20 problems

Let ${\frak g}$ be a finite-dimensional Lie algebra with a metric and orthonormal basis $\{{\frak g}_a\}_{a=1}^{\dim{\frak g}}$, and let $R$ be a finite-dimensional representation.…

Let a weight system be associated with a Vassiliev invariant, and let the natural cabling operation on knots have an order parameter. Cabling-order conjecture. The order of the nat…

Let $W_n$ be the weight system defined in the paper, let $Z$ denote the Kontsevich integral, and let $\nabla_n$ denote the degree-$n$ coefficient of the multivariable Conway potent…

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…

A weight system is a mapping from trivalent graphs, or chord diagrams modulo the 4-term relation, to ${\bf R}$ that satisfies the defining relations, including the 4-term relation.…

Suppose that $\xi(G)$ is a 4-invariant of graphs extending the specialization of the $\mathfrak{sl}(2)$-weight system at $c=-\frac{3}{32}$. Let $G$ be a graph, let $x$ be a vertex…

Let $\mathbf{G}$ be the set of isomorphism classes of finite simple graphs. For $G\in\mathbf{G}$, let $\varphi(G)$ be the graph invariant defined by … where $\chi_3(G)$ is the numb…

Lando's conjecture. There exists a unique 4-invariant that extends the $\mathfrak{sl}(2)$-weight system.

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 $\mathcal{J}_n$…

Let ${\bf w}$ be the weight system of an isolated quasihomogeneous singularity, and let $\psi_{\bf w}$ and $d_{\bf w}$ be the functions and parameter associated with the weight sys…

Let ${\bf w}=(w_1,\ldots,w_n;1)$ be the weight system of an isolated quasihomogeneous singularity of multiplicity at least $3$, and let $\psi_{\bf w}$ and $d_{\bf w}$ be the functi…

Lando's complete-bipartite reformulation. The value of the $\mathfrak{sl}_2$ weight system at $K_{l,n}$ is a polynomial of degree $\min(l,n)$.

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…

Kulakova–Lando conjecture.

Weight-system coefficient conjecture. For any chord diagram $d$ with $2k$ chords,

A weight system is a linear functional on the relevant graph or chord-diagram space satisfying the corresponding graph relations. Bar-Natan's conjecture. All weight systems come fr…

Consider virtual-knot theories with skeleton equal to a circle (round), a line (long), or a line with all real crossings descending; with R2 and R3 treated in the standard, braid-l…

For virtual knots, finite type invariants are defined by the schematic difference relation … A weight system is a linear functional on virtual-knot arrow diagrams satisfying the re…