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

A weight system is a function on chord diagrams that satisfies the four-term relations. The Lie algebra $\mathfrak{sl}(2)$ gives a weight system with values in the polynomial ring $\mathbb{Q}[c]$, and its value on a chord diagram depends only on the intersection graph of the diagram. Fomichev, Karev, Pavutnitskiy and Usanov showed that this weight system does not come from a function on graphs, and that a specialisation at a value $c=c_0$ can come from one only for $c_0=0$, $3/8$, $1$ or $-3/32$. The first three values were settled before. They conjectured that the last one, $c=-3/32$, which is the eigenvalue of the Casimir element on the oscillator representation, also extends. We prove this. The extension is the function $\xi(G)=(-32)^{-n}S_G(-2)$, where $G$ is a graph with $n$ vertices and $S_G(N)$ is a signed count of the maps that send each vertex of $G$ to a two-element subset of $\{1,\dots,N\}$: a map counts with the sign $-1$ for every edge whose two endpoints are sent to subsets meeting in exactly one element. The count is a polynomial in $N$, so its value at $N=-2$ makes sense. The four-term relation for $\xi$ is proved by an involution on these maps. The identity with the weight system is proved by letting the chord diagram act on Verma modules, then on the oscillator representation, where a supertrace on the Weyl algebra turns the value into a sum over perfect matchings, and finally by reading the same sum as a trace in a Clifford algebra on $N$ generators.