Geometric Significance of The Polynomial
Since the Alexander ideal is principal, if and only if the commutator subgroup of the knot group is perfect (i.e. equal to its own commutator subgroup).
For a topologically slice knot, the Alexander polynomial satisfies the Fox–Milnor condition where is some other integral Laurent polynomial.
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 (Freedman and Quinn, 1990).
There are other relations with surfaces and smooth 4-dimensional topology. 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.
Knots with symmetries are known to have restricted Alexander polynomials. See the symmetry section in (Kawauchi 1996). Although, 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 (highest and lowest order terms equal to ). In fact, if is a fiber bundle where is the knot complement, let represent the monodromy, then where is the induced map on homology.
Read more about this topic: Alexander Polynomial
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