Algebraic concordance of knots

Two Seifert forms are algebraically concordant when their difference is metabolic. Stable equivalence classes form the algebraic concordance group.

A nonsingular Seifert form on a $2g$-dimensional space is metabolic when it vanishes on a $g$-dimensional subspace called a metabolizer.

The algebraic concordance group over a field $F$ is the Witt group of nonsingular Seifert forms over $F$, modulo metabolic forms.

An isometric structure consists of a finite-dimensional vector space $V$, a nonsingular symmetric bilinear form $Q$, and a $Q$-isometry $T$ with the required nondegeneracy at $\pm 1$. Metabolic isometric structures are quotiented out to form ${\mathcal{W}}_F$.

The Witt group of isometric structures identifies two isometric structures when their orthogonal difference is metabolic.

For an irreducible symmetric Laurent polynomial $\delta$, the primary component isfor sufficiently large $N$. Distinct symmetric primary components are orthogonal, so restriction defines a projection ${\mathcal{W}}_F\to {\mathcal{W}}_F^{\delta }$.

Extending an algebraic-concordance class from $\mathbb{Q}$ to a p-adic field ${\mathbb{Q}}_p$ detects torsion invisible over the real numbers. For $p\equiv 3(\mathrm{mod}4)$, an odd-dimensional second-residue form generates the order-four part of the local Witt group.