The $\mathfrak{sl}(2)$ weight system at $c=-3/32$ extends to a graph 4-invariant

Note. The earlier versions and their checking programs are archived on Zenodo under the concept DOI 10.5281/zenodo.23312529 (versions 10.5281/zenodo.23312664, 10.5281/zenodo.23384025, 10.5281/zenodo.23541142 and 10.5281/zenodo.23630528). The source of this revision is in the repository https://github.com/creelie/Paper_7, release v1.0.1, archived on Zenodo as 10.5281/zenodo.23658136 (concept DOI 10.5281/zenodo.23658071).

Introduction

A weight system is a function on chord diagrams that satisfies the four-term relations. Weight systems are the symbols of finite type knot invariants [1, 2, 3].³ A Lie algebra with an invariant form gives a weight system with values in the centre of its universal enveloping algebra. For $\mathfrak{sl}(2)$ with the Killing form the centre is the polynomial ring $\mathbb{Q}[c]$ in the Casimir element $c$, and Chmutov and Lando proved that the value $w_D \in\mathbb{Q}[c]$ of this weight system on a chord diagram $D$ depends only on the intersection graph $\Gamma(D)$ of $D$ [4]. Lando’s four-term relations for graphs [5] are such that every function $f$ on graphs satisfying them gives a weight system $D \mapsto f(\Gamma(D))$; such a function is called a 4-invariant. Lando asked whether $w$ comes from a 4-invariant [4]. For graphs with at most 8 vertices this is true [7]; see [6] for a survey.

Fomichev, Karev, Pavutnitskiy and Usanov [8] answered the question in the negative. They found a linear combination of intersection graphs that vanishes modulo the four-term relations of graphs and on which $w$ takes the value $c\left(c-\frac{3}{8}\right)(c-1)\left(c+\frac{3}{32}\right)$. So the specialisation of $w$ at a value $c=c_0$ can extend to a 4-invariant only if $c_0$ is one of $0$, $\frac{3}{8}$, $1$ and $-\frac{3}{32}$. The first three values are the eigenvalues of the Casimir element on the irreducible representations of dimension 1, 2 and 3, and for them extensions are known [8, 9, 10].⁴ The value $-\frac{3}{32}$ is the eigenvalue of the Casimir element on the oscillator representation, the representation of $\mathfrak{sl}(2)$ on polynomials in one variable by second order differential operators. The authors of [8] built an extension for graphs with at most 10 vertices and stated the following conjecture.

We prove Conjecture 1.1, with an explicit formula for $\xi$. All graphs are finite and simple. For $N \in\mathbb{N}$ write $[N]=\{1,\ldots,N\}$ and $\binom{[N]}{2}$ for the set of its 2-element subsets.

(c) For every chord diagram $D$, $w_D\left(-\frac{3}{32}\right)=\xi(\Gamma(D))$.

Parts (b) and (c) are Conjecture 1.1. The two expressions for $\xi$ agree by (a), because $\binom{-2}{k}=(-1)^k(k+1)$.

Example 1.4. For one vertex, $S_{K_1}(N)=\binom{N}{2}$ and $\xi(K_1)=3/(-32)=-\frac{3}{32}$, the value of $w$ on one chord. Since $S_{G\sqcup H}=S_GS_H$, $\xi$ is multiplicative, as $w$ is. For two adjacent vertices, the pairs $(p(x),p(y))$ with $|p(x)\cap p(y)|=2,1,0$ number $\binom{N}{2}$, $\binom{N}{2}\cdot2(N-2)$ and $\binom{N}{2}\binom{N-2}{2}$, so $S_{K_2}(N)=\binom{N}{2}(1-2(N-2)+\binom{N-2}{2})$ and $\xi(K_2)=3\cdot19/1024=\frac{57}{1024}$. This is $c(c-\frac{1}{2})$ at $c=-\frac{3}{32}$, the value of $w$ on two crossing chords.⁵

The proof of (b) is an involution on assignments (§ 6). For (c) we let the element $C_D\in U(\mathfrak{sl}(2))$ of a chord diagram act on the Verma module of highest weight $\mu$; it acts by a polynomial $\lambda_D(\mu)$, and $w_D(-\frac{3}{32})=\lambda_D(-\frac{1}{2})$ (§ 2). The oscillator representation on $\mathbb{Q}[z]$ is the sum of the Verma modules of highest weights $-\frac{1}{2}$ and $-\frac{3}{2}$, so the image of $C_D$ in the Weyl algebra is the scalar $\lambda_D(-\frac{1}{2})$. A supertrace on the Weyl algebra with a Wick formula writes this scalar as a sum over perfect matchings, in which every cycle contributes a factor $-2$ (§ 3 and § 4). With $N$ in place of $-2$ the same sum is a trace in the Clifford algebra on $N$ generators, and that trace equals $4^nS_\Gamma(N)$ (§ 5). Figure 2 shows the two halves of the argument.⁶

The combinatorial objects of the proof, chord diagrams with their legs, perfect matchings, their cycle and crossing counts, are finite and the proof uses nothing beyond linear algebra over $\mathbb{Q}$ and counting. Related constructions in which a finite complex or a group action is studied through counts of this kind appear in the authors’ earlier work [13, 14, 15].

The weight system and Verma modules

Chord diagrams and the weight system

A chord diagram with $n$ chords is a fixed-point-free involution $\sigma$ of $[2n]$; its chords are the pairs $\{p,\sigma(p)\}$. It is a chord diagram on an oriented circle cut at a point that is not an endpoint; cutting at another point rotates $[2n]$ cyclically. Chords $\{p<q\}$ and $\{p'<q'\}$ cross if $p<p'<q<q'$ or $p'<p<q'<q$. The intersection graph $\Gamma(D)$ has the chords as vertices and the crossing pairs as edges; it does not depend on the cut.⁷

Let $\mathfrak{g}=\mathfrak{sl}(2)$ over $\mathbb{Q}$ with basis $E,F,H$, $[H,E]=2E$, $[H,F]=-2F$, $[E,F]=H$, and let

be the Casimir tensor of the Killing form.⁸ It is symmetric and $\mathfrak{g}$-invariant: $(\operatorname{ad}_{x}\otimes1+1\otimes\operatorname{ad}_{x})t=0$ for $x\in\mathfrak{g}$. For a chord diagram $D$ let $t_D\in\mathfrak{g}^{\otimes2n}$ be the tensor with a copy of $t$ in the factors $p,\sigma(p)$ of every chord, and let $C_D\in U(\mathfrak{g})$ be the image of $t_D$ under $u_1\otimes\cdots\otimes u_{2n}\mapsto u_1\cdots u_{2n}$. For one chord $C_D=C=\frac{1}{4}(EF+FE+\frac{1}{2}H^2)$.

For $d\geq1$ let $V_d$ be the irreducible representation of dimension $d$, with basis $e_0,\ldots,e_{d-1}$ and

where $e_{-1}=e_d=0$. The Casimir element $C$ acts on $V_d$ by $c_d=(d^2-1)/8$.⁹ Put $W_d(D)=\operatorname{tr}_{V_d}(C_D)/d$. Rotating the cut permutes the factors of $t_D$ cyclically, so $W_d(D)$ depends only on the diagram on the circle.

By Proposition 2.3 below there is a unique $w_D\in\mathbb{Q}[c]$ with $w_D(c_d)=W_d(D)$ for all $d\ge1$. It is the $\mathfrak{sl}(2)$-weight system of [11, 8]: by Lemma 2.1 $C_D$ lies in the centre of $U(\mathfrak{g})$, which is $\mathbb{Q}[C]$ [12], so $C_D=w(C)$ for a polynomial $w$, and taking traces on $V_d$ gives $w(c_d)=W_d(D)$; the weight system of [8] is defined with the Killing form and takes the value $c$ on one chord.

Verma modules

Let $R=\mathbb{Q}[\mu]$ and let $M$ be the free $R$-module with basis $v_0,v_1,\ldots$ and

These operators satisfy the relations of $\mathfrak{g}$:

and $(HE-EH)v_j=2Ev_j$, $(HF-FH)v_j=-2Fv_j$, because $H$ has eigenvalue $\mu-2j+2$ on $v_{j-1}$ and $\mu-2j-2$ on $v_{j+1}$. So $M$ is a $U(\mathfrak{g})$-module on which $U(\mathfrak{g})$ acts $R$-linearly. For $\mu_0\in\mathbb{Q}$, specialising $\mu=\mu_0$ gives the Verma module $V(\mu_0)$ of highest weight $\mu_0$.

Proof. By Proposition 2.2(b), $\nu\mapsto\lambda_D(\nu-1)$ is even, so $\lambda_D(\nu-1)=g(\nu^2)$ for some $g\in\mathbb{Q}[x]$. Put $w_D(c)=g(8c+1)$. Then $w_D(\mu(\mu+2)/8)=g((\mu+1)^2)=\lambda_D(\mu)$, and $w_D(c_d)=\lambda_D(d-1)=W_d(D)$ by Proposition 2.2(a). Uniqueness holds because the $c_d$ are pairwise distinct. Each term of $C_Dv_0$ is obtained by applying $2n$ operators from (1); their coefficients have degree at most one in $\mu$. So $\deg\lambda_D\leq2n$. Translation by 1 preserves this degree bound, so $\lambda_D(\nu-1)=g(\nu^2)$ gives $\deg g\leq n$, hence $\deg w_D\leq n$. In particular the values $W_d(D)$ for $1\leq d\leq n+1$ determine $w_D$ by polynomial interpolation. Finally $\mu(\mu+2)/8=-3/32$ for $\mu=-1/2$ and for $\mu=-3/2$.

The oscillator representation and a supertrace

Let $A$ be the Weyl algebra over $\mathbb{Q}$ generated by $z$ and $\partial$ with $\partial z-z\partial=1$. It acts on $\mathbb{Q}[z]$, with $z$ acting by multiplication and $\partial$ by differentiation, and the monomials $z^i\partial^j$ $(i,j\geq0)$ span $A$. In $A$,

Proof. The relation $\partial z=z\partial+1$ rewrites every word as a linear combination of these monomials. Suppose a finite sum $B=\sum_{i,j}a_{ij}z^i\partial^j$ acts by zero. On $z^k$, for all large enough integers $k$, the coefficient of $z^{k+r}$ is

It vanishes for infinitely many $k$, so it is the zero polynomial in $k$. For each fixed $r$ the falling factorials in this sum have distinct degrees $j$ and leading coefficient 1; hence every $a_{ij}$ is zero. This also shows that the normal forms are unique, so the action of the abstract Weyl algebra is faithful.

Proof. On $\mathbb{Q}[z]$,

Hence $(EF-FE)z^k=-\frac{1}{4}((k+2)(k+1)-k(k-1))z^k=Hz^k$, and $(HE-EH)z^k=2Ez^k$, $(HF-FH)z^k=-2Fz^k$, because $H$ has eigenvalue $-(k-\frac{3}{2})$ on $z^{k-2}$ and $-(k+\frac{5}{2})$ on $z^{k+2}$. The relations of $\mathfrak{g}$ hold on $\mathbb{Q}[z]$, hence in $A$ by Lemma 3.1. For $k=2j$ the formulas read $Ez^{2j}=j(-\frac{1}{2}-j+1)z^{2j-2}$ and $Hz^{2j}=(-\frac{1}{2}-2j)z^{2j}$, which is (1) at $\mu=-\frac{1}{2}$; for $k=2j+1$ they read $Ez^{2j+1}=j(-\frac{3}{2}-j+1)z^{2j-1}$ and $Hz^{2j+1}=(-\frac{3}{2}-2j)z^{2j+1}$, which is (1) at $\mu=-\frac{3}{2}$.

Proposition 3.3. $\varphi(C_D)=w_D(-\frac{3}{32})\cdot1$ in $A$.¹¹

Proof. By Proposition 2.2 and Lemma 3.2, $\varphi(C_D)$ acts on $\mathbb{Q}[z]_{\mathrm{even}}$ by $\lambda_D(-\frac{1}{2})$ and on $\mathbb{Q}[z]_{\mathrm{odd}}$ by $\lambda_D(-\frac{3}{2})$. Both equal $w_D(-\frac{3}{32})$ by Proposition 2.3, and Lemma 3.1 gives faithfulness.

Let $\tau:A\to\mathbb{Q}$ be the linear map with

Since $z^j\partial^j$ multiplies $z^k$ by $k(k-1)\cdots(k-j+1)$ and $\sum_k k(k-1)\cdots(k-j+1)x^k=j!x^j(1-x)^{-j-1}$, the number $\tau(a)$ is twice the Abel sum $\lim_{x\to-1^+}\sum_k a_{kk}x^k$ of the supertrace of $a$ on $\mathbb{Q}[z]$, where $az^k=a_{kk}z^k+\text{(other powers)}$.10 Only the following properties are used.

Proof. (a) is the case $i=j=0$. For (b) it is enough to take $a=z^i\partial^j$. By (2), $\tau(z^iz^i\partial^j)=\tau(z^{i+1}\partial^j)+j\tau(z^i\partial^{j-1})$. Both terms vanish unless $j=i+1$, and then the sum is $(-1)^j\frac{j!}{2^j}+j(-1)^{j-1}\frac{(j-1)!}{2^{j-1}}=-(-1)^j\frac{j!}{2^j}=-\tau(z\cdot z^i\partial^j)$. In the same way $\tau(\partial z^i\partial^j)=\tau(z^i\partial^{j+1})+i\tau(z^{i-1}\partial^j)$ vanishes unless $i=j+1$, and then it equals $(-1)^{j+1}\frac{(j+1)!}{2^{j+1}}+(j+1)(-1)^j\frac{j!}{2^j}=(-1)^j\frac{(j+1)!}{2^{j+1}}=-\tau(z^i\partial^j\cdot\partial)$.

(c) Induction on $m$. For $m=0$ use (a); for $m=1$, $\tau(z)=\tau(\partial)=0$. For $m\ge2$, as the $[u_1,u_s]$ are scalars,

Applying $\tau$ and (b) to the last term gives

which is the expansion of the right side of (c) according to the partner $s$ of 1.

Corollary 3.5. $w_D(-\frac{3}{32})=\tau(\varphi(C_D))$.

Expansion over matchings

Matchings

Let $X$ be a finite totally ordered set and $P,M$ perfect matchings of $X$. Two edges $\{a<b\}$ and $\{a'<b'\}$ cross if $a<a'<b<b'$ or $a'<a<b'<b$, and $\operatorname{cr}(P)$ is the number of crossing pairs of edges of $P$. The multigraph $P\cup M$ is a disjoint union of cycles alternating between edges of $P$ and of $M$ (a common edge gives a cycle of length 2); $\operatorname{cyc}(P,M)$ is their number. Orient every cycle; an edge oriented from a larger to a smaller element is a descent, and $\operatorname{des}(P,M)$ is the number of descents. Reversing a cycle of length $2r$ turns $\delta$ descents into $2r-\delta$, so the parity of $\operatorname{des}(P,M)$ does not depend on the orientations. Put

the sum over all perfect matchings $M$ of $X$.

Proof. Let $Q(P,M)$ be the sum. Let $x<y$ be consecutive in $X$, and exchange their positions in the order, keeping $P$ and $M$. If $\{x,y\}\notin P$, let $e\ni x$ and $f\ni y$ be the edges of $P$; the exchange moves $y$ to the other side of the endpoint $x$ of $e$ and keeps the side of the other endpoint of $f$, so it changes whether $e$ and $f$ cross, and no other pair. So $\operatorname{cr}(P)$ changes by $1-[\{x,y\}\in P]$ modulo 2, and likewise $\operatorname{cr}(M)$. $\operatorname{cyc}(P,M)$ does not change. A descent status changes only on edges joining $x$ and $y$, so $\operatorname{des}(P,M)$ changes by $[\{x,y\}\in P]+[\{x,y\}\in M]$. The total change is 2. Any two total orders of $X$ are related by exchanges of consecutive elements, so $Q$ modulo 2 is the same for all orders. Choose an order in which every cycle $x_1x_2\cdots x_{2r}$ (with $x_1x_2\in P$) occupies consecutive positions in this order. Then $P$ and $M$ have no crossings, and each cycle, oriented as listed, has exactly one descent, from $x_{2r}$ to $x_1$. Hence $Q\equiv2\operatorname{cyc}(P,M)\equiv0$.

The bosonic expansion

For a perfect matching $P$ of $X$ with $k$ edges let

the sum over the $2^k$ orientations $o$ of the edges of $P$, where $u_x^o=z$ if $x$ is the tail of its edge and $u_x^o=\partial$ if it is the head, the product is taken in the order of $X$, and $\operatorname{sgn}(o)=(-1)^{\text{number of descents of }o}$.

Proof. By Lemma 3.4(c),

A term vanishes unless every edge of $M$ joins a head and a tail of $o$. Orient such an edge from the head to the tail; then $\frac{1}{2}[u_a^o,u_b^o]$ is $\frac{1}{2}$ if it is oriented from $a$ to $b$ ($u_a^o=\partial$, $u_b^o=z$) and $-\frac{1}{2}$ if it is a descent. As $\operatorname{sgn}(o)$ is $-1$ to the number of descents among the edges of $P$, the term equals $2^{-k}(-1)^{\operatorname{des}(P,M)}$. Every element of $X$ is then entered by exactly one oriented edge and left by exactly one, so the cycles of $P\cup M$ are directed cycles. Conversely, each of the $2^{\operatorname{cyc}(P,M)}$ orientations of the cycles of $P\cup M$ as directed cycles restricts on $P$ to an orientation $o$ for which every edge of $M$ joins a head to a tail. Hence

by Lemma 4.1.

Legs

Let $D$ be a chord diagram with $n$ chords. Every point $p \in[2n]$ carries two legs $(p,1) < (p,2)$, and the set $X_D$ of the $4n$ legs is ordered lexicographically. For a chord $c = \{p < q\}$ and $\beta\in\{0,1\}$ let $P_c^\beta$ be the pair of edges

and for $\beta:\{\text{chords}\} \to\{0,1\}$ let $P_\beta= \bigcup_c P_c^{\beta(c)}$, a perfect matching of $X_D$. Let $\pi= z \otimes\partial- \partial\otimes z \in A \otimes A$. Writing $\pi= \sum_r \varepsilon_r x_r \otimes y_r$, put $\pi_{13}\pi_{24} = \sum_{r,s} \varepsilon_r\varepsilon_s x_rx_s \otimes y_ry_s$ and $\pi_{14}\pi_{23} = \sum_{r,s} \varepsilon_r\varepsilon_s x_rx_s \otimes y_sy_r$.

Clifford algebras

For $N \ge1$ let $\operatorname{Cl}_N$ be the $\mathbb{Q}$-algebra generated by $e_1,\ldots,e_N$ with $e_i e_j + e_j e_i = 2\delta_{ij}$. For $I = \{i_1 < \cdots< i_r\} \subseteq[N]$ put $e_I = e_{i_1}\cdots e_{i_r}$; these form a basis, and $e_I e_J = \pm e_{I \triangle J}$. Let $\operatorname{tr}:\operatorname{Cl}_N \to\mathbb{Q}$ be linear with $\operatorname{tr}(e_I) = \delta_{I\varnothing}$. Then $\operatorname{tr}(ab) = \operatorname{tr}(ba)$: for basis elements both sides vanish unless $I=J$.¹³

The edge $\{1,s\}$ crosses exactly the edges with one endpoint in $\{2,\ldots,s-1\}$; the elements of this set matched among themselves come in pairs, so the number of such edges is congruent to $s-2$ modulo 2. By induction both sides satisfy the same recursion.

Proof. By Lemma 5.1 the left side is $\sum_M(-1)^{\operatorname{cr}(M)}$ times the number of maps $X\to[N]$ constant on the edges of $P$ and of $M$, which is $N^{\operatorname{cyc}(P,M)}$.

For $I=\{i<j\}\in\binom{[N]}{2}$ write $b_I=e_i e_j$. Then

Proof. Both sides are polynomials (Theorem 1.3(a) is proved in Section 6 on its own), so it is enough to take $N\geq 1.

The two edges of $P^\beta_c$ cross if and only if $\beta=0$. If chords $c$ and $c'$ cross, every edge of $P^\beta_c$ crosses every edge of $P^{\beta'}_{c'}$; otherwise none does. Hence

A map $\kappa$ constant on the edges of $P_\beta$ is given by a pair $(i_c,j_c)\in[N]^2$ for every chord $c=\{p<q\}$: $i_c$ on the edge through $(p,1)$ and $j_c$ on the edge through $(p,2)$. The legs at $p$ then read $e_{i_c}e_{j_c}$, and the legs at $q$ read $e_{i_c}e_{j_c}$ if $\beta(c)=0$ and $e_{j_c}e_{i_c}$ if $\beta(c)=1$. By Corollary 5.2, summing over $\beta(c)$ with the sign $(-1)^{\beta(c)}$ replaces the factor at $q$ by $e_{i_c}e_{j_c}-e_{j_c}e_{i_c}$, which is 0 if $i_c=j_c$ and $2e_{i_c}e_{j_c}$ otherwise. Exchanging $i_c$ and $j_c$ changes the sign of both factors of $c$, so every 2-element set $s(c)=\{i_c,j_c\}$ arises twice with the same value. Therefore

the sum over $s:\{\text{chords}\}\to\binom{[N]}{2}$, where $c_p$ is the chord through $p$.

We show $\operatorname{tr}(B_s)=(-1)^n\prod_{cc'\in E(\Gamma)}(-1)^{\lvert s(c)\cap s(c')\rvert}$ by induction on $n$. Let $c=\{p<q\}$. Move the factor at $p$ to the right until it is next to the factor at $q$. By (3) it passes the factors at the points strictly between $p$ and $q$; a chord with both endpoints there contributes the square of a sign, and the chords with one endpoint there are the neighbours of $c$ in $\Gamma$. Then $b_{s(c)}^2=-1$, and the remaining word is $B_{s'}$ for the diagram $D\setminus c$, whose intersection graph is $\Gamma-c$, with $s'=s|_{\Gamma-c}$. Summing over $s$ gives $\sum_s\operatorname{tr}(B_s)=(-1)^nS_\Gamma(N)$.

Proof of Theorem 1.3

Proof of (a). Group the maps $p: V \to\binom{[N]}{2}$ by $U=\bigcup_x p(x)$. Relabelling $U$ in increasing order by $[\lvert U\rvert]$ preserves all intersection sizes, so the maps with $\lvert U\rvert=k$ contribute $\binom{N}{k}T_k(G)$; and $T_k(G)=0$ for $k>2n$.

Proof of (c). By Corollaries 3.5 and 4.4, Proposition 4.2 with $k=2n$, and Proposition 5.3,

For a graph $G$ and distinct vertices $a,b$ let $G'_{ab}$ be $G$ with the adjacency of $a$ and $b$ reversed, and $\widetilde{G}_{ab}$ be $G$ with the adjacency of $a$ and $y$ reversed for every neighbour $y\ne a$ of $b$. The four-term relation for a function $f$ on graphs is

as in Figure 3¹⁴

Proof. For $p: V\to\binom{[N]}{2}$ write

Since $\epsilon(G'_{ab},p)=\epsilon(G,p)(-1)^{\lvert p(a)\cap p(b)\rvert}$ and $\lvert p(a)\cap p(b)\rvert\in\{0,1,2\}$,

The construction $\widetilde{G}_{ab}$ does not change the adjacency of $a$ and $b$, so the same holds for $\widetilde{G}_{ab}$. On $\Pi$ define $\iota(p)$ by $\iota(p)(b)=p(a)\mathbin{\triangle}p(b)$ and $\iota(p)(x)=p(x)$ for $x\ne b$. As $\lvert p(a)\cap p(b)\rvert=1$, $\iota(p)(b)$ is a 2-element set meeting $p(a)$ in one element, and $\iota$ is an involution of $\Pi$.¹⁵ Let $Y$ be the set of neighbours $y \ne a$ of $b$ in $G$, which is also the set of such neighbours in $\widetilde{G}_{ab}$. For $y \in Y$, $\lvert(p(a)\mathbin{\triangle}p(b)) \cap p(y)\rvert\equiv\lvert p(a) \cap p(y)\rvert+ \lvert p(b) \cap p(y)\rvert$ modulo 2, and the edge $ab$, if present, keeps its sign because $\lvert p(a) \cap\iota(p)(b)\rvert= 1$. Hence

the second for every $q$, since $\widetilde{G}_{ab}$ differs from $G$ exactly in the pairs $ay$, $y \in Y$. With $q=\iota(p)$, which agrees with $p$ at $a$ and on $Y$, this gives $\epsilon(\widetilde{G}_{ab},\iota(p)) = \epsilon(G,p)$. Therefore $\sum_{p\in\Pi}\epsilon(\widetilde{G}_{ab},p)=\sum_{p\in\Pi}\epsilon(G,p)$, and the two differences agree.

Proof of (b). $\xi$ depends only on the isomorphism class of $G$. The four graphs of a four-term relation have the same vertex set, so by Proposition 6.1 and (a) the four-term relation holds for the polynomials $S_G$, hence at $N=-2$, and after division by $(-32)^n$ for $\xi$.

Remark 6.2. Proposition 6.1 applied to the sets $\Pi$ restricted to the maps with $\bigcup_x p(x)=[k]$, which $\iota$ preserves, shows that every $T_k$ satisfies the four-term relation. By the proof of Proposition 5.3, for $N\ge1$ the 4-invariant $S_\Gamma(N)$ is $(-4)^{-n}$ times the Clifford trace of the element obtained by placing the tensor $\sum_{i\ne j}e_i e_j\otimes2e_i e_j$ on the chords as in the definition of $C_D$, and Theorem 1.3(c) identifies $w_D\left(-\frac{3}{2}\right)$ with $(-32)^{-n}$ times its value at $N=-2$.

Remark 6.3. The formula for $\xi$ is a finite formula for every finite graph, with no bound on the number of vertices, since $T_k(G)=0$ for $k>2\lvert V(G)\rvert$. The involution of Proposition 6.1 is defined on every assignment with $\lvert p(a)\cap p(b)\rvert=1$, so the proof of (b) does not rest on a table of graphs. The identities of § 2 to § 5 hold for every chord diagram. The evaluation at $N=-2$ in the proof of (c) is substitution into a polynomial; no limit or analytic continuation is involved. The finite-dimensional normalised traces $W_d(D)$ are the starting definition, and the Verma module argument proves that they interpolate to a unique polynomial before its value at the oscillator representation is taken. The theorem says nothing about the unspecialised weight system, which does not extend to graphs by [8].

Declarations

Funding: none. Conflict of interest: none.