267 problems

Ooguri–Vafa conjecture. There exists a corresponding Lagrangian submanifold $\mathcal{D}_{\mathcal{K}}$ such that

Determinant-square conjecture. In general, the maximum determinant of a prime symmetric ribbon knot $K$ with $r_s(K)=n$ is the square of the maximum determinant of a prime knot $J$…

Gukov–Manolescu conjecture. The invariant $F_K$ is obtained by reorganizing this perturbative expansion into an integral two-variable $q$-series.

Let $(C,p)$ be a reduced irreducible plane curve singularity, let $C^{[l]}_p$ be its punctual Hilbert scheme of length-$l$ subschemes, let $\boldsymbol{\mu}$ be its Milnor number,…

Let the inclusive Racah matrices and the corresponding $\mathcal{R}$-matrices be those associated with three-strand knots, with the $\mathcal{R}$-matrix eigenvalues normalized by a…

Let $K$ and $K'$ be virtual knots, and let $\mathcal{B}(K)$ and $\mathcal{B}(K')$ denote their fundamental biquandles. Two virtual knots are identified up to orientation-reversed m…

Melvin–Morton–Rozansky conjecture. The semi-classical expansion is

Rozansky's rationality conjecture. Each term of the expansion of the Kontsevich integral should be given by a series of trivalent graphs with rational functions attached to their e…

Let $\beta$ be a braid and let $L$ be its closure. Let ${\large {\Huge {\Gamma^R}}}(L)\in\mathbb L^R$ denote the refined universal ADO intersection, and let…

A virtual knot has an associated fundamental biquandle, an algebraic invariant defined from a virtual-knot diagram, and two virtual knots are identified up to mirror reflection whe…

For an oriented virtual knot $K$, let $\mathcal{B}(K)$ denote its fundamental biquandle, the algebraic structure associated with $K$ that encodes its Reidemeister moves. Fundamenta…

Let $M$ be a rational homology $3$-sphere and let $K\subset M$ be a null-homologous knot. The Kricker and Lescop invariants are defined for such knots, and their equivalence is alr…

Let $K$ be a knot in an integral homology $3$-sphere $M$, and let $Q_n(M,K)$ denote the $n$-loop part of the logarithm of the LMO invariant, for $n\geq 1$. Rozansky's rationality c…

Let $[K]$ be a knot class represented by lattice knots, and let $dV[K]$ denote its vertex distortion, defined as the infimum of $dV(K)$ over all lattice-knot conformations represen…

Let $K\subseteq S^3$ be a knot. Write $F_K(x,q)$ for the knot invariant, set $q={\rm e}^{\hbar}$, and keep $x=e^{n\hbar}$ fixed, where $n$ is the color of $K$. Let $\Delta_K(x)$ be…

A knot's superbridge index is the minimum, over all embeddings of the knot in Euclidean three-space, of the maximum number of local maxima of the height function over all direction…

Let $\mathcal{K}_{\tau}$ be a knot in $S^3$ with framing $\tau\in\mathbb{Z}$, and let $Z_{CS}^{(S^3,\mathcal{K}_{\tau})}(q,a,\mathbf{x})$ be its Chern–Simons partition function. De…

The original Kontsevich integral is a universal finite-type invariant of 3-manifolds, and $h_\infty$ denotes the Chern–Simons invariant constructed earlier in the paper. Equality c…

Let $F8$ denote the figure-eight knot, viewed as a genus-one fibered knot, and let $Z_p(F8)$ be the map induced by the monodromy of its zero-framed surgery, acting on the correspon…

Let $g_{2n}$ be the representation-dependent factors in the coefficients $w_{2n}$ of the Wilson-loop expansion, and let $c_2$ and $c_v$ denote the quadratic Casimir factors appeari…

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

Resolution conjecture. The generalized Kauffman bracket is given by

Let $Z^K$ denote the Kontsevich integral and $Z^{BT}$ the Bott–Taubes invariant. Kontsevich–Bott–Taubes conjecture. It is conjectured that … Both invariants are integral-formula co…

Let a round diagram be a diagram with a Wilson loop, and let a symmetrized diagram be its corresponding diagram in the symmetrized-diagram basis. For a positive integer $l$, consid…