Every integral Seifert form is realized by a knot

If $g=0$, $V$ is the empty matrix and a disk with its empty homology basis realizes it. Assume $g>0$. Put $J=V-V^T$ and $H={\mathbb{Z}}^{2g}$. Choose a primitive vector $e\in H$. Since $J:H\to \mathrm{Hom}(H,\mathbb{Z})$ is an isomorphism, the functional $J(e,-)$ is primitive, so Bezout's identity gives $f\in H$ with $J(e,f)=1$. For every $x\in H$, the vector $x^{\mathrm{\prime }}=x-J(x,f)e+J(x,e)f$ satisfies $J(x^{\mathrm{\prime }},e)=J(x^{\mathrm{\prime }},f)=0$. Thus $H=⟨e,f⟩\oplus ⟨e,f{⟩}^{\perp }$, and the restricted alternating form on the second summand is unimodular. Induction gives a basis with matrix $J_0={⨁}_{k=1}^g\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$. Let $P\in {\mathrm{G}\mathrm{L}}_{2g}(\mathbb{Z})$ be the corresponding change of basis and put $W=P^{T}VP$; then $W-W^T=J_0$.

To apply [F2] to any disjoint oriented smooth core/push-off pair $J,K$, first use [F4] to obtain a spanning surface $\mathrm{\Sigma }$ for $K$. Apply [F5] to the inclusion of $\mathrm{\Sigma }$ with closed target $J$. Its boundary $K$ is disjoint from $J$, so boundary transversality is vacuous; the isotopy fixes a neighbourhood of $K$ and yields an embedded oriented spanning surface transverse to the unchanged curve $J$. These auxiliary surfaces do not modify the band surface. All constructed curves lie in ${\mathbb{R}}^3\subset S^3$; [F6] supplies a regular diagram of each pair. Thus every hypothesis of [F2] is met, both before and after a local band modification, and the crossing formula computes the topological linking entries in [F3].

Start with a disk and attach $2g$ oriented bands in the standard one-boundary handle pattern. Their core curves form an integral basis with intersection matrix $J_0$. The result is a connected compact smooth oriented surface with one boundary component. Write $W_0$ for its positive-push matrix; by [F1], $W_0-W_0^T=J_0$.

For each band core, inserting a full twist in a small subarc adds two same-sign mixed crossings between that core and its positive push-off, changing its self-linking by $+1$ or $-1$ by step 1.2. The twist box is disjoint from the other core arcs, so the other entries and the surface intersection matrix are unchanged. Perform finitely many such twists to make every diagonal entry of $W_0$ equal the corresponding diagonal entry of $W$.

For each pair $i<j$, choose subarcs of bands $i$ and $j$ in a ball disjoint from every other core and every earlier modification. Insert the oriented local clasp whose two new mixed crossings between $a_i^{+}$ and $a_j$ have the same sign $\epsilon \in \{+1,-1\}$. By step 1.2, their signed sum changes by $2\epsilon$, so $W_{ij}$ changes by $\epsilon$. The clasp preserves the abstract surface intersection form; by [F1], $W_{ij}-W_{ji}$ is unchanged, and therefore $W_{ji}$ also changes by $\epsilon$. Choose the bands' framings through the clasp without adding individual full twists; each band alone is unchanged up to framed isotopy. Thus all diagonal entries stay fixed, and support away from the other cores leaves all other pair entries fixed. The target equation $W-W_0=(W-W_0{)}^T$ off the diagonal holds because both skew parts are $J_0$. A finite number of positive or negative clasps thus sets both entries to their target values for each pair, without changing previously treated pairs. The resulting surface has positive-push matrix exactly $W$.

The basis change $P^{-1}\in {\mathrm{G}\mathrm{L}}_{2g}(\mathbb{Z})$ on this surface changes its matrix from $W$ to $(P^{-1}{)}^{T}W{P}^{-1}=V$. The constructed surface has one boundary component, so its boundary is an oriented knot and the matrix is a Seifert matrix for that knot.