42 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…

Let … where $\beta_{m,u}$ counts the relevant chinese-character diagrams of degree $m$ and loop number $u$. Pseudopolynomial conjecture. The Taylor coefficients of $b(x,y)$ in $x$…

Let $\Lambda$ be the graded algebra of fixed diagrams defined by the antisymmetry condition under $S_3$ and the relation $\phi_1(u)=\phi_2(u)$, and let $t,x_3,x_5,\ldots$ be the el…

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…

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

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…

A finite type knot invariant is a link invariant of finite type, restricted to knots; a knot is non-invertible if it is not equivalent to the knot obtained by reversing its orienta…

Let $v$ be an arbitrary Vassiliev invariant, and let the coefficients of the Alexander–Conway polynomial generate an algebra. Bar-Natan–Garoufalidis' highest-order conjecture. Ther…

Let $K$ be a knot, and let $V_K(t)$ denote its Jones polynomial. A knot is nontrivial if it is not equivalent to the unknot. Jones polynomial detection conjecture. Every nontrivial…

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 $V_n^{\mathrm{prim}}$ denote the space of primitive Vassiliev invariants of degree $n$. Kontsevich–Bar-Natan conjecture. The dimension of $V_n^{\mathrm{prim}}$ grows, as $n$ te…

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…

A Jacobi diagram is a uni-trivalent graph equipped with the additional structure used in the theory of Vassiliev invariants. Its legs are its univalent vertices. Odd-leg vanishing…

A Vassiliev invariant is a knot invariant extended to singular knots by resolving each double point as the difference between a positive and a negative crossing; an invariant is of…

Birman's desingularization-map conjecture. The map $\psi$ is an embedding. This conjecture connects singular braids with Vassiliev invariants and quantum groups; it was proved affi…

Weight-system classification conjecture. Every weight system should arise from a complex semisimple Lie algebra or superalgebra in this way.

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…

Let $\phi$ be a knot invariant, let $v_n$ be Vassiliev invariants, and let $f_n$ be functions defined on their values. Functional approximation conjecture. For every knot invariant…

Let $\phi$ be a knot invariant, and let $v_n$ be Vassiliev invariants, that is, finite-type knot invariants. Approximation conjecture. For every knot invariant $\phi$, there is a s…

Let $K_1$ and $K_2$ be knots, and let $v$ be a Vassiliev invariant, meaning a finite-type knot invariant. Separation conjecture. For any two knots $K_1$ and $K_2$, there is a Vassi…

A knot invariant is a numerical function on knots, and a Vassiliev invariant is a knot invariant of finite type. Equip the space of numerical knot invariants with the pointwise top…