Alexander polynomial

James Waddell Alexander II discovered this, the first knot polynomial, in 1923.

Soon after Conway's reworking of the Alexander polynomial, it was realized that a similar skein relation was exhibited in Alexander's paper on his polynomial.

Let X be the infinite cyclic cover of the knot complement of K. This covering can be obtained by cutting the knot complement along a Seifert surface of K and gluing together infinitely many copies of the resulting manifold with boundary in a cyclic manner.

Consider the first homology (with integer coefficients) of X, denoted

Alexander's choice of normalization is to make the polynomial have a positive constant term.

Thus an Alexander polynomial always exists, and is clearly a knot invariant, denoted

To work out the Alexander polynomial, first one must create an incidence matrix of size

Consider the entry corresponding to a particular region and crossing.

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.

Remove two columns corresponding to adjacent regions from the matrix, and work out the determinant of the new

Depending on the columns removed, the answer will differ by multiplication by

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

for all knots K. Furthermore, the Alexander polynomial evaluates to a unit on 1:

For a topologically slice knot, the Alexander polynomial satisfies the Fox–Milnor condition

Twice the knot genus is bounded below by the degree of the Alexander polynomial.

Michael Freedman proved that a knot in the 3-sphere is topologically slice; i.e., bounds a "locally-flat" topological disc in the 4-ball, if the Alexander polynomial of the knot is trivial.

[5] Kauffman describes the first construction of the Alexander polynomial via state sums derived from physical models.

For example, under certain assumptions, there is a way of modifying a smooth 4-manifold by performing a surgery that consists of removing a neighborhood of a two-dimensional torus and replacing it with a knot complement crossed with S1.

The result is a smooth 4-manifold homeomorphic to the original, though now the Seiberg–Witten invariant has been modified by multiplication with the Alexander polynomial of the knot.

[9] Nonetheless, the Alexander polynomial can fail to detect some symmetries, such as strong invertibility.

If the knot complement fibers over the circle, then the Alexander polynomial of the knot is known to be monic (the coefficients of the highest and lowest order terms are equal to

John Conway later rediscovered this in a different form and showed that the skein relation together with a choice of value on the unknot was enough to determine the polynomial.

Conway's version is a polynomial in z with integer coefficients, denoted

Here are Conway's skein relations: The relationship to the standard Alexander polynomial is given by

See knot theory for an example computing the Conway polynomial of the trefoil.

Using pseudo-holomorphic curves, Ozsváth-Szabó[10] and Rasmussen[11] associated a bigraded abelian group, called knot Floer homology, to each isotopy class of knots.

The graded Euler characteristic of knot Floer homology is the Alexander polynomial.

While the Alexander polynomial gives a lower bound on the genus of a knot, [12] showed that knot Floer homology detects the genus.