4. "Perfect" fullerenes Using the mentioned connection between alternating knot or
link diagrams and 4regular (chemical) Schlegel diagrams of fullerenes,
it is interesting to consider all of them after such conversion. For example,
two chemical isomers of C_{20} will result in knots, and from 7
isomers of C_{24 }we obtain four knots, one 3component, one 4component
and one 5component link. Among the links obtained, two of them (3component
and 5component one) contain a minimal possible component: hexagonal carbon
ring (or simply, a circle). It is interesting that C_{60} consists
only of such regularly arranged carbon rings, so maybe this could be another
additional reason for its stability (Fig.
7). Therefore, it will be interesting to consider the infinite class
of 5/6 fullerenes with that property, that will be called "perfect". Some
of "perfect" fullerenes are modeled with hexastrips by P.Gerdes [13],
and similar structures: buckling patterns of shells and spherical honeycomb
structures are considered by different authors (e.g. T.Tarnai [14]).
To obtain them, we will start from some 5/6 fullerene
given in geometrical form (i.e. by a 3regular graph). Than we could use
"midedgetruncation" and vertex bifurcation in all vertices of the triangular
faces obtained that way, transforming them into hexagons with alternating
digonal edges. Let is given some fullerene (e.g. C_{20}) in its
geometrical form (i.e. as 3regular graph). By connecting the midpoints
of all adjacent edges we obtain from it the 3/5 fullerene covered by connected
triangular net and pentagonal faces preserved from C_{20}. After
that, in all the vertices of the truncated polyhedron we introduce digons,
to transform all triangles into hexagonal faces. This way, from C_{20}
we derived C_{60 }(in its chemical form) (Fig.
7). The midedgetruncation we could apply to any 5/6 (geometrical)
fullerene, to obtain new "perfect" (chemical) fullerene, formed by carbon
rings. This way, from a 5/6 fullerene with v vertices we always
may derive new "perfect" 5/6 fullerene with 3v vertices (Fig.
8). Moreover, the symmetry of new fullerene is preserved from its generating
fullerene. According to the theorem by Grünbaum & Motzkin [15],
for every nonnegative n_{6} unequal to 1, there exists
3valent convex polyhedron having n_{6} hexagonal faces.
Hence, from the infinite class of 3regular 5/6 polyhedra with v=20+n_{6}
vertices, we obtain the infinite class of "perfect" fullerenes with v=60+3n_{6}
vertices. The "perfect" fullerenes satisfy two important chemical conditions:
(a) the isolated pentagon rule (IPR); (b) hollow pentagon rule (HPR). The
IPR rule means that there are no adjacent pentagons, and HPR means that
all the pentagons are "holes", i.e. that every pentagon could have only
external double bonds. The first 5/6 fullerene satisfying IPR is C_{60},
and it also satisfies HPR. The IPR is well known as the stability criterion:
all fullerenes of lower order (less than 60) are unstable, because they
don't satisfy IPR. On the other hand, C_{70} satisfies IPR, but
cannot satisfy HPR (Fig. 9).
The same situation is with C_{80}, possessing
the same icosahedral geometrical symmetry as C_{60}, but not able
to preserve it after edgecoloring, because HPR cannot be satisfied (Fig.
9). This is the reason that only "perfect" fullerenes, with G=G'=
[3,5]=I_{h}=S_{5},
satisfying both IPR and HPR_{ }will be C_{60}, C_{180},
C_{240}, etc. We need also to notice that for n_{6}=
0,2,3 we have always one 3regular 5/6 polyhedron (i.e. geometrical form
of C_{20}, C_{24}, C_{26}), but for some larger
values (e.g. n_{6}= 4,5,7,9) there are several geometrical
isomers of the generating fullerene, and consequently, the same number
of "perfect" fullerenes derived from them (Fig.
10). Hence, considering the fullerene isomers, we could distinguish
"geometrical isomers", this means, different geometrical forms of some
fullerene treated as 3regular 5/6 polyhedron, and "chemical isomers" –
different arrangements of double bonds, obtained from the same 3regular
graph by its edgecoloring.
