3.3.3  Knot theory and fullerenes

The function fKLfromGraph (webMathematica fKLfromGraph) converts any 4-valent graph into the corresponding alternating KL projection and calculates its Dowker code in the Knotscape format. From this Dowker code, the function fPDataFromDow (webMathematica fPDataFromDow) computes P-data, that can be used for further computations. For example, from the graph  

G = {{1,2},{2,3},{3,4},{4,5},{1,5},{6,7},{7,8},{7,8},{8,9},{9,10},{10,11},{10,11},{11,12},{12,13},{13,14},{14,15},{15,16},{16,17},{16,17},{19,20},{17,18},{18,19},{19,20},{6,20},{20,21},{21,22}, 

{22,23},{23,24},{24,25},{25,26},{26,27},{27,28},{28,29},{29,30}, {30,31},{31,32},{32,33},{33,34},{34,35},{35,36},{36,37},{37,38},{38,39},{39,40},{21,40},{7,22},{8,25},{10,26},{11,29},{13,30},

{14,33},{16,34},{17,37},{19,38},{40,41},{41,42},{42,43},{43,44}, {44,45},{45,46},{46,47},{47,48},{48,49},{49,50},{50,51},{51,52},{52,53},{53,54},{54,55},{41,55},{24,44},{27,45},{28,47},{31,48}, 

{32,50},{35,51},{36,53},{39,54},{55,56},{56,57},{57,58},{58,59},{59,60},{56,60},{13,14},{1,9},{1,9},{2,12},{2,12},{3,15},{3,15},{4,18},{4,18},{5,6},{5,6},{21,40},{22,23},{24,25},{26,27},{28,29}, 

{30,31},{32,33},{34,35},{36,37},{38,39},{41,42},{44,45},{47,48},{50,51},{53,54},{55,56},{43,57},{43,57},{46,58},{46,58},{49,59}, {49,59},{52,60}, {52,60},{23,42}}

of the fullerene C60, we obtain the Dowker code of the corresponding link in the Knotscape format  

{{3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3},{28,36,10,18,42,6,48,24, 12,54,30,16,60,2,22,4,72,66,68,78, 8,74,84,14,82,90,20,88,64,26,34, 

96,58,32,38,102,40, 44,106,112,50,46,118,56,52,62,100,116,70,108,92, 114,76,98,120,80,104,94,86,110}}  

with the carbon rings as the components. 

For the recognition of fullerene isomers converted into alternating KL projections we can use polynomial invariants of KL projections (see page 197). The LinKnot functions JablanPoly (webMathematica JablanPoly) and LiangPoly (webMathematica LiangPoly) calculate these invariants. For example, let us show that two isomers of C20 are different . After converting their chemical Schlegel diagrams into alternating KL diagrams, denoting their generators, and calculating the corresponding projection polynomials, we obtain 

D1 = t20 -10t18+45t16-120t14 +200t12-197t10 +105t8-40t6+25t4 -10t2,

D2 = t20 -10t18+45t16-120t14 +208t12-250t10 +217t8-130t6+49t4 -10t2,
proving their difference. Using the same multivariable invariant for link projections, we conclude that one can obtain from the fullerene C24 seven non-isomorphic diagrams. The same results can be obtained by using the Liang polynomial. 

All 4-valent (chemical) Schlegel diagrams of fullerenes can be converted into alternating KL diagrams. For example, two chemical isomers of C20 will give knots, and from 7 isomers of C24 we obtain four knots, one 3-component, one 4-component and one 5-component link. Among the links obtained, two of them (3-component and 5-component link) contain a minimal possible component: hexagonal carbon ring. It is interesting that C60 consists only of regularly arranged hexagonal carbon rings, which might be additional reason for its stability. Therefore, it will be interesting to consider the infinite class of 5/6 fullerenes with this property, called perfect fullerenes. Some of these perfect fullerenes are modelled with hexastrips by P. Gerdes (1998). Similar structures, buckling patterns of shells and spherical honeycomb structures have been considered by different authors (e.g., T. Tarnai (1989)). 

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