An Alexander-one knot supplies good-group disk-embedding data

Use [F1] to fix the actual geometric trivial Alexander basis. The following integral shear preserves its short–long intersection convention and makes its long–long block zero.

The current general disk carrier demands vanishing mutual/self Wall numbers for its dual spheres, not merely the triangular disk–sphere matrix in the short GT application. The following integral operation removes a potential extra obstruction rather than assuming those two interfaces coincide.

Suppose a supplied geometric trivial Alexander basis has matrix V=[[0,I+U],[U^T,W]], with U strictly upper triangular and W symmetric. Put N(C)=CU+U^T C on symmetric integral g-by-g matrices. Left multiplication by U^T and right multiplication by U commute and each has gth power zero. The binomial expansion therefore gives N^(2g−1)=0: in every summand one of the two exponents is at least g. Hence I+N has the integral inverse sum_{r=0}^{2g−2}(−N)^r (the g=0 case is empty). Set C=−(I+N)^(−1)W. This is an explicit symmetric INTEGRAL matrix, since N preserves symmetry. The block change S=[[I,C],[0,I]] is symplectic: its new long basis vectors are ell'_j=ell_j+sum_i s_i C_ij, and their intersections are −C_ij+C_ji=0; short–long intersections remain the identity. Direct multiplication gives S^T V S = [[0,I+U],[U^T,W+C(I+U)+U^T C]] = [[0,I+U],[U^T,0]].

Thus the same triangular block admits an entirely zero long–long Seifert block. This does NOT assert minimal Seifert rank: U may be nonzero. The initial basis is supplied by [F1]. This particular shear has an explicit geometric realization, so no general surjectivity assertion about the mapping-class group is needed for THIS step. For any vector v among e_i and e_i+e_j, realize its short-class sum by an embedded circle: use s_i for e_i; for e_i+e_j use a band sum of parallel copies of s_i,s_j along a simple arc avoiding all short curves. Such an arc exists because cutting along all the short curves leaves a connected surface. If that cut were disconnected, the oriented boundary of one complementary component would give a nontrivial integral relation among the independent short classes, a contradiction. The band sum and its small annular neighborhood can be chosen disjoint from every original short curve. A Dehn twist in that annulus, written in annular coordinates as (theta,r) to (theta+2pi rho(r),r), is a homeomorphism fixed on the annulus boundary; choose rho constant zero/one near its two ends. Tracking an oriented transverse arc gives on homology the transvection x to x+(x intersection v)v, with sign reversed by the inverse twist. It fixes every short class and sends the long classes by the symmetric rank-one shear vv^T, with the sign chosen according to the intersection convention. Now E_ij+E_ji = (e_i+e_j)(e_i+e_j)^T−E_ii−E_jj.

Every symmetric integral C is a finite integer sum of these rank-one matrices. Compose the corresponding positive/negative twist powers. Each action fixes the short Lagrangian, so their shear matrices ADD even when their supporting annuli are not disjoint. The composition is an actual orientation-preserving boundary-fixed surface homeomorphism. Applying it to the entire original geometric basis preserves all its geometric intersection counts and realizes S. In the smooth category choose smooth annular cutoffs to obtain a diffeomorphism. Thus the zero-W basis is genuinely geometric and has exactly the same triangular U. The basis supplier [F1] supplies that initial input.

Apply the preceding shear to [F1]. We obtain a smooth geometric basis s_i,ell_i with V=[[0,I+U],[U^T,0]], not only the existential lattice statement. The following construction retains the ambient ball and all attaching data. It is stated before invoking any disk-embedding conclusion.

Use the literal ball of radius1.5 and its unit sphere. Place the original surface F in that unit sphere and add the exterior collar K times [1,1.5]. Round its junction with F, retaining the outer K product collar; call this proper smooth surface Sigma. Near F its normal coordinates are the radial coordinate x and the positive normal coordinate y of F in S3. Choose its open normal thickening there to be .9<x<1.1, |y|<epsilon, with the compatible product continuation near K. This is an ACTUAL smooth thickening, supplied by the radial product and the normal line of the oriented surface in S3. Put N=the complement of that open thickening, with corners rounded away from the marked circles. It is compact, connected, smooth, and the original ball/surface have not been replaced by an abstract four-manifold.

A disk-and-band handle decomposition of F has one disk and2g bands. Make its height increase from that disk across the bands to the boundary, and realize that height as a small radial displacement in the displayed collar. The complement has one0handle, one1handle from the surface minimum and2g2handles from its bands, with no higher handles. This local complement recipe follows by taking the product normal2disk to a surface0handle or1handle: its complement adds respectively the normal meridian1handle or the2handle whose core crosses the band. The product collars identify the attaching pieces; no arbitrary embedded4ball is identified with the standard ball. The resulting complement deformation retracts onto a2complex with one vertex, one1cell and2g2cells. Name the positive normal meridian of its one1handle a. The local2handle at an oriented surface band identifies its endpoint meridians by transport through the band: its attaching relation is m_1 w m_2^−1 w^−1. To see the signs, take the boundary of a normal2disk and transport it along the band core; the two boundary ends of the resulting normal-annulus strip have opposite induced boundary orientations. Both endpoint meridians come from the same original disk and represent a in the one-generator1skeleton. Any winding of a knotted band only changes the transport word w, which is a power of a there. Thus the attaching word is a a^k a^−1 a^−k=1, for every band. Consequently pi1(N)=Z, and the same2complex is homotopy equivalent to a circle wedge2g spheres. This is the actual group/attaching-word computation, not an inference from H1 alone.

Choose generic immersed disks Delta(s_i) in the inner unit ball with boundaries s_i-down. Run their boundary collars in the negative-y side of the marked normal strip, bending into the interior of N immediately off each boundary circle. Choose disks Delta(ell_i) bounding the long curves and trim their radial collars to x=.8; they then have boundaries ell_i at x=.8 and lie in the ball x<=.8. The short disks consist of disks in that inner ball and their mutually disjoint negative-y collars. Their union lies in a simply connected part of N: the inner ball together with those product collars retracts onto the inner ball. Thus all their equivariant intersection labels are1. Their ordinary intersections are the boundary linking numbers, which are V(s_i,s_j)=0. To justify this relative4-ball identity, compare one filling disk with a Seifert surface for the same boundary curve pushed into a thin boundary collar. Their difference is a two-cycle, null in the ball. Its intersection with the other proper disk is therefore zero, by the boundary-of-intersection identity for a bounding three-chain. The collar surface counts exactly the linking number on the boundary sphere, with the outward-normal-first orientation. This proves the identity for immersed fillings, rather than assuming disjoint embedded disk fillings. The boundary framings are the prescribed surface pushoffs. A generic filling disk with that zero longitude can be made framed: the parallel-intersection identity gives 0=2mu+e, so its relative normal Euler number e is even; an interior cusp changes e by2. Use finitely many cusps to make e0. The same identity then gives mu0. These changes are local in the inner ball and preserve the boundary circle/frame. Mutual intersection numbers are invariant under those rel-boundary homotopies. Accordingly lambda(Delta(s_i),Delta(s_j))=0 and mu(Delta(s_i))=0 in the precise integral group-ring convention, not merely after augmentation.

For each long curve construct the torus T_i=ell_i times the boundary of the normal rectangle [.8,1.2] times [−2epsilon,2epsilon]. The long curves are geometrically disjoint, so these tori are disjoint. They lie in N. A short disk meets T_i at the point of ell_i intersection s_j on its negative-y radial collar, hence with geometric signed count delta_ij. Remove an annular neighborhood of the long curve at x=.8,y=0 in T_i, and cap its two boundary circles by opposite oriented framed parallel copies of the trimmed Delta(ell_i). This is a smooth immersed sphere g_i in N. The long–long Seifert block is ZERO, so these long disks have zero boundary linking and can be chosen with the zero extending normal framing as above. The torus product framing matches both caps explicitly: along x=.8 the first common normal is the direction in F transverse to ell_i; the second rotates from the radial direction on the torus to the y-direction on each disk across the chosen rectangle corner. The opposite upper/lower cap orientations give opposite half-rotations, not an extra full twist. The long disk's extending zero frame supplies the remainder. Hence the untwisted attaching annuli give a trivial normal2plane bundle on g_i. Thus these are framed spheres, not tori called spheres without a framing check.

Fix orientations explicitly before the coefficient calculation. At a positive short–long intersection let u be the short tangent and v the long tangent, so (u,v) orients F. The ambient ball orientation is (x,y,u,v), with x outward radial and y the positive surface normal in S3. Orient the rectangle boundary counterclockwise in (x,y), and orient T_i by its rectangle tangent FIRST and its long tangent SECOND. At its x=.8 side the torus tangent basis is (−y,v), while the outward-radial collar of the short disk has basis(x,u); their ordered determinant relative to(x,y,u,v) is positive. Thus the old torus–short point contributes +delta_ij. Removing the annulus |y|<epsilon' at that side leaves its upper boundary oriented +ell_i and its lower boundary −ell_i. Consequently the surgery uses −Delta(ell_i) on the upper cap and +Delta(ell_i) on the lower cap.

Take t to be the counterclockwise meridian in the oriented (x,y) normal plane and base through the inner ball on the lower side. Traveling in the retained torus from the lower cap to the upper one goes around the rectangle's outer three sides counterclockwise; returning through the inner ball closes the positive meridian. Thus the upper cap path differs from the lower by t, with no arbitrary relabeling of the prescribed positive meridian. Use common basing paths through the inner ball. The old torus–short-disk point contributes delta_ij. Every intersection of the long disk with a short disk gives two sphere–short intersections of opposite signs. Their paths differ by precisely the short meridian, so lambda(g_i,Delta(s_j))=delta_ij+(1−t)lk(ell_i,s_j-down) =delta_ij+(1−t)(U^T)_ij.

For two DIFFERENT g spheres the tori and their annular surgery pieces contribute no points, because the long curves are geometrically disjoint. Each long-disk intersection contributes four points from the two cap parallels, with factors1,−t,−t^−1,1. Hence lambda(g_i,g_j)=(1−t)lk(ell_i,ell_j-down)(1−t^−1)=0.

For a self-double point the same parallel-sheet accounting gives two identity-label contributions and the negative t,t^−1 contributions, modulo the sheet-exchange relation. The long disk has framed relative Euler number0 and self-intersection sum0, so mu(g_i)=0. No extra intersections occur at the surgery seam: its two parallel attaching annuli are disjoint in the explicit product torus, with their normal framings already matched. This verifies the additional dual-sphere vanishings demanded by the ACTUAL local disk-data Definition, not just the weaker duality assertion quoted in AppendixA.

The displayed disk–sphere matrix is lower unitriangular. Set f_i=g_i−sum_(k<i) T_ik f_k, inductively, using whiskered parallel sphere copies and framed connected sums. For j<i the sum cancels T_ij; for j=i the earlier rows vanish and the diagonal stays1; for j>i both terms vanish. A whisker around t^r left-multiplies the first-variable coefficient by t^r in [F2], so each Laurent coefficient is realized with its stated sign. Its result has lambda(f_i,Delta(s_j))=delta_ij, and the zero lambda/mu conditions among the spheres persist by direct sesquilinearity and the self-sum formula (all cross terms vanish). Taking conjugate transpose gives lambda(Delta(s_i),f_j)=delta_ij in the original disk-data order. The output is therefore the ACTUAL good-group disk problem in this particular N, with exactly the short-circle boundaries and product normal framings needed to compress Sigma. Goodness of Z is [F3]; disjoint embedded replacement disks still require the separate tower/standardness proof. This construction makes no such replacement by definition.

The relative smooth filling step used above is justified without TOP smoothing. Approximate a filling map in the smooth ball, fix its supplied boundary collar by cutoff interpolation in the Euclidean embedding/retraction from [F5], and choose finite affine coordinate bumps which independently vary its first jets. Rank-one jets have codimension3>2, so parametric transversality gives an immersion. Independent bumps at distinct pairs make the double locus zero-dimensional; triple configurations have codimension8>6 and are absent. Compactness away from the uniform local-injectivity diagonal gives finitely many double points. Boundary collars are fixed throughout. The explicit cusp and sphere operations above are in those smooth charts.