Alexander polynomial

《Symplectic Geometry》 · 第 157 页 · Chapter 7，Definition 7.1.1（本书采用参数 x 的规范化 Laurent 多项式版本）

The Alexander polynomial of an oriented link \(L\), written variously as \(\Delta(L)\) or \(\Delta_L(x)\) depending on the required emphasis, is the determinant \(\det(xM-x^{-1}M^{\mathrm T})\) where \(M\) is any Seifert matrix for \(L\). It is a Laurent polynomial in \(\mathbb Z[x^{\pm1}]\), that is a polynomial in positive and negative powers of \(x\) with integer coefficients.

《Mathematical Physics: Classical Mechanics》 · 第 79 页 · Chapter 8 开头（Conway 为消除单位歧义的规范化 Alexander 不变量）

The Conway polynomial [20] for an oriented link is really just the Alexander polynomial without the ambiguity concerning multiplication by units of \(\mathbb Z[t^{-1},t]\).

《Symplectic Geometry》 · 第 165–166 页 · Section 7.4，变量替换

However, making the substitution \(z=x^{-1}-x\), we recover the Alexander polynomial, and all the axioms are known to hold in this case.