No, I mean that there is NO perspective that enables the in-proportion knot (parts) to be seen unambiguously --the "explosion" thus will pull some things out of being hidden from the perspective view to disclose their place.

Imagine the usual mathematical knots representation, in a 2-D surface.

1. All "crossings" are crossings of two, only lines.

2. All crossings are visible, i.e., they are apart the one from each other. There are no two crossing points in one point of the drawing, because, if this would had happened, this point would had not be a crossing point of two lines, but a crossing point of four lines.

3. In each and every crossing point, there is a "first" line going "over" a "second" = a "second" line going over a "first".

Now, go one step further.

*Imagine that this 2-D surface is not a plane, but a sphere*.

So, now, the crossing points are not arranged on a the surface of a flat plane, where every one is visible, but on the surface of a curved space. In other words, the crossing points are arranged on the surface of a "shell", so now one can be "over" or "under" another one. ( I had called it " the outer shell", for reasons I will explain in a while ).

If that happens, we get a 'perspective" problem, like the one you probably mean. Two crossing points can fall on the same point of the 2-D image of this shell, i.e., two crossing points can be on the same line of sight, so the one is "over" and the other is "under" = the one hides the other.

**The "explosion" is the means by which this problem is solved.** When the shell is "inflated", the 2-D surface is "expanded", like the elastic surface of a balloon when it is inflated. The probability of two points which happened to be on one line of sight in the deflated state ( so the one was hiding the other ), to remain in one line of sight in the inflated state, is almost zero. Even in the rare case it happens, a small rotation of the shell around one axis going through its centre corrects the problem.

So, we have a knot with many crossing points arranged on the surface of a sphere, and we manage to make all those crossing points visible, by inflating this sphere, so any two crossing points which happened to be in the same line of sight in the deflated sphere, now they are no more. We can see clearly all of them, and see which one of the two lines that crossing each other at each and every point is "over", and which is "under".

In

*this* sense, you are right, the 'explosion" solves a problem of perspective.

Now, why I say "the outer sell" ? Because there can be complex knots, where, to keep the geometric proportions as accurate as possible, the crossing points of the deflated / compact knot can only be arranged on two, or more concentric shells, which are connected, through some "necks", to each other. However, the crossing points of most, if not all of the simple practical knots we are interested in can be arranged on one shell, so this "outer" adjective is redundant.

Clear as mud ? Perhaps, but here it comes the images ! The whole idea I was trying to make you imagine with words, becomes transparent with images ( see the attached image ). THAT is why I tell you that you should learn to start to use less words, and more images !

I am using the word "

*sphere*" in the topological, not the geometrical sense. In fact, any continuous "closed" surface, topological equivalent to the sphere, is "spherical". An egg-shaped surface, for example.

So, here comes the cooking recipe :

1. Take your compact knot, hold it in your palm, and start looking at it from any angle. Find a proper view that will help you do the following, more easily :

2. Start imagining the crossing points of any pair of lines been arranged on a transparent and elastic "spherical" surface, on a soap bubble for example.

3. Start imagining this surface been "exploded", that is, start imagining the bubble been inflated.

4. At some point, you will "see" all the crossing points : there will be no points of three or more lines on the same line of sight.

3. At that point, you will have a view where all the crossing points of all lines are clearly visible, and you can say if the one line goes "over" or "under" the other.

4. That s all, folks !