really cares about the details.  So we think of them as different, since they come from different ideas often.  We don't usually deal with the parametric equations of a knot, since we like to move and bend them.  Anyway, hope this helped.
