One of the reasons knots have given mathematicians fits is that the same knot can appear in very different guises. Tug here, tug there, and soon a knot will become unrecognizable, but remain fundamentally unchanged. To allow a knotted string to wiggle around without danger of untying, mathematicians seal its two ends together, making it a knotted circle. The first question mathematicians have to answer is simply, when are two knots really, secretly the same?
The dream is to create a sort of machine: Send in one of these looped knots, and out pops some result that would be the same regardless of the particular configuration of the knot. Because the answer wouldn’t vary with the arrangement of the knot, such a machine is called a “knot invariant.” And indeed, in 1927, mathematician J.W. Alexander created just such a “machine,” a method that produces a polynomial (an expression like 3×2 + 4x + 1) from any knot. The good news is that Alexander’s method always gives the same polynomial for a particular knot, even if the knot has been wiggled around to look very different. The bad news is that it can also give the identical answer for knots that really are different. For example, the granny knot and the square knot have identical Alexander polynomials.