Symmetry and central-circuit conjectures for ({1,2,3},6)- and ({2,3},6)-spheres

A $(\{1,2,3\},6)$-sphere is a sphere whose face sizes belong to $\{1,2,3\}$ and whose vertex degree is $6$; a $(\{2,3\},6)$-sphere is defined similarly with face sizes in $\{2,3\}$. A sphere is $z$-knotted or $c$-knotted when it has only one zigzag or only one central circuit, respectively. Its symmetry is denoted by the corresponding point-group symbol, and its $c$- and $z$-vectors record the lengths of its central circuits and zigzags.

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 $(\{2,3\},6)$-sphere with only simple central circuits has symmetry $T_d$, $T_h$, $D_{6h}$, $D_{3d}$, $D_{2d}$, $D_{2h}$, $D_3$, $C_{2h}$ or $C_{3v}$. (iii) A $(\{1,2,3\},6)$-sphere of symmetry $D_{6h}$, $T_h$ or $T_d$ has only simple central circuits and zigzags. (iv) A $(\{2,3\},6)$-sphere of symmetry $T_d$ has $v=4x^2$ vertices, $c$-vector $(3x)^{4x}$ and $z$-vector $(6x)^{4x}$. (v) A $(\{2,3\},6)$-sphere of symmetry $T_h$ has $12x^2$ vertices, $c$-vector $(6x)^{6x}$ and $z$-vector $(12x)^{6x}$.

These assertions classify the possible symmetries and circuit structures in several highly symmetric or knotted families of discrete spheres. The supplied text gives no evidence that the conjecture has been proved or refuted.