14 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 $W_n$ 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…

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.

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,

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…