Metabolic Seifert forms and algebraic knot concordance

Assume the full Axiom of Choice when a knot's Seifert pairing is formed. An integral Seifert lattice is a pair $(H,V)$ in which $H$ is a free abelian group of rank $2g$ for some $g\ge 0$, and $V:H\times H\to \mathbb{Z}$ is bilinear with $V-V^T$ unimodular. The rank-zero lattice is allowed, with the determinant of its empty skew matrix equal to $1$. A metabolizer is a primitive direct summand $M\subset H$ of rank $g$ such that $V(x,y)=0$ for every $x,y\in M$; the lattice is metabolic when it has a metabolizer. A chosen Seifert surface of an oriented knot gives such a lattice through the positive-push pairing. Its form is algebraically slice when that lattice is metabolic.

An isomorphism of integral Seifert lattices $(H,V)$ and $(H^{\mathrm{\prime }},V^{\mathrm{\prime }})$ is an integral group isomorphism $P:H\to H^{\mathrm{\prime }}$ satisfying $V^{\mathrm{\prime }}(Px,Py)=V(x,y)$ for all $x,y\in H$. Let $\mathcal{S}$ be the commutative monoid of these isomorphism classes under orthogonal direct sum, where $(H,V)\oplus (H^{\mathrm{\prime }},V^{\mathrm{\prime }})$ has underlying group $H\oplus H^{\mathrm{\prime }}$ and pairing $V\oplus V^{\mathrm{\prime }}$, with the rank-zero lattice as identity. The integral algebraic concordance group $G_{\mathbb{Z}}$ is the abelian group generated by symbols $[V]$ for $[V]\in \mathcal{S}$, subject to $[V\oplus W]=[V]+[W]$ and $[M]=0$ for every metabolic lattice $M$. Two selected Seifert forms are algebraically concordant when they represent the same element of $G_{\mathbb{Z}}$. This form relation does not assert the existence of a smooth or locally flat topological concordance annulus between the knots.

The positive-push linking form on the integral first homology of a chosen Seifert surface

The skew part of the Seifert form is the unimodular surface intersection form

Smooth concordance is an oriented proper smooth annulus

Topological concordance is an oriented proper locally flat annulus