 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: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,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.