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
Famous quotes containing the words basic and/or properties:
“It is a strange fact that freedom and equality, the two basic ideas of democracy, are to some extent contradictory. Logically considered, freedom and equality are mutually exclusive, just as society and the individual are mutually exclusive.”
—Thomas Mann (1875–1955)
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)