Alexander in his paper. Take an oriented diagram of the knot with crossings; there are regions of the knot diagram. To work out the Alexander polynomial, first one must create an incidence matrix of size . The rows correspond to the crossings, and the columns to the regions. The values for the matrix entries are either . 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: on the right before undercrossing: on the left after undercrossing: on the right after undercrossing: Remove two columns corresponding to adjacent regions from the matrix, and work out the determinant of the new matrix. Depending on the columns removed, the answer will differ by multiplication by , where the power of is not necessarily the number of crossings in the knot. To resolve this ambiguity, divide out the largest possible power of and multiply by 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. After the work of J. W. Alexander, Ralph Fox considered a copresentation of the knot group , and introduced non-commutative differential calculus , which also permits one to compute . Detailed exposition of this approach about higher Alexander polynomials can be found in the book . Basic properties of the polynomial The Alexander polynomial is symmetric: for all knots K. From the point of view of the definition, this is an expression of the Poincaré Duality isomorphism where is the quotient of the field of fractions of by , considered as a -module, and where is the conjugate -module to ie: as an abelian group it is identical to but the covering transformation acts by . Furthermore, the Alexander polynomial evaluates to a unit on 1: . From the point of view of the definition, this is an expression of the fact that the knot complement is a homology circle, generated by the covering transformation . More generally if is a 3-manifold such that it has an Alexander polynomial defined as the order ideal of its infinite-cyclic