Vassiliev invariants and chord diagrams

A degree-$n$ chord diagram is a pairing of $2n$ cyclically ordered endpoints, so only finitely many such diagrams exist. A 4T relation is determined by a background diagram and finitely many endpoint positions, and a 1T relation by one diagram; hence there are finitely many relation vectors. By [F1], each has integer coefficients, so they form the rows of a finite integer matrix $A$. Write $A_n(F):=F^{C_n}\mathrm{/}{\mathrm{rowspan}}_F(A)$ and $W_n(F):={\mathrm{Hom}}_F(A_n(F),F)$. Since $k$ has characteristic zero, $\mathbb{Q}$ embeds in $k$. The rank of $A$ is unchanged by this field extension: a nonzero maximal minor over $\mathbb{Q}$ remains nonzero in $k$, while every larger minor that vanishes over $\mathbb{Q}$ still vanishes in $k$. Thus the row space over $k$ is the scalar extension of the rational row space, and $A_n(k)\cong A_n(\mathbb{Q}){\otimes }_{\mathbb{Q}}k$.

The quotient is finite-dimensional, so extending a rational basis and its dual basis gives $W_n(k)\cong W_n(\mathbb{Q}){\otimes }_{\mathbb{Q}}k$. If $n=0$, the sole diagram is the empty diagram and any weight system is its value $c\in k$; the constant invariant $V(K)=c$ has top symbol $c$. For $n>0$, choose a finite $\mathbb{Q}$-basis ${\varphi }_1,\dots ,{\varphi }_d$ of $W_n(\mathbb{Q})$; finite-dimensional row reduction chooses this basis without a choice axiom. By [F2], for each $i$ there is a rational-valued finite-type invariant $V_i$ with top symbol ${\varphi }_i$. Write the given weight system as $W={\sum }_{i=1}^d{a}_i{\varphi }_i$ with $a_i\in k$, and define $V={\sum }_{i=1}^d{a}_i{V}_i$, viewing each rational value in $k$. Linearity of the finite-difference operator gives top symbol $W$. If $W_n(\mathbb{Q})=0$, the sum is empty and the zero invariant supplies the zero weight system.

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.

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 $D_{n}V(K)=0$.

For each degree-$n$ diagram $D$, choose an $n$-singular representative $K_D$ and set $w_V(D)=D_{n}V(K_D)$. By [F3] the value is independent of the representative and, in the framed case, of its zero-free framing class. Thus this defines a function on degree-$n$ diagrams. Extending on the free $R$-module by $R$-linearity defines a linear functional.

By [F3], $w_V(D_1)-w_V(D_2)+w_V(D_3)-w_V(D_4)=0$ for every 4T generator. Thus the functional annihilates the 4T submodule and factors through the framed quotient by the definition in [F2].

If $V$ is unframed, [F4] makes the functional vanish on every 1T generator as well, so it factors through the unframed quotient in [F2]. In the framed branch under ${\mathrm{A}\mathrm{C}}_{\omega }$, [F4] and [A1] give a type-at-most-one framed self-linking example with one-chord value $2$; since $2\mathrm{\ne }0$ in $\mathbb{Z}$, no universal framed 1T relation can be added.