The four-term relation from singular Reidemeister III

To realize a degree-$n$ diagram, take its finite cyclic list of chord endpoints, identify each paired pair as a four-valent vertex, and join consecutive endpoints by arcs in $S^3$. Choose disjoint small vertex balls first. Place the finitely many intervening arcs successively in general position relative to those already placed; two one-dimensional arcs in a three-manifold can be separated by a small endpoint-fixed perturbation, so there are no unintended intersections. Smoothing the joins gives exactly the prescribed double points. In the framed case choose a nonzero normal section along the regular arcs and the two branches at every vertex. To close the section around the parameter circle, cut at one regular parameter; the two endpoint unit normals lie in an oriented two-plane and can be joined because its unit circle is connected. Smooth the finitely many joins. This constructs a zero-free framed representative. The framed branch uses [A1] for the normal-bundle structures.

Let $K,K^{\mathrm{\prime }}$ be representatives of the same diagram. Identify their finite marked graphs and fix matching local models in disjoint vertex balls. Join corresponding edge arcs by a homotopy relative to those balls, and homotope the normal sections over the edge strips, allowing zeroes during the homotopy. A finite relative perturbation makes the coincidence equations for two distinct curve parameters transverse: they are three scalar equations in the two curve parameters and time, so their solutions are isolated. It also makes the normal-section zero equation transverse: it is two scalar equations in the curve parameter and time, so its solutions are isolated. The parameter domains can be made compact after deleting a neighborhood of the diagonal, where each curve is locally embedded; the fixed vertex neighborhoods contain no new events. The higher-codimension loci for tangencies, triple points, degenerate section zeroes, or simultaneous events are avoided by the same finite perturbation. Hence only finitely many isolated double-point or framing-zero events occur. Passing any one event changes $D_{n}V$ by, up to sign, $D_{n+1}V$ on the object with that extra singularity, which vanishes by type at most $n$ in [F1]. Therefore $D_{n}V(K)=D_{n}V(K^{\mathrm{\prime }})$. These are finite-dimensional perturbations of a compact one-parameter family; the framed branch additionally uses [A1], and no countable choice is used in the unframed argument.

For a 4T generator choose the four local singular representatives with the same outside $(n-2)$ marked double points and, in the framed case, the same zero-free framing outside the move ball. Resolving the local marked double point with the positive-minus-negative convention gives $w_V(D_1)=a-b$, $w_V(D_2)=c-d$, $w_V(D_3)=c-a$, and $w_V(D_4)=d-b$, where $a,b,c,d$ are the four values of $D_{n-1}V$ on the remaining local configurations. The repeated configurations are identified by the oriented Reidemeister III isotopies in [F3]: $a$ occurs in the positive resolution of $D_1$ and negative resolution of $D_3$, $b$ in the negative resolutions of $D_1,D_4$, $c$ in the positive resolutions of $D_2,D_3$, and $d$ in the negative resolution of $D_2$ and positive resolution of $D_4$. Their resolution signs follow [F4]. Therefore $w_V(D_1)-w_V(D_2)+w_V(D_3)-w_V(D_4)=(a-b)-(c-d)+(c-a)-(d-b)=0$. The same local calculation applies to zero-free framed representatives because the framing outside the move ball is unchanged. This local calculation is choice-free once the framed objects are defined.