Make a knot with a string and "fix" it by joining its ends. Next, let the string fall on a table: you get a "picture" of a curve which is almost flat. It is not totally flat - unless it is the unknot - because there are some points where the string does not lie on the surface of the table but it has some crossings.

You can represent this situation with a diagram where the discontinous piece of the curve stands for the branch of the knot that has an undercrossing.
This corresponds to project the knot on a plane, which results in a plane curve with some crossings. This curve is not enough to reconstruct the three-dimensional knot unless you specify every time the branch that goes above and the one that passes below.
Thus, the three-dimensional knot descends to a two-dimensional environment (a plane curve with some crossings and a specialization of over/under for each crossing). It is quite natural to imagine that the number of crossings of the curve can give some information about the complexity of the knot.

This is quite right; yet you need to be cautious. Indeed, the number of crossings may depend on the projection we choose, that is to say, how the string falls on the plane. It happens that a knot may seem to have three crossings under a given projection, and four from another view

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There is even more to ponder! The investigation of knots is topological. This means that you do not have to think about knots like rigid objects.You can imagine that they are as long and extensible as you wish. You can also manipulate them in any way you want; so you obtain knots which can’t be distinguished from the original one as long as the knot stays tied. It is strictly forbidden to cut the string and tie it after making some operations on the knot. To summarize, you may have "legal" projections of the same knot where the number of crossings is much higher than the original one.

This is what happens for the knot represented by the exhibit from the exhibition matemilano. It is a trefoil as you may realize after you manipulate it for a bit.

In fact, when we speak about the “crossing number” of a knot, this does not mean (and could not mean!) the number of crossings of an arbitrary projection, but it refers to the MINIMUM number of crossings after ANY manipulation of the knot, followed by ANY possible projection on a plane.

It is not easy at all to compute this number. If you let a string fall on a plane and find out that the knot diagram has seven crossings, you need to be sure there is no other way of letting the string fall on the plane with a smaller number of crossings before you can say that the crossing number is indeed seven.

The crossing number of the unknot is 0. The crossing number of the trefoil is 3; there is no knot with crossing number 1 or 2. In fact, if you draw a knot with one or two crossings, you actually get the unknot, i.e., an object which can be manipulated into a circle.

The two trefoils ( right trefoil and left trefoil) are the only ones with crossing number 3. The figure eight knot is the only one with crossing number 4. The number of different possible knots with a given crossing number increases a lot as this crossing number gets higher: with nine crossings you have more or less one hundred possibilities!

The history of knot theory is very interesting. Some forerunners of this theory - P.G. Tait and W. Thomson (or Lord Kelvin) - began to investigate knots at the end of the XIXth century. Their study was based on a theory regarding the structure of matter, which appears very imaginative these days. According to this theory, the atoms (or vortex atoms) were not points but small knots which interlinked and formed molecules. With atoms in the back of their mind, Tait and Thomson aimed at classifying knots. Very likely, they undervalued the problem, which in fact has not been hitherto solved. The first "knot tables" date back to their trials, i.e., the first lists of knots which were ordered with respect to the crossing number.

In the compilation of these lists (with knots up to crossing number 10), Tait made some claims that he thought they were so obvious they did not need any proofs - such claims were related to the number of crossings. As time went by, the claims turned into conjectures - the so-called "Tait conjectures". They have been proved only in recent years in spite of the growing specialization of the techniques available in knot theory.