Alexander polynomials from Seifert matrices — Examples and Counterexamples

Given: Full AC and the standard unknot $U$ in the oriented three-sphere.

Full AC is the choice-function axiom used by the Alexander-module and Seifert-matrix conventions. (The Axiom of Choice)

The standard unknot is the oriented unit circle in a fixed coordinate two-plane of $S^3\subset {\mathbb{R}}^4$. (Oriented connected sum and prime knot)

A Seifert surface is a connected compact smooth oriented embedded surface with the given knot as oriented boundary; a disk has Euler characteristic $1$ and genus $0$. (Oriented Seifert surfaces in the three-sphere)

A nonempty contractible space has the singular homology of a point. The singular chain complex of a point has ${\mathrm{\partial }}_1=0$ and ${\mathrm{\partial }}_2=1$, so $H_1(\ast ;\mathbb{Z})=0$. (Contractible nonempty spaces have the homology of a point)

The positive-push Seifert matrix is defined on an ordered basis of $H_1(F;\mathbb{Z})$, and the determinant of the empty $0\times 0$ matrix is $1$. (Seifert pairing and matrix of an oriented knot)

The positive-meridian module is presented by $V-t{V}^T$, the square presentation generates $E_0(A_K)$ by its determinant, and that determinant is associated to ${\mathrm{\Delta }}_K(t)$. (The one-variable Alexander module of an oriented link, A Seifert matrix presents the one-variable Alexander module, The square Seifert presentation generates the knot Alexander polynomial, The Seifert determinant is the knot Alexander polynomial)

For a knot, the separate knot-normalized fraction is $D_K(t)={\mathrm{\Delta }}_K(t)\mathrm{/}(1-t)$. (The Alexander polynomial from the zeroth elementary ideal)

The assignment $t\mapsto 1$ extends to a ring homomorphism ${\mathrm{ev}}_1:\mathrm{\Lambda }\to \mathbb{Z}$, since $1$ is invertible in the Laurent ring. (The Laurent polynomial ring as the principal localisation of Z[t] at t)

Connected-sum multiplicativity gives ${\mathrm{\Delta }}_{K\mathrm{\#}J}\doteq {\mathrm{\Delta }}_K{\mathrm{\Delta }}_J$. (The knot Alexander polynomial multiplies under connected sum)

Full AC implies the Axiom of Countable Choice, the stated hypothesis of the Seifert-surface definition. (AC implies DC implies countable choice, The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$))

Every generator ${\mathrm{\Delta }}_K$ satisfies ${\mathrm{\Delta }}_K(1)=\pm 1$. (Knot Alexander normalization, symmetry, and genus bound)

The Alexander polynomial is unchanged up to a unit under oriented ambient isotopy, and a transported Seifert matrix is related by unimodular congruence. (The Alexander polynomial is an oriented link invariant, Seifert matrices of ambient-isotopic pairs are S-equivalent)