Finite standard-skew Seifert matrices have explicit disk-band realizations

We first describe the finite linking calculation used in the construction. For disjoint oriented polygonal circles with a specified regular plane projection, the linking number is half the signed mixed-crossing sum. This can be proved with finite chains as follows. For the second curve $D$, extend each projected edge vertically downward from its actual height to a plane lying below both curves, and fill its resulting planar projected cycle by the finitely many oriented cone triangles from a plane point. The resulting finite two-chain has boundary $D$. Its intersections with the first curve $C$ occur exactly at the mixed crossings where $C$ is below $D$, with the usual oriented crossing signs. It therefore computes linking as the sum of that subset of signs. The full mixed sum is twice this sum: the difference between the sums where $C$ is below and above $D$ is the algebraic plane intersection of the two projected cycles, which is zero. To check the latter directly, fill one projected cycle by its finite planar cone chain and intersect the other with that chain; the signed endpoints of the resulting finite one-chain cancel. Independence of a finite spanning chain is equally finite: the difference of two such chains is a two-cycle in ${\mathbb{R}}^3$, bounded by its cone three-chain. Intersecting that three-chain with the closed curve gives a finite one-chain whose signed boundary sum is zero, so the two intersection numbers agree. Finite subdivisions and perturbations needed to avoid vertices or nontransverse affine intersections avoid only finitely many bad affine conditions; a rational point outside their union exists, for example by trying rational points in a fixed enumeration. Thus no transversality selection on an infinite family is involved. The same computation applies after small local smoothing or to parallel pushed curves, because these changes take place in disjoint boxes without introducing intersections between the two curves.

If $g=0$, use an oriented disk. Otherwise start with an oriented disk and $2g$ labeled orientable rectangular bands. In the cyclic order of their attachment intervals on the disk, put the endpoints of each pair of bands $(2j-1,2j)$ in the alternating order $2j-1,2j,2j-1,2j$, and put the pairs in disjoint successive sectors. Attaching the first band of a pair splits the current boundary into two components; attaching the second joins those two, as direct traversal of the four alternating attachment intervals shows. Each pair therefore preserves one boundary component and raises genus by one. The finite relative cell calculation for the disk and bands has one degree-one generator per band and no relative degree-two cells: collapse the disk, and collapse each band across its width. The relative degree-one boundary is zero because both band ends lie in the connected disk. The pair sequence, with the disk contractible and the degree-zero inclusion an isomorphism, therefore identifies $H_1(F;\mathbb{Z})$ with ${\mathbb{Z}}^{2g}$. The band cores completed by disk arcs map to its coordinate basis. Choose the disk arcs so that only paired arcs meet, once in their sector. The embedded bands can be routed through a finite regular ribbon diagram: give each band a private twist box and, for every unordered pair, give those two bands one private common box with two parallel strands. Connect these finitely many specified visits by polygonal plane arcs, avoiding the finite existing vertices and arranging transverse double crossings without triple crossings by finitely many rational perturbations. At every other diagram crossing assign one strand above the other, then lift it in a small crossing box. Distinct crossing boxes are disjoint; sufficiently narrow ribbons following these finitely many arcs, and the specified orientable attachments, give an embedded disk-band surface. Arbitrary incidental diagram crossings merely determine its initial matrix and cause no restriction.

Let $W_0$ be the positive-push matrix of this initial surface, computed by step 1.1. Its skew part has zero entries for core pairs whose disk arcs are disjoint. Indeed the annulus sweeping a core from its negative to its positive push misses the other core in that case. In the explicit normal ribbons, sliding both members of the pair $(a_i^{-},a_j)$ a small equal distance in the positive normal direction identifies its linking with that of $(a_i,a_j^{+})$: their normal heights stay distinct throughout, including over a disk crossing. Linking is symmetric by step 1.1, so the signed-intersection calculation gives equality of the two ordered entries. For a paired pair the sweep meets the other core once in the disk, with sign $+1$ or $-1$, so the difference of its two ordered entries is that sign. In local disk coordinates the sweep is the product of the core arc with a normal interval; its single transverse intersection has multiplicity one. Orient the second core of each pair so that this difference is $+1$. Thus $W_0-W_0^T$ is exactly the displayed standard skew matrix. This calculation is entirely within the given disk and finite ribbon chart; it does not assume a surface-classification or duality theorem. Consequently $W-W_0$ is symmetric.

In the private box of band $i$, insert one full ribbon twist with sign chosen so that the two mixed crossings between its core and its positive parallel have signed sum $+2$. Step 1.1 says this increases $(W_0{)}_{ii}$ by one. The opposite twist decreases it by one. Every other core and its push miss that box, so its pairings with this modification are unchanged, as are all other matrix entries. Insert the finite integer number of twists required for each $i$ to make the diagonal equal that of $W$. Full twists preserve orientability and the attachment intervals, hence the abstract disk-band surface and its one boundary component.

In the private common box of bands $i<j$, replace the two parallel ribbon strands by the two-strand full braid whose two mixed crossings have sign $\epsilon \in \{+1,-1\}$. Give each ribbon its diagram framing, with no individual full twist. The two crossings between the positive parallel of the first core and the second core have signed sum $2\epsilon$, so step 1.1 changes entry $(i,j)$ by $\epsilon$. The same two crossings between the positive parallel of the second and the first give the same change in entry $(j,i)$. Each ribbon has no self crossing or extra rotation of its framing in this box, so its diagonal entry is unchanged. All other cores and parallels miss the box. Thus this modification changes exactly those two symmetric entries. Since the remaining difference from $W$ is symmetric with zero diagonal by steps 3.1–4.1, insert the prescribed finite number of positive or negative full braids in each pair box. None affects another pair box or previously corrected entries. The resulting surface has matrix exactly $W$.

All polygonal bends, band attachments and diagram height changes in this description lie in finitely many disjoint small coordinate boxes. Replace each polygonal bend by a smooth arc in its plane, agreeing with its incident straight pieces outside the box; a fixed smooth cutoff flat at the endpoints supplies this replacement. Along a band use the same interpolation for its two edges and framing. At a disk attachment the construction is the product of its attachment interval with a rounded planar corner, so it too has an explicit local smooth replacement. Choose the smoothing boxes and ribbon widths smaller than the minimum of the finitely many positive separations between nonincident pieces. The replacements remain embedded, retain the band parametrizations and preserve every chosen crossing sign and push-off separation. They therefore preserve the exact linking computations in steps 3.1–5.1. The resulting surface is smooth and oriented with one smooth boundary component, which is a knot in $S^3$. The band collapse still provides the displayed basis, and its positive normal is the actual oriented smooth normal.

Every list in the construction has length bounded in terms of the finite matrix entries: the bands, visits, vertices, crossings, twist counts and pair-clasp counts are finite. Rational perturbations exclude finitely many affine conditions at each of finitely many stages, and local smoothing uses an explicit cutoff and a finite minimum. No arbitrary sequence of embeddings, bases, surfaces, generic projections or normal fibers is selected. The final surface and ordered basis realize every entry of the supplied $W$ with the positive push convention in the Statement. This proves the claimed finite, choice-free realization.