Let $F = F(x_1,\ldots,x_n)$ be the free group on generators $x_1,\ldots,x_n$. The Fox derivatives $\frac{\partial}{\partial x_j}: \mathbb{Z}[F] \to \mathbb{Z}[F]$ are the unique $\mathbb{Z}$-linear maps satisfying: (1) $\frac{\partial x_i}{\partial x_j} = \delta_{ij}$, (2) $\frac{\partial(uv)}{\partial x_j} = \frac{\partial u}{\partial x_j} + u\frac{\partial v}{\partial x_j}$ (Leibniz rule), and (3) $\frac{\partial 1}{\partial x_j} = 0$. From (2): $\frac{\partial x_i^{-1}}{\partial x_j} = -x_i^{-1}\delta_{ij}$. For any word $w = x_{i_1}^{\varepsilon_1}\cdots x_{i_k}^{\varepsilon_k}$, Fox derivatives compute the "infinitesimal effect" of each generator.

Given a Wirtinger presentation $\pi_K = \langle x_1,\ldots,x_n \mid r_1,\ldots,r_n\rangle$, the Alexander matrix $A$ is the $(n-1)\times(n-1)$ matrix obtained as follows: (1) Compute the Jacobian matrix $J_{ij} = \frac{\partial r_i}{\partial x_j} \in \mathbb{Z}[F]$. (2) Apply the abelianization map $\phi: \mathbb{Z}[\pi_K] \to \mathbb{Z}[t^{\pm 1}]$ sending each generator $x_j \mapsto t$ (since $\pi_K^{\mathrm{ab}} \cong \mathbb{Z}$). (3) Delete one row and one column from $\phi(J)$. The Alexander polynomial $\Delta_K(t)$ is the determinant $\det A$, defined up to units $\pm t^k$ in $\mathbb{Z}[t^{\pm 1}]$.

The Alexander polynomial satisfies: (1) Symmetry: $\Delta_K(t^{-1}) = \pm t^k\Delta_K(t)$ for some $k$ (from the duality of the knot exterior). (2) Normalization: $\Delta_K(1) = \pm 1$ (since $H_1(S^3\setminus K;\mathbb{Z}) = \mathbb{Z}$). (3) Skein relation: $\Delta_{K_+}(t) - \Delta_{K_-}(t) = (t^{1/2}-t^{-1/2})\Delta_{K_0}(t)$ where $K_+$, $K_-$, $K_0$ are links related by a crossing change. (4) Genus bound: the degree of $\Delta_K$ satisfies $\deg\Delta_K \leq 2g(K)$ where $g(K)$ is the Seifert genus. For fibered knots, equality holds and $\Delta_K$ is monic.