= Solution
Choose an oriented <Seifert surface> $F$ of <genus> $g$ and an integral basis $a_1,\ldots,a_{2g}$ of $H_1(F;\mathbb Z)$. Push $a_i$ in the positive normal direction to $F$. The <Seifert matrix> is
$$
\boxed{V_{ij}=\operatorname{lk}(a_i^+,a_j).}
$$
It represents the <Seifert form>. Its skew-symmetrization $V-V^T$ is the integral <intersection form> of $F$. The relation to the <Alexander polynomial of a knot> is
$$
\boxed{\Delta_K(t)\doteq\det(tV-V^T).}
$$
This follows from the presentation of the <Alexander module of a knot> obtained by cutting the <knot exterior> along $F$ and stacking the resulting copies; a proof is not needed here.
\b[The printed genus inequality is false.] The useful consequence of the determinant relation is the <Alexander breadth bound on Seifert genus>:
$$
\boxed{\operatorname{br}\Delta_K\leq2g_s(K).}
$$
Here the <breadth of a Laurent polynomial> is its largest exponent minus its smallest exponent, so it is unchanged by multiplying by $\pm t^m$. Indeed, for a minimal-<genus> <Seifert surface>, the matrix has size $2g_s(K)$ and each entry of $tV-V^T$ has degree at most one. Its nonzero <determinant> is an ordinary polynomial of degree at most $2g_s(K)$; its breadth is no larger than that degree. Equivalently, the highest exponent of a symmetrically normalized <Alexander polynomial of a knot> is at most $g_s(K)$. An unnormalized degree is not invariant under Laurent units.
For a counterexample to the printed inequality and the requested example, take the \b[untwisted Whitehead double of a trefoil knot]. The picture specifies the two-strand <tangle> in the zero <Seifert framing>; the two exterior caps form a <Whitehead double> clasp. The inset identifies the <trefoil knot> used as companion.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2012/iii/paper-16-whitehead-double.png]
{title=Untwisted Whitehead double: zero-framed trefoil tangle and Whitehead clasp}
Its usual genus-one <Seifert surface> is a zero-framed annulus following the <trefoil knot>, joined by the clasp band. Choose the annulus core and a curve traversing the clasp band as the basis of its first <homology>. Zero annulus twisting gives the first self-<linking number> $0$. The clasp contributes a <Hopf band> with self-<linking number> $-1$ in the positive-clasp convention used here. Plumbing the bands contributes one linking in one push-off direction and none in the other. Orienting the basis suitably gives
$$
V=\begin{pmatrix}0&1\\0&-1\end{pmatrix},\qquad
tV-V^T=\begin{pmatrix}0&t\\-1&1-t\end{pmatrix}.
$$
Tying the annulus into the <trefoil knot> changes neither these local linking counts nor its prescribed zero <Seifert framing>. Therefore
$$
\boxed{\Delta_K(t)\doteq t\doteq1.}
$$
This <Whitehead double> is not the <unknot>, as allowed without proof in the question. Its <Seifert genus> is consequently at least one and at most one, hence exactly one. Thus $g_s(K)=1$ while every normalized constant <Alexander polynomial of a knot> has degree zero: a direct counterexample to the printed inequality.
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