if you draw a line on a piece of paper (not a perfectly straight one but a line that can curve as it likes) and if that line bends around and crosses itself once there are two possible configurations either it will hide both the start and end of the line inside the loop or it will leave them outside the loop if it crosses itself twice there are twelve possible configurations these twelve form three families if A is a time that the line passes through the first crossing and B is a time it passes through the second crossing these three families are AABB (which has four members) ABAB (which has two) ABBA (which has six) each letter appears twice because that is what a crossing is, a time the line comes to the same place twice. there is no such family as BAAB for example since it is just the wrong way to write ABBA using this notation it would appear that their are 15 possible families of three-crossing figures but two are equivalent to each other (if the start and end of the line are the same) and two have no members because they are impossible to actually draw. -these are very interesting shapes to play with i encourage you to take a stroll in this little world and draw them yourself (i'll post a picture if i can get my scanner working) -has this been studied? if so do you know what this set of figures is called? or is it equivalent to some other system that has been studied? -the closest thing i could find was knot theory but that is quite different, as far as i can tell it has only a cosmetic similarity.