Lego-like spheres and tori

Abstract

Given a connected surface ${\mathbb {F}}^2$ with Euler characteristic $\chi$ and three integers $b>a\ge 1<k$, an $(\{a,b\};k)$-${\mathbb {F}}^2$ is a ${\mathbb {F}}^2$-embedded graph, having vertices of degree only k and only a- and b-gonal faces. The main case are (geometric) fullerenes (5, 6; 3)-${\mathbb {S}}^2$. By $p_a$, $p_b$ we denote the number of a-gonal, b-gonal faces. Call an $(\{a,b\};k)$-map lego-admissible if either $\frac{p_b}{p_a}$, or $\frac{p_a}{p_b}$ is integer. Call it lego-like if it is either $ab^f$-lego map, or $a^fb$-lego map, i.e., the face-set is partitioned into $\min (p_a,p_b)$ isomorphic clusters, legos, consisting either one a-gon and $f=\frac{p_b}{p_a}\,b$-gons, or, respectively, $f=\frac{p_a}{p_b}\,a$-gons and one b-gon; the case $f=1$ we denote also by ab. Call a $(\{a,b\};k)$-map elliptic, parabolic or hyperbolic if the curvature $\kappa _b=1+\frac{b}{k}-\frac{b}{2}$ of b-gons is positive, zero or negative, respectively. There are 14 lego-like elliptic $(\{a,b\};k)$-${\mathbb {S}}^2$ with $(a,b)\ne (1,2)$. No $(\{1,3\};6)$-${\mathbb {S}}^2$ is lego-admissible. For other 7 families of parabolic $(\{a,b\};k)$-${\mathbb {S}}^2$, each lego-admissible sphere with $p_a\le p_b$ is $a^fb$ and an infinity (by Goldberg–Coxeter operation) of $ab^f$-spheres exist. The number of hyperbolic $ab^f\,(\{a,b\};k)$-${\mathbb {S}}^2$ with $(a,b)\ne (1,3)$ is finite. Such $a^f b$-spheres with $a\ge 3$ have $(a,k)=(3,4),(3,5),(4,3),(5,3)$ or (3, 3); their number is finite for each b, but infinite for each of 5 cases (a, k). Any lego-admissible $(\{a,b\};k)$-${\mathbb {S}}^2$ with $p_b=2\le a$ is $a^f b$. We list, explicitly or by parameters, lego-admissible $(\{a,b\};k)$-maps among: hyperbolic spheres, spheres with $a\in \{1,2\}$, spheres with $p_b\in \{2,\frac{p_a}{2}\}$, Goldberg–Coxeter’s spheres and $(\{a,b\};k)$-tori. We present extensive computer search of lego-like spheres: 7 parabolic ($p_b$-dependent) families, basic examples of all 5 hyperbolic $a^fb$ (b-dependent) families with $a\ge 3$, and lego-like $(\{a,b\};3)$-tori.