18 problems

A Vassiliev invariant is a finite-type invariant of knots, and a quantum Lie group invariant is a knot invariant obtained from quantum-group representations. Bar-Natan's conjecture…

A Vassiliev invariant is a finite-type invariant of knots; a knot is prime if it cannot be expressed as a nontrivial connected sum, and unoriented knots are considered without a ch…

For each $m$, let ${\cal D}^*_m$ be the dual diagram complex with differential $\partial$, and let $H^1_\partial({\cal D}^*_m)$ denote its first cohomology group. The graph-homolog…

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…

Asymptotic Vassiliev-invariant conjecture. The numbers $\log_{v_2(K_i)}v_3(K_i)$, which are well-defined for $K_i\ne !3_1$, converge to $2$ as $i\to\infty$. This predicts a precise…

Let $K$ be a knot, and let a Vassiliev (finite-type) invariant mean any finite-type invariant of knots. Vassiliev's stabilization conjecture. If every Vassiliev invariant of $K$ is…

Linked-pair lower-bound conjecture. One has

Positive twist-knot minimization conjecture. Positive twist knot diagrams minimize $v_3$ among all connected irreducible positive diagrams with odd crossing number.

A knot is 2-almost positive if the minimal number of negative crossings among all its diagrams is $2$. Let $L$ be a 2-almost positive diagram with even crossing number that minimiz…

Let $D$ be a positive bireduced knot diagram, and let $lk(D)$ denote the number of linked pairs in $D$. Let $v_2$ be the second Vassiliev invariant. The linked-pair lower-bound con…

Let $CK^*$ be the space obtained from $CK_b(S^3)$ by attaching an $(n+1)$-cell for every knot $K_{(n-2,1)}$, with $n\geq 2$, whose boundary is given by resolving the double points…

A Vassiliev invariant of order $k$ is a knot invariant that vanishes on higher-order differences of knots. Let $v_k$ be any Vassiliev invariant of order $k$, let $K$ be a knot with…

An even, degree-one Vassiliev invariant is a function $V$ on virtual knots such that $V(K_+)-V(K_-)=0$ for even crossings and $V(K+)-V(K-) e 0$ for odd crossings. Nonexistence conj…

Let $\alpha$ be a Vassiliev $1$-cocycle of degree $m$, and let $\tilde{\alpha}$ be the $1$-cocycle obtained by composing the weight system associated to the principal part of…

Let $H_{d,\text{far}}^i$ denote the submodule of $i$-th cohomology classes in degree $d$ represented by cocycles in the far subcomplex, and let $H^i(\mathcal{K}\setminus\Sigma)$ be…

Let ${\cal A}^m_n$ denote the corresponding degree-$n$ chord-diagram space for links with $m$ components, with the induced symmetric-group action on components. An equivariant basi…

Let ${\cal C}^m_n$ denote the space of chord diagrams with $m$ distinguishable components and degree $n$, modulo the 4T-relation. The symmetric group on the components acts on this…

Geometric characterization conjecture. A knot $K$ is $n$-trivial for all $n\in\mathbb{N}$ if and only if it is $n$-hyperbolic for all $n\in\mathbb{N}$.