It is a hard mathematical question to completely codify all possible knots. Given two knots, it is hard to tell whether they are the same. It is harder still to tell for sure that they are different.
Many simple knots can be arranged in a certain form, as illustrated below, which is described by a string of positive integers along with a sign.
Figure 4: Here are drawings of some examples of knots that Conway `names' by a string of positive integers. The drawings use the convention that when one strand crosses under another strand, it is broken. Notice that as you run along the knot, the strand alternates going over and under at its crossings. Knots with this property are called alternating knots. Can you find any examples of knots with more than one name of this type?
Figure 5: Here are the knots with up to six crossings. The names follow an old system, used widely in knot tables, where the th knot with crossings is called . Mirror images are not included: some of these knots are equivalent to their mirror images, and some are not. Can you tell which are which?