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 Alexander polynomial ： ウィキペディア英語版
Alexander polynomial
In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered this, the first knot polynomial, in 1923. In 1969, John Conway showed a version of this polynomial, now called the Alexander–Conway polynomial, could be computed using a skein relation, although its significance was not realized until the discovery of the Jones polynomial in 1984. Soon after Conway's reworking of the Alexander polynomial, it was realized that a similar skein relation was exhibited in Alexander's paper on his polynomial.〔Alexander describes his skein relation toward the end of his paper under the heading "miscellaneous theorems", which is possibly why it got lost. Joan Birman mentions in her paper ''New points of view in knot theory'' (Bull. Amer. Math. Soc. (N.S.) 28 (1993), no. 2, 253–287) that Mark Kidwell brought her attention to Alexander's relation in 1970.〕
==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 $H_1\left(X\right)$. The transformation ''t'' acts on the homology and so we can consider $H_1\left(X\right)$ a module over $\mathbb\left(t^\right)$. 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 $\pm t^n$, 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 $\Delta_K\left(t\right)$. The Alexander polynomial for the knot configured by only one string is a polynomial of t2 and then it is the same polynomial for the mirror image knot. Namely, it can not distinguish between the knot and one for its mirror image.

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