Keeping track of crossings in knots

If you begin at one point on a knot and follow the rope all the way around until you come back to the starting point, you can record whether the strand you are following passes over or under at each crossing. Mathematicians sometimes use + to indicate that the strand passes over, and - to show that it passes under.

For example, if you start at the top right of the figure-8 knot above and follow the strand going down and to the right, the crossings are: + - + - + - + - . If you reshape and tangle the knot, will the list of over/under crossings show the same pattern?

Experiment with crossings and crossing numer on a variety of knots and decide how useful you think it is to study crossings when you are classifying knots, or trying to determine if two knots are the same.