3. Knot theory and fullerenesLet be given an oriented knot or link diagram g, _{i}g, _{j}g
are related as in Fig. 6a, then
_{k}a= -_{ii}t, a= 1, _{ij}a=
-1; if they are related as in Fig. 6b,
then _{ik}a= _{ii}t, a= 1, _{ij}a=
-1; in all the other cases _{ik}a=0. The determinant _{ij}d(t)=|a|
is the polynomial invariant of _{ij}D. In the case of links, in this
polynomial invariant we use for generators of different components different
variables (up to a permutation of variables). For example, let us show
that two isomers of C_{20} are different (Fig. 3-4,
6c). After converting their chemical
Schlegel diagrams into the alternating knot diagrams, denoting their generators,
and calculating the corresponding determinants we obtain, respectively,
D_{1}=t^{20} -10t^{18}+45t^{16}-120t^{14}
+200t^{12}-197 t^{10} +105t^{8}-40t^{6}+25t^{4}
-10t^{2} and D_{2}=t^{20}
-10t^{18}+45t^{16}-120t^{14}
+208t^{12}-250t^{10} +217t^{8}-130t^{6}+49t^{4}
-10t^{2}, proving their difference. On the other hand, for
three-component alternating links corresponding to the diagrams (Fig.
6d), using the same multivariable invariant for link projections, we
conclude that they both represent the isomorphic Schlegel diagrams of the
same fullerene C_{24}.
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