= Metabolic Seifert form
An integral <Seifert form> on a <free abelian group> $L$ of rank $2g$ is metabolic if there is a rank-$g$ <direct summand> $M\subset L$ on which the entire <bilinear form> vanishes:
$$
\theta(x,y)=0\qquad(x,y\in M).
$$
Over a <field>, replace the <direct summand> by a $g$-dimensional <vector subspace>. Such a half-rank <vector subspace> is a <metabolizer>. In a <basis> starting with a <basis> of the <metabolizer>, the <Seifert matrix> has a zero upper-left $g$-by-$g$ block.
The knot-theoretic nonsingularity condition concerns the <matrix> $V-V^T$, which defines a <unimodular intersection pairing>; $V$ itself need not be an <invertible matrix>. For example,
$$
V=\begin{pmatrix}0&1\\0&0\end{pmatrix}
$$
has a <metabolizer> spanned by the first <basis vector> and has $\det(V-V^T)=1$, despite $\det V=0$. Requiring $V$ itself to be nonsingular would exclude this legitimate metabolic example. <Algebraic concordance of knots> identifies two forms when their difference is metabolic.
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