S-equivalence of integral Seifert forms

Let $V$ be an integral Seifert matrix in the positive-push convention $V_{ij}=\mathrm{lk}(a_i^{+},a_j)$. Two integral Seifert forms are S-equivalent when they are related by a finite sequence of integral unimodular basis changes and algebraic tube additions or their inverses. A basis change replaces $V$ by $P^{T}VP$ for $P\in {\mathrm{GL}}_m(\mathbb{Z})$. A tube addition replaces $V$ by $\left(\begin{array}{ccc}V& 0& \rho \\ 0& 0& 1\\ {\rho }^T& 0& 0\end{array}\right),\rho \in {\mathbb{Z}}^m,$ where the two new basis elements are the short and long curves of the added tube. The empty Seifert form is the unique bilinear form on the zero module, represented by the $0\times 0$ matrix.

The positive-push convention and integral Seifert matrices

Stavros Garoufalidis and Peter Teichner, “On Knots with Trivial Alexander Polynomial,” §3, Definition 3.2 and the tube-addition matrix, https://people.math.gatech.edu/~stavros/publications/teichner1.pdf

H. F. Trotter, “On S-equivalence of Seifert Matrices,” Inventiones Mathematicae 20 (1973), 173–207, https://gdz.sub.uni-goettingen.de/download/pdf/PPN356556735_0020/LOG_0020.pdf