Algebraic concordance of knots

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

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.

A metabolizer for an integral Seifert form $\theta$ on a free abelian group $L$ of rank $2g$ is a rank-$g$ direct summand $M\subseteq L$ on which the form vanishes:This is the data witnessing a metabolic Seifert form. Over a field, the corresponding definition uses a half-dimensional vector subspace. The same term also denotes a half-dimensional isotropic subspace for a metabolic bilinear form.

For $L={\mathbb{Z}}^2$ with Seifert matrix $V=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)$, the subgroup spanned by the first basis vector is a metabolizer. Its self-pairing is zero and it has the complementary summand spanned by the second basis vector. The subgroup generated by twice the first basis vector is also isotropic and has rank one, but is not a direct summand: its quotient in $L$ has torsion.

Thus half rank alone does not suffice over the integers. The direct-summand condition ensures that a basis of the metabolizer extends to a basis of the whole lattice, exposing a zero $g$-by-$g$ block in the Seifert matrix.

Let $F$ be a field of characteristic different from two. In the convention corresponding to isometric structures with no eigenvalues $\pm 1$, use finite even-sized matrices $V$ for which both $V-V^T$ and $V+V^T$ are nonsingular. Their Witt group under direct sum, with metabolic Seifert forms set to zero, is the algebraic concordance group over $F$. Thus $V$ and $W$ define the same class when $V\oplus (-W)$ is metabolic.^[ref]

Invertibility of $V$ itself is a separate condition: in this Witt classification one can choose invertible representatives before converting to isometric structures. For $F=\mathbb{Q}$, integral Seifert matrices of knots satisfy the symmetric and skew nondegeneracy conditions, and define rational algebraic-concordance classes. Extension to $\mathbb{R}$ detects signature information, while extension to p-adic fields can retain torsion subgroup information invisible over $\mathbb{R}$.

Over a field $F$ of characteristic different from two, an isometric structure in the knot-theoretic convention is a triple $(V,Q,T)$: $V$ is a finite-dimensional vector space, $Q$ is a nonsingular symmetric bilinear form, and $T$ is a linear automorphism satisfyingThus $T$ is a $Q$-isometry, with neither $1$ nor $-1$ as an eigenvalue. The structure is metabolic if $V$ has a half-dimensional $T$-invariant vector subspace on which $Q$ vanishes. The invariance requirement is part of this definition: an arbitrary isotropic subspace for $Q$ alone does not suffice.

For an invertible Seifert matrix $A$ with $A\pm A^T$ nonsingular, the associated matrices areDirect multiplication verifies $T^{T}QT=Q$, while $\det (T\pm I)=\det (A{)}^{-1}\det (A^T\pm A)$ verifies the excluded eigenvalues. Conversely $A=Q(I+T{)}^{-1}$ recovers the Seifert form in this convention. Quotienting metabolic structures gives the Witt group of isometric structures.

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.