Above are rectilinear realizations of the trefoil knot.  The trefoil is the simplest of all knots and is fundamental to the study of mathematical knot theory.  It is chiral, meaning that a trefoil knot can be distinguished from its own mirror image.  Thus it exists in two distinct forms.  Not all knots are chiral.  For example, the next simplest knot, the figure-eight knot exists in only one form and is identical to its mirror image.

Mathematicians have tried to make a list of all possible knots.  The list grows rapidly as the “number of crossings” grows.  Thus, while the trefoil is the only knot with 3 crossings, there are 552 distinct knots with 10 crossings and 2176 with 11 crossings.  Probably there is no mathematical formula for the number of knots in the general case.  

It is not even known if the number grows exponentially as the crossing number increases.

There are deep connections between knot theory and quantum physics.