Alexander polynomial (${\Delta }_K(t)$)

Up to multiplication by a unit $\pm t^n$, the Alexander polynomial isIt satisfies ${\Delta }_K(t^{-1})\doteq {\Delta }_K(t)$ and ${\Delta }_K(1)=\pm 1$.

The Conway polynomial is normalized by ${\nabla }_U=1$ for the unknot, zero for a split unlink with more than one component, and the skein relation ${\nabla }_{L_{+}}-{\nabla }_{L_{-}}=z{\nabla }_{L_0}$. For a knot, the symmetric Alexander polynomial with value one at $t=1$ is ${\Delta }_K(t)={\nabla }_K(t^{1/2}-t^{-1/2})$. This fixes the Laurent-unit ambiguity of the Seifert-matrix presentation of the Alexander module.

The symmetric Alexander polynomial normalized by the Conway polynomial is uniquely fixed for a knot. Its sign matters in the signed Jones evaluation at minus one, whereas the knot determinant retains only the absolute value.

A minimal-genus Seifert surface has a $2g_s(K)$-by-$2g_s(K)$ Seifert matrix $V$. Each entry of $tV-V^T$ has degree at most one, so its determinant has breadth of a Laurent polynomial at most $2g_s(K)$. This lower bound on Seifert genus need not be sharp: an untwisted Whitehead double can have Alexander polynomial of a knot $1$ and positive Seifert genus.

Assume $g_s(K)>0$. A nontrivial connected sum of knots has positive genus summands, and additivity of Seifert genus applies. Its Alexander polynomial of a knot is the product of the summand polynomials. If the polynomial is irreducible, one summand has unit polynomial; the other must then have polynomial breadth $2g_s(K)$, exceeding twice its strictly smaller genus. This contradicts the Alexander breadth bound on Seifert genus. Irreducibility without the full-degree hypothesis alone does not imply primeness.

Every Laurent polynomial $f(t)\in \mathbb{Z}[t^{\pm 1}]$ satisfying $f(t^{-1})=f(t)$ and $f(1)=1$ is the Alexander polynomial of a knot up to multiplication by a unit $\pm t^k$.

The Alexander module is the first integral homology of the infinite cyclic cover of a knot exterior. Its deck transformation makes it a module over $\mathbb{Z}[t^{\pm 1}]$, and its first elementary ideal determines the Alexander polynomial of a knot.

Boundary-connected-summing Seifert surfaces gives a block-diagonal Seifert matrix, because basis curves in the separated summands have zero linking. The Seifert-matrix presentation of the Alexander module is block diagonal too. Thus the Alexander modules form a direct sum and their Alexander polynomials of a knot multiply up to a unit.

Given a presentation $⟨x_1,\dots ,x_n\mid r_1,\dots ,r_m⟩$ of a knot group, apply abelianization to the Fox derivatives $\partial r_j/\partial x_i$. The resulting matrix over $\mathbb{Z}[t^{\pm 1}]$ is an Alexander matrix; suitable maximal minors recover the Alexander polynomial of a knot.

For a reduced knot diagram with two adjacent starred regions, a Kauffman state chooses one corner at every crossing so that every unstarred region contains exactly one chosen corner. Terms in a determinant expansion of a Dehn-presentation Alexander matrix correspond bijectively to these states.

The knot determinant is $|{\Delta }_K(-1)|=|\det (A+A^T)|$. It is also the order of the first homology of the two-fold cover of $S^3$ branched over $K$.