Chmutov–Duzhin–Lando conjecture on intersection graphs of chord diagrams

Let $D$ be a chord diagram, and let its intersection graph $\Gamma(D)$ be the simple graph whose vertices are the chords of $D$, with two vertices adjacent exactly when the corresponding chords intersect, meaning that their endpoints on the bounding circle alternate. Two chord diagrams are equivalent modulo the 4-term relation when they represent the same element under that relation. Chmutov–Duzhin–Lando conjecture. If $D_1$ and $D_2$ are two chord diagrams with the same intersection graph, that is,

$$\Gamma(D_1)=\Gamma(D_2),$$

then for every weight system $W$,

$$W(D_1)=W(D_2).$$

Equivalently, intersection graphs determine chord diagrams up to the 4-term relation. Although a circle graph can arise from more than one chord diagram, the examples discussed in the source are equivalent modulo the 4-term relation; the conjecture asserts that this always holds.