Integral Witt classes and rational Seifert isometric structures

Assume the full Axiom of Choice when an integral Seifert lattice comes from a knot's positive-push pairing. For an integral Seifert lattice $(H,V)$, its integral Witt class is its class in $G_{\mathbb{Z}}$ from the definition of metabolic Seifert form and algebraic knot concordance. An isomorphism $(W,B,T)\to (W^{\mathrm{\prime }},B^{\mathrm{\prime }},T^{\mathrm{\prime }})$ of rational isometric structures is a rational-linear bijection $P:W\to W^{\mathrm{\prime }}$ such that $B^{\mathrm{\prime }}(Px,Py)=B(x,y)$ and $PT=T^{\mathrm{\prime }}P$. A rational isometric structure is a triple $(W,B,T)$ in which $W$ is a finite-dimensional even-dimensional $\mathbb{Q}$-vector space, $B:W\times W\to \mathbb{Q}$ is a nonsingular symmetric bilinear form, $T:W\to W$ is an invertible linear map satisfying $B(Tx,Ty)=B(x,y)$ for all $x,y\in W$, and $I-T$ is invertible. Such a structure is metabolic if it has a $T$-invariant subspace $L\subset W$ with ${\dim }_{\mathbb{Q}}L=\frac{1}{2}{\dim }_{\mathbb{Q}}W$ and $B{\mathrm{\mid }}_{L\times L}=0$. Define the rational Witt group $G_{\mathbb{Q}}$ by direct sum of isomorphism classes, with metabolic structures set equal to zero. For a nonsingular integral Seifert matrix $V$, the associated matrix pair is $(B,T)=(V+V^T,V^{-1}{V}^T)$; whether this pair is a rational isometric structure is established separately.

The definition concerns finite-rank bilinear forms and isometry data. It does not identify the unrelated Witt Lie algebra with either $G_{\mathbb{Z}}$ or $G_{\mathbb{Q}}$.

The integral metabolic relation and algebraic concordance group

The integral Seifert matrix convention