21 problems

Let $\operatorname{Sym}(S^3,K)$ denote the group of isotopy classes of self-homeomorphisms of the pair $(S^3,K)$, and let $\operatorname{Th}(p,q)$ be a Turk's head knot with $p$ an…

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 $\Sigma K_{p,n}$…

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…

The isogonal–isohedral polycycle conjecture. All polycycles that are isogonal and isohedral, but not isotoxal, are the infinite $(3,4)$-polycycle from the third column in Figure 1…

Let $V_n=\{1,\ldots,n\}$. For $m>1$, consider the cones $P_n^m$, $NHM_n^m$, and $HM_n^m$ of $m$-hemimetrics on $V_n$, together with their faces and symmetries. Symmetry-group conje…

Given two disjoint $S^3$ Laves networks inscribed on a 600-cell, consider the surface dividing the two networks of the same chirality. The four networks arise from four disjoint 24…

Let the terminating ${}_2\phi_1$ correspond to the 2-variable symmetric polynomials in the $q$-Askey scheme, and let its symmetry group be viewed as a subgroup of $S_6$. Symmetry-g…

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…

Automorphism-generation conjecture. Having excluded Galilean transformations, $\operatorname{Aut}(A,B,\sigma)$ is generated by the three types of subgroups, the Euclidean groups ac…

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…

Consider a quantum field theory with a finite group global symmetry $\mathrm{G}$, of order $|\mathrm{G}|$, on a compact spatial manifold. Let $\alpha$ label an irrep of…

Let a quantum field theory have a finite group global symmetry $\mathrm{G}$ and be defined on a compact spatial manifold. For each irreducible representation (irrep) $\alpha$ of…

For a three-dimensional polytope $P$, let $f(P)$ be its $f$-vector and let $\operatorname{Symm}(P)=\{A\in\mathbb{R}^{3\times 3}:A\cdot P=P\}$ be its linear symmetry group. For a gr…

Let $F(G)$ denote the set of $f$-vectors realizable by three-dimensional polytopes with symmetry group $G$, and let the certificates used to characterize these sets be those constr…

Let $G$ be a Lie symmetry group acting by a linear representation $T$ on a suitable Hilbert or Banach phase space $\mathcal E$, and consider a generic $G$-invariant autonomous Hami…

Symmetric surface embedding conjecture. (1) All genera $g$ realizing the upper bound $E^+_+(\Sigma_g)=12(g-1)$ are listed in Theorem 1(1), and all examples realizing those genera a…

All-singular-orbits conjecture. For this purpose, “all concentrations” should include concentrations on all singular orbits, namely, orbits of dimension less than $|\gamma|-m$, of…

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 $(\{2,3\},6)$…