10 problems

Symmetry-group conjecture. (i) $3_n(D_{2h})$ exists if and only if $\frac{n}{4}$ is even and $n\geq 16$; there are no tight $3_n(D_{2h})$. (ii) $3_n(D_{2d})$ exists if and only if…

Let $H=\bigoplus_{p,q}H^{p,q}$ be the state space of a conformal field theory, with $H^{1,0}$ its subspace of states of conformal weights $(1,0)$, and let the group of symmetries m…

Let $p,q$ be positive integers with $\gcd(p,q)=1$, so that $Th(p,q)$ is a Turk’s head knot, and let $D_{2q}$ denote the dihedral group of order $2q$. Symmetry-group conjecture. … T…

Let $p$ be a normalized polynomial, let $n\geq 2$, and let $K_{p,n}$ be the König root-finding map. Write $\Sigma p$ for the symmetry group associated with $p$ and $\Sigma K_{p,n}$…

Let $G$ be a finite cyclic or dihedral group. A $G$-symmetry type of a prime knot is a symmetry type arising from an action of $G$ on the knot. Hyperbolic realization conjecture. F…

Let $G$ be the point-symmetry pseudogroup under consideration, with splitting $G=G^{\rm ess}\ltimes G^{\rm lin}$ and Lie algebras $\mathfrak g$ and $\mathfrak g^{\rm ess}$. For…

Edge-only orbit coloring conjecture. The edge-only orbit coloring is symmetry faithful for both the linear symmetry group and the orthogonal symmetry group:

Let $L$ be a prime, nonsplit two-component link, and let $\Sigma(L)$ denote its intrinsic symmetry group. The link has full symmetry when $\Sigma(L)=\Gamma_2$, where $\Gamma_2$ is…

General-order linear ODE symmetry-generator conjecture. The vector field $X^0$ is the infinitesimal generator of $G_c$.

Symmetry and circuit conjecture. (i) A $z$- or $c$-knotted $(\{1,2,3\},6)$-sphere, except Trifolium $C_{3v}$, has symmetry $C_1$, $C_2$, $C_3$, $D_2$ or $D_3$. (ii) A $(\{2,3\},6)$…