Take a length of string and tie the ends together. The resulting loop is called a "knot" in mathematics. If the loop can be deformed (without cutting the string) to a circle then it is called the "trivial" knot. If the loop has twists and turns that can not be removed without cutting the string, then it is a non-trivial knot.

Given a loop of string, how can we tell if it can be untwisted to a circle (that is, it is a trivial knot) or not. One idea is to flatten the loop against a table, follow the string around the loop and keep count of the times the string crosses itself. If the piece you are following passed under another piece of the string, add 1 to the count. If the string crosses over another piece of string, subtract 1 from the count. This is called the "crossing number".

  Getting Started

Does the crossing number tell you anything about which loops of string are really trivial knots. Make some trivial knots, twist them up, put the on a table and count? What do you notice and why does it work?

  Going Deeper

What if you make the loop into a non-trivial knot. What do you notice about the crossing number? How does the crossing number depend on how you place the tangle on the table? Are there other ways to assign integers to the knot that tell you things about how tangled the knot is?

Communicated by G. Hall.