An Alexander-one knot has a trivial Alexander basis

Full AC implies Countable Choice, so choose the actual connected smooth Seifert surface supplied by [F2]. The proved reduction [F3] splits its given lattice by symplectic pairs and the exact inverse tube enlargements of [F4].

Reversing these reductions also produces the claimed trivial Alexander block. Keep all short vectors first and all long vectors second. A reverse tube addition introduces a new short x with zero column and only the entry 1 in its own long column, and a new long y with zero diagonal. Its couplings to earlier short vectors occupy precisely the new LAST column of U, so U stays strictly upper triangular. Its couplings to earlier long vectors give a symmetric new row/column of W, with zero new diagonal. Starting from the empty matrix therefore gives [[0,I+U],[U^T,W]] with exactly the advertised conditions. This is an integral symplectic basis of the original lattice; the finite construction below realizes it geometrically rather than assuming that arbitrary integral basis changes are surface homeomorphisms.

Fix a standard geometric basis a_i,b_i on the actual compact oriented genus-g surface with one boundary circle. Its existence follows by cutting a finite disk-and-handle model of the surface, not from an arbitrary homology basis. Put omega(a_i,b_j)=delta_ij. An annular Dehn twist about an embedded curve v acts by the transvection T_v(z)=z+omega(z,v)v, choosing the twist sign accordingly. This formula is proved by cutting each transverse arc at the annulus: the twisted arc acquires one oriented copy of its core. Curves representing a_i+a_j, b_i+b_j and a_i+b_j for i≠j are embedded band sums on distinct handles. The joining arc lies in the complement of the other standard handles; the band gives the indicated homology class. Thus these transvections and their inverses are actual boundary-fixed surface homeomorphisms (smooth diffeomorphisms in the smooth model).

For orthogonal u,v, direct expansion gives T_(u+v) T_u^−1 T_v^−1(z) =z+omega(z,u)v+omega(z,v)u.

Taking u=a_i,v=b_j, i≠j, realizes the elementary block-GL operation a_j to a_j+a_i and b_i to b_i−b_j. Its powers realize arbitrary integral elementary additions. Handle permutations and simultaneous sign reversal of a_i,b_i are also realized; the latter is the square of the two-twist rotation on that handle. Euclidean elimination consequently realizes every block action (x,y) to (A x,A^−T y) for A in GL_g(Z). Likewise the a_i and a_i+a_j twists realize every symmetric upper shear, because the diagonal rank-one matrices and the identity (e_i+e_j)(e_i+e_j)^T−E_ii−E_jj=E_ij+E_ji generate all symmetric integral matrices; the b_i analogues realize every symmetric lower shear.

Here is a complete generation argument, so this is not an appeal to mapping-class surjectivity. Given a primitive vector v=(x,y), first interchange the a and b directions on a handle if necessary to make x nonzero. A block-GL operation makes x=(d,0,...,0), d>0. If some y_j is not divisible by d, reduce d strictly: for j=1 use the ordinary Euclidean two-twist SL_2 algorithm on (d,y_1); for j≠1 an upper diagonal shear inserts y_j in x_j, then block-GL Euclidean reduction replaces d by gcd(d,y_j). Repeat. The positive integer d decreases until every y_j is divisible by d; primitivity then forces d=1. A lower symmetric shear with first column −y clears y while keeping x=e_1. Thus an actual finite product of the specified twists takes every primitive vector to a_1.

Let A be an arbitrary symplectic integral matrix. Apply this procedure to its first column, so the resulting matrix fixes a_1. Its image of b_1 has the form b_1+c a_1+z, with z in the span W of the remaining handles. The displayed orthogonal-transvection combination for u=a_1 and each basis vector v of W fixes a_1. Its k-th power subtracts k v from that image of b_1 and only changes its a_1 coefficient in addition. Use these powers to clear all coordinates of z. Then a power of T_(a_1) clears c. The resulting matrix fixes both a_1,b_1, and hence restricts symplectically to W. Induct on g. We have reduced A to the identity by actual twists; reversing the finite list realizes A. In particular the symplectic lattice basis from the preceding primitive-vector/tube reduction is represented by an ACTUAL geometric basis, with disjoint short curves, disjoint long curves and exactly one transverse short–long intersection on each matching pair.