Alexander polynomial

Builds on Knot Seifert surface Determinant

Definition

Let $K \subset S^3$ be an oriented tame knot, and let $F$ be an oriented Seifert surface of $K$ of genus $g$ . Let $a_1, \ldots, a_{2g}$ be oriented simple closed curves on $F$ whose classes form a basis of $H_1(F; \Z)$ . For a curve $a$ on $F$ , let $a^+$ be the curve $a$ pushed off $F$ slightly in the positive normal direction. The Seifert matrix is the $2g \times 2g$ integer matrix $V$ with entries $V_{ij} = \operatorname{lk}(a_i, a_j^+)$ , where $\operatorname{lk}$ is the linking number. The Alexander polynomial of $K$ is $\Delta_K(t) = \det\big(V - t V^{T}\big) \in \Z[t, t^{-1}],$ taken up to multiplication by a unit $\pm t^{k}$ with $k \in \Z$ .

Remarks

Different Seifert surfaces and bases change $\Delta_K$ only by such a unit. The usual normalization, due to Conway, is the representative with $\Delta_K(t^{-1}) = \Delta_K(t)$ and $\Delta_K(1) = 1$ . Example: the trefoil knot has a genus-one Seifert surface with $V = \begin{pmatrix} -1 & 1 \\ 0 & -1 \end{pmatrix}$ . Then $\det(V - tV^T) = (t - 1)^2 + t = t^2 - t + 1$ , normalized to $t - 1 + t^{-1}$ . The unknot bounds a disk with $g = 0$ , so its polynomial is the empty determinant $1$ . Non-example of detection: the Conway knot is knotted but also has $\Delta = 1$ , so $\Delta_K = 1$ does not imply that $K$ is the unknot. The figure-eight knot has $\Delta = -t + 3 - t^{-1}$ . The degree span of $\Delta_K$ is at most $2g(K)$ .

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Determinant concept
Knot concept
Seifert surface concept
Alexander polynomial definition

Tag	`WKJ4EP6E1X`
URI	`m2://WKJ4EP6E1X`
Kind	definition
Writing	W0 stub
MSC	57K14, 57K10
Defines	Alexander polynomial
Uses	Knot, Seifert surface, Determinant
Changed	2 November 2026

Alexander polynomial

Definition

Remarks

Reading path

Further reading

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