Minimal-volume conjecture for fullerenes with 10k atoms

Let $C_N$ be the class of fullerene isomers with $N$ atoms, and let $N=10k$ with $k\geq 2$. A nanotubical fullerene has a tubular part and two caps; a cap of type (a) is one of the four cap types considered in the paper. Let $D$ be a right-angled hyperbolic dodecahedron. Minimal-volume conjecture for fullerenes with 10k atoms. If a fullerene with $N=10k$ atoms, $k\geq 2$, has the minimal hyperbolic volume in the class $C_N$, then it is a nanotubical fullerene with caps of type (a), and its volume is given by

$$\operatorname{vol}(F)=\left(\frac{N}{10}-1\right)\cdot\operatorname{vol}(D).$$

The preceding proposition proves this volume formula for nanotubical fullerenes with caps of type (a), but does not prove that these fullerenes minimize volume in all of $C_N$.