Suppose we take a length of small gauge rope, loop it into some sort of knot, splice the ends to form a closed loop, and lay the knotted loop ‘flat’ in the plane. Then there are various interesting questions which can be asked. The knotted rope will have a number of simple crossovers. What is the smallest possible number of crossovers (> 0!)? If you have a closed loop with say 6 crossovers, will it necessarily be knotted? If we rearrange the loop to remove any unnecessary crossings, the number of crossovers is called the crossing number. Do different knots necessarily have different crossing numbers? This nice little bit of theory belongs to the branch of mathematics known as topology: it is independent of the length and exact shape of the loop.
