covering space. In this case is, up to sign, equal to the order of the torsion subgroup of . It is known that every integral Laurent polynomial which is both symmetric and evaluates to a unit at 1 is the Alexander polynomial of a knot (Kawauchi 1996). 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). Kauffman describes the first construction of the Alexander polynomial via state sums derived from physical models. A survey of these topic and other connections with physics are given in. 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). 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 ). In fact, if is a fiber bundle where is the knot complement, let represent the monodromy, then where is the induced map on homology. Relations to satellite operations If a knot is a satellite knot with pattern knot (there exists an embedding such that , where is an unknotted solid torus containing ), then , where is the integer that represents in . Examples: For a connect-sum . If is an untwisted Whitehead double, then . Alexander–Conway polynomial Alexander proved the Alexander polynomial satisfies a skein relation.