33 problems

Let $\mathcal{C}_{n}(T_{\geqslant 4},B_{\geqslant 4})$ be the set of connected chord diagrams of size $n$ avoiding top and bottom cycles of size at least $4$, and let $C_n$ denote…

Let an invariant of chord diagrams satisfy the $T4T$ and $TFI$ relations. An invariant is integrated one step when it is lifted to an invariant at the next diagrammatic level. The…

Let $\mathcal{M}^{G}$ be the moduli space associated with a simple group $G$, and let the star product be the product defined by equation on the chord-diagram algebra. Descent conj…

Elementary-transformation generation conjecture. The group of elementary transformations is generated by transformations of the following two types: reflecting a share across anoth…

Conway–Kinoshita–Terasaka conjecture. The chord diagrams $C$ and $KT$ are not equivalent.

Intersection graph conjecture. If

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 corresp…

Chmutov–Duzhin conjecture. If two chord diagrams have the same intersection graph, then they are equivalent modulo the four-term relation.

Let $C$ be a chord diagram with $n$ chords, let $g(C)$ be its intersection graph, and let $\pi(C)$ be the projection of $C$ to the subspace of primitive elements of $\mathcal{J}_n$…

A chord diagram is $k$-terminal-minimal if it is $k$-terminal and minimal with respect to that property under the reduction considered in the paper. The forbidden-subdiagram charac…

Let $\mathcal{T}_{n}(\\{T_{2k+1},B_{2k+1}\\}_{k\geqslant 1})$ be the set of $1$-terminal chord diagrams of size $n$ avoiding all odd top and bottom cycles, and let a Baxter permuta…

Let $\mathcal{T}_{n}(T_3)$ be the set of $T_3$-free $1$-terminal chord diagrams of size $n$, and let a semi-Baxter permutation be a permutation in the standard semi-Baxter class. T…

Let $\mathcal{C}_{n}(B_{\geqslant 3})$ be the set of connected chord diagrams of size $n$ avoiding all bottom cycles of size at least $3$. The Fuss–Catalan conjecture. Its cardinal…

Let $\mathcal{C}_{n}(\\{T_{2k+1},B_{2k+1}\\}_{k\geqslant 1})$ be the set of connected chord diagrams of size $n$ avoiding all odd top and bottom cycles. The cubic-map enumeration c…

Let $C_n(T_{\geqslant 3},B_{\geqslant 3})$ be the set of connected chord diagrams of size $n$ avoiding all top and bottom cycles of size at least $3$, and let…

Lando–Yang conjecture. The projection $u(D)$ is a linear combination of connected Jacobi diagrams with at most $tau(D)$ legs. In particular, for any Lie algebra $bfg$ and its asso…

Let $D_n(x)$ denote the polynomial associated with the $n$-chord diagram, and define … For $k\geq 0$, set $b_k(x)=x-k(k+1)$, and for $k\geq 1$ set…

For each $ngeq 1$, let $\mathcal{D}_n$ be the $n$-chord diagram in which every chord intersects every other chord, and let $D_n=varphi(\mathcal{D}_n)inmathbb{Z}[x]$. Write…

Let $D_0\in \mathcal{D}^n_0$ with $n\in \mathbb{N}^{\ell}$ and $N=\sum n_i$. For the chord diagrams $D$ in $\mathcal{D}^n_N$, let $s(D)$ be the number of chords and let $E_D^0$ den…

Uniqueness conjecture for genus range $\{1,2\}$. For any $n\neq 4$, there is a unique, up to equivalence, chord diagram with genus range

Uniqueness conjecture for genus range $\{0,1\}$. For any $n\neq 2$, there is a unique, up to equivalence, double-occurrence word

Two-value genus-range conjecture. If the genus range of $D$ consists of two numbers, then it is either

Asymptotic enumeration conjecture. For $\mu-\nu=O(\nu^{\varepsilon})$, where $\varepsilon>0$ is small,

Let $m=m(n)$ be the number of crossings in a uniformly random chord diagram on $2n$ vertices, and let the second largest component mean the component with the second greatest numbe…

Kulakova–Lando conjecture.