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 .
and it 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 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).
Read more about this topic: Alexander Polynomial
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