The Alexander Polynomial was discovered by James Waddell Alexander II in 1928 as a generalization of linear color tests. Designed to be calculated through matrices, the original form differs from more recent polynomial invariants that use skein relations. John Conway, however, found a skein relation that lead to the Alexander Polynomial in the 1960's, bridging the gap between Alexander's polynomial and today's widely used polynomial invariants.
The first step in calculating the Alexander Polynomial for a knot is to draw an oriented knot diagram on a two-dimensional plane like the one below. A knot with c crossings will divide the plane into c + 2 regions, each of which we label. This includes the region outside of the knot! Then it is time to perform the calculations.
The heart of the Alexander Polynomial is the matrix. This matrix is indexed in rows by crossing number and in columns by region letter. The values of the elements in this matrix in the row of a crossing are filled in as follows:
Then, any two columns representing regions that share an arc of the knot diagram are removed, leaving a c x c matrix. The preliminary Alexander Polynomial is the determinant of the matrix. To get the final Alexander Polynomial, t is factored out and removed as many times as possible.
Unfortunately, the Alexander Polynomial is not an ideal polynomial invariant. It distinguishes between different isotopic classes with crossings fewer than nine. Being the first Polynomial Invariant, the Alexander Polynomial was not without its problems, but it is still worthy of note.