 Isometric structure

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.