Alexander Polynomial - Computing The Polynomial

Computing The Polynomial

The following procedure for computing the Alexander polynomial was given by J. W. Alexander in his paper.

Take an oriented diagram of the knot with n crossings; there are n + 2 regions of the knot diagram. To work out the Alexander polynomial, first one must create an incidence matrix of size (n, n + 2). The n rows correspond to the n crossings, and the n + 2 columns to the regions. The values for the matrix entries are either 0, 1, −1, t, −t.

Consider the entry corresponding to a particular region and crossing. If the region is not adjacent to the crossing, the entry is 0. If the region is adjacent to the crossing, the entry depends on its location. The following table gives the entry, determined by the location of the region at the crossing from the perspective of the incoming undercrossing line.

on the left before undercrossing: −t

on the right before undercrossing: 1

on the left after undercrossing: t

on the right after undercrossing: −1

Remove two columns corresponding to adjacent regions from the matrix, and work out the determinant of the new n by n matrix. Depending on the columns removed, the answer will differ by multiplication by . To resolve this ambiguity, divide out the largest possible power of t and multiply by −1 if necessary, so that the constant term is positive. This gives the Alexander polynomial.

The Alexander polynomial can also be computed from the Seifert matrix.