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.

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

(

X

)

{\displaystyle H_{1}(X)}

. The transformation t acts on the homology and so we can consider

H

1

(

X

)

{\displaystyle H_{1}(X)}

a module over the ring of Laurent polynomials

Z

[

t

,

t

−

1

]

{\displaystyle \mathbb {Z} [t,t^{-1}]}

. 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.