The one-term relation and framing

Let $K$ be an $n$-singular knot with an isolated chord corresponding to the double point $p$. Its two preimages split the parameter circle into arcs $A$ and $B$. Isolation means no other marked double point has one preimage on each of $A$ and $B$. Choose a regular projection. There are finitely many ordinary crossings between the arcs. Switch them one at a time until every crossing has the $A$-branch over the $B$-branch. Each switch changes $D_{n}V$ by, up to sign, the $(n+1)$-fold difference on the knot with that ordinary crossing newly marked; this is zero by [F1]. Thus these switches preserve $D_{n}V(K)$.

For the framed branch, let $V$ be integer-valued self-linking. On a regular framed diagram compare a unit representative of its framing section $f$ with the unit blackboard normal section $b$. Their oriented fibrewise angle defines a map $S^1\to S^1$; let $\tau (f,b)\in \mathbb{Z}$ be its degree. In the tubular coordinates determined by $(b,Jb)$, the push-off along $f$ winds $\tau (f,b)$ times meridionally relative to the blackboard push-off. A positive meridian links the core once, since its oriented normal disk meets the core once with the sign fixed by the positive frame $(T,b,Jb)$. Thus $\mathrm{sl}(K,f)=\mathrm{sl}(K,b)+\tau (f,b)$; [F9] supplies the unit-twist normalization, and this local orientation calculation fixes its sign. Equivalently, each positive turn creates a local clasp with two positive mixed crossings, and [F7]–[F8] give the same linking increment. By [F6], $\mathrm{sl}(K,b)=\mathrm{wr}(D)$. For a framed singular knot with two disjoint singularities, all changes outside their local balls are fixed. Inside each ball, resolving a double point changes only that crossing's writhe contribution and local angle path; resolving a framing zero changes only the local angle path. Winding degrees add under concatenation, so the four self-linking values have the form $C+c_1{ϵ}_1+c_2{ϵ}_2$ for constants independent of $({ϵ}_1,{ϵ}_2)$. Their alternating second difference is $(C+c_1+c_2)-(C+c_1-c_2)-(C-c_1+c_2)+(C-c_1-c_2)=0$. This covers any pair of double-point or framing-zero singularities, so self-linking has type at most $1$. On a one-chord diagram with blackboard framing, the two crossing resolutions have writhes $+1$ and $-1$ and zero relative twist; their derivative is $1-(-1)=2$ by [F3] and [F6]. This is nonzero over $\mathbb{Z}$, so no universal framed 1T vanishing law is valid. The ${\mathrm{A}\mathrm{C}}_{\omega }$ assumption enters only through [F4]–[F9] and [A1].

The consistent over-under assignment lets the lifts of the interiors of $A$ and $B$ lie in opposite open half-spaces bounded by a plane through $p$. Cap each arc on its own side by a short boundary-parallel arc to obtain oriented singular knots $A^{\circ }$ and $B^{\circ }$ carrying the remaining marked double points. The two resolutions at $p$ are the connected sums $A^{\circ }\mathrm{\#}{B}^{\circ }$ and $B^{\circ }\mathrm{\#}{A}^{\circ }$. An orientation-preserving half-turn of the round $S^3$ that exchanges the two balls of the connected-sum decomposition carries the first knot, with its orientation, to the second; the path of rotations through angles from $0$ to $\pi$ is an ambient isotopy. Hence the two resolutions are ambient-isotopic, so their $V$-values agree and $D_{n}V(K)=0$.