Definition
Let K be a knot in the 3-sphere. 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. There is a covering transformation t acting on X. Consider the first homology (with integer coefficients) of X, denoted . The transformation t acts on the homology and so we can consider a module over . This is called the Alexander invariant or Alexander module.
The module is finitely presentable; a presentation matrix for this module is called the Alexander matrix. If the number of generators, r, is less than or equal to the number of relations, s, then we consider the ideal generated by all r by r minors of the matrix; this is the zero'th Fitting ideal or Alexander ideal and does not depend on choice of presentation matrix. If r > s, set the ideal equal to 0. If the Alexander ideal is principal, take a generator; this is called an Alexander polynomial of the knot. Since this is only unique up to multiplication by the Laurent monomial, one often fixes a particular unique form. Alexander's choice of normalization is to make the polynomial have a positive constant term.
Alexander proved that the Alexander ideal is nonzero and always principal. Thus an Alexander polynomial always exists, and is clearly a knot invariant, denoted .
Read more about this topic: Alexander Polynomial
Famous quotes containing the word definition:
“... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
—George Eliot [Mary Ann (or Marian)
“Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.”
—Walter Pater (1839–1894)
“According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
—Ana Castillo (b. 1953)