Fullerenes and Polygonal Partitions of Surfaces
Victor Buchstaber Steklov Mathematical Institute Lomonosov Moscow State University [email protected]
3rd Workshop on Analysis, Geometry and Probability
September 28 - October, 2, 2015 Ulm University, Germany 1 / 54 Abstract
The talk is devoted to classical and recent mathematical results, which had drawn attention of physicists and chemists in connection with problems of nanotechnology. Fullerenes Nobel Prize 1996 in Chemistry R. F.Kurl, H. Kroto, R. E. Smalley «for their discovery of fullerenes». Graphene Nobel Prize 2010 in Physics A. K. Geim and K. S. Novoselov «for groundbreaking experiments regarding the two-dimensional material graphene».
2 / 54 Abstract
We will formulate combinatorial, geometrical, and topological tasks concerning molecular carbon structures in terms of convex polytopes, polygonal partitions of 2-surfaces. The connection of these tasks with known and new problems in graph theory and toric topology will be discussed. The talk is prepared jointly with N.Yu.Erokhovets.
3 / 54 Fullerenes
A fullerene is a spherical-shaped molecule of carbon such that any atom belongs to exactly three carbon rings, which are pentagons or hexagons.
Fullerenes have been the subject of intense research, both for their unique chemistry and for their technological applications, especially in materials science, electronics, and nanotechnology.
Fullerene C60
4 / 54 Fullerenes
Fullerenes were named after Richard Buckminster Fuller – a noted american architectural modeler.
Are also called buckyballs Fuller’s Biosphere USA Pavilion, Expo-67 Montreal, Canada
5 / 54 Graphene
The plane R2 can be tiled by regular hexagons.
The schematic picture of the graphene. Nodes are atoms of carbon, edges – bonds that hold atoms in the graphene sheet. 6 / 54 Carbon nanotubes
Carbon nanotubes are obtained when the graphene sheet is rolled up into the infinite cylinder. They are characterized by the chiral vector c = na + mb. The shift by the vector c moves the lattice to itself. The nodes that differ by the shift are identified during the rolling up.
(0,0) zigzag (9,0)
a (7,2) chiral b
(5,5)
armchair c=na+mb 7 / 54 Carbon nanotubes
Because of the symmetry and unique electronic structure of graphene, the structure of a nanotube strongly affects it’s electrical properties. For a given (n, m)-nanotube, if n = m, the nanotube is metallic; if n − m is a multiple of 3, then the nanotube is semiconducting with a very small band gap, otherwise the nanotube is a moderate semiconductor.
To obtain a finite nanotube one should cut off the infinite cylinder along two simple edge cycles. Any of the cycles should divide the cylinder into two infinite parts. The cycles should not intersect and not selfintersect.
8 / 54 Carbon nanotubes
(n, 0)-nanotubes are called zigzag nanotubes, while (n, n)-nanotubes are called armchair. These tubes have mirror symmetry. All other types are chiral, that is they do not coincide with the mirror image.
9 / 54 Carbon nanotubes
By a capped nanotube we mean a surface obtained by gluing up the ends of a finite nanotube by disks subdivided into pentagons and hexagons.
10 / 54 Carbon nanobuds
A nanobud is a concatenation of a fullerene with a nanotube. Nanotubes are chemically neutral, therefore it is difficult to combine them with other materials. Fullerenes are chemically active and give good conditions for this.
11 / 54 Convex polytopes
A convex polytope P is a bounded set of the form
n P = {x ∈ R : ai x + bi > 0, i = 1,..., m} Let this representation be irredundant, that is a deletion of any inequality changes the set. Then each hyperplane n Hi = {x ∈ R : ai x + bi = 0} defines a facet Fi = P ∩ Hi .
Archimedean solid – snub dodecahedron
12 / 54 Schlegel diagram (Victor Schlegel, 1843 - 1905)
A Schlegel diagram (1886) of a convex polytope P in R3 is a projection of P from R3 into R2 through a point beyond one of its facets.
A Shlegel diagram is a polygonal subdivision of the facet. The graph of edges drown on the diagram is a complete combinatorial invariant of 3-polytopes.
Schlegel diagram of the dodecahedron 13 / 54 Euler’s formula (Leonhard Euler, 1707-1783)
Let f0, f1, and f2 be numbers of vertices, edges, and 2-faces of a 3-polytope. Then
f0 − f1 + f2 = 2
f0 f1 f2 Tetrahedron 4 6 4 Cube 8 12 6 Octahedron 6 12 8 Dodecahedron 20 30 12 Icosahedron 12 30 20
14 / 54 Simple polytopes
An n-polytope is simple if any its vertex is contained in exactly n facets.
3 of 5 Platonic solids are simple. 7 of 13 Archimedean solids are simple.
For any simple 3-polytope f0 = 2(f2 − 2), f1 = 3(f2 − 2).
15 / 54 Consequence of Euler’s formula for simple 3-polytopes
Let pk be a number of k-gonal 2-faces.
For any simple 3-polytope P X 3p3 + 2p4 + p5 = 12 + (k − 6)pk k>7
Corollary If p = 0 for k = 3, 4, then p = 12 + P (k − 6)p . k 5 k>7 k If pk = 0 for k 6= 5, 6, then p5 = 12. There is no simple 3-polytopes with all faces hexagons.
3p3 + 2p4 + p5 > 12.
16 / 54 Realization theorems
Theorem (Eberhard, 1891)
For every sequence (pk |3 6 k 6= 6) of nonnegative integers satisfying X 3p3 + 2p4 + p5 = 12 + (k − 6)pk , k>7
3 3 there exist a simple 3-polytope P with pk = pk (P ), k 6= 6.
For a fixed sequence (pk |3 6 k 6= 6) There are infinitely many realizable values of p6. ! P There exists realizable p6 6 3 pk (J.C. Fisher, 1974) k6=6 If p3 = p4 = 0 then any p6 > 8 is realizable (B. Grunbaum, 1968). 17 / 54 Flag polytopes
A simple polytope is called flag if any set of pairwise
intersecting facets Fi1 ,..., Fik :Fis ∩ Fit 6= ∅, s, t = 1,..., k, has
a nonempty intersection Fi1 ∩ · · · ∩ Fik 6= ∅.
Flag polytope Non-flag polytope
18 / 54 Fullerenes
A (mathematical) fullerene is a simple 3-polytope with all 2-facets pentagons and hexagons.
Fullerene C60 Truncated icosahedron
For any fullerene p5 = 12,
f0 = 2(10 + p6), f1 = 3(10 + p6), f2 = (10 + p6) + 2
There exist fullerenes with any p6 6= 1. 19 / 54 Schlegel diagrams of fullerenes
Fullerene barrel Schlegel diagram of the barrel
20 / 54 Endo-Kroto operations
The Endo-Kroto operation increases p6 by 1. Starting from the Barrel and applying a sequence of Endo-Kroto operations it is possible to obtain a fullerene with arbitrary p6 = k, k > 2.
21 / 54 Number of combinatorial types of fullerenes
Two combinatorially nonequivalent fullerenes with the same number p6 are called combinatorial isomers.
Let F(p6) be the number of combinatorial isomers with given p6. 9 It is known that F(p6) = O(p6).
There is an effective algorithm of combinatorial enumeration of fullerenes using supercomputer (Brinkman, Dress, 1997).
p6 0 1 2 3 4 5 6 7 8 ... 75 F(p6) 1 0 1 1 2 3 6 6 15 ... 46.088.157
22 / 54 The function F(n − 10).
The sequence A007894(n) see in on-line encyclopedia of integer sequences, founded in 1964 by N.J.A. Sloane.
23 / 54 The function ln(F(n − 10)).
See in on-line encyclopedia of integer sequences, founded in 1964 by N.J.A. Sloane.
24 / 54 IPR-fullerenes
Definition An IPR-fullerene (Isolated Pentagon Rule) is a fullerene without pairs of adjacent pentagons.
Let P be some IPR-fullerene. Then p6 > 20. An IPR-fullerene with p6 = 20 is combinatorially equivalent to Buckminsterfullerene C60.
The number FIPR(p6) of combinatorial isomers of IPR-fullerenes also grows fast as a function of p6.
p6 20 21 22 23 24 25 26 27 28 ... 97 FIPR 1 0 0 0 0 1 1 1 2 ... 36.173.081
25 / 54 The Brinkman-Dress conjecture, 1997
Conjecture (Brinkman, Dress)
The functions F(p6) and FIPR(p6 + 24) are in some sense good approximations of each other asymptotically.
p6 0 1 2 3 4 5 6 7 8 ... 73 F(p6) 1 0 1 1 2 3 6 6 15 ... 36.798.433 FIPR(p6 + 24) 0 1 1 1 2 5 7 9 24 ... 36.173.081
26 / 54 Construction of simple 3-polytopes
Theorem (Eberhard (1891), Brückner (1900)) Any simple 3-polytope is combinatorially equivalent to a polytope that is obtained from the tetrahedron by a sequence of vertex, edge and (2, k)-truncations.
27 / 54 (2, 7)-truncation
28 / 54 (2, 7)-truncation
29 / 54 (2, 7)-truncation
30 / 54 Properties of an edge-truncation
F' F1 1
F4 E F2 F'4 F F'2
F3 F'3
Combinatorially the edge-truncation is a local operation and affects only the facets F1 and F3, the edge E, and it’s vertices: the edge E is transformed into the quadrangle F;
the number of edges of F1 and F3 increases by 1.
31 / 54 Properties of a (2, k)-truncation
F1 F'1
F F'2 E1 2 E'2 E2 F F' F'' E3 E'3 E4 F3 F'3
F4 F'4 Combinatorially the (2, k)-truncation is a local operation and affects only the facets F, F1, and F4, and the edges E1 = F1 ∩ F and E4 = F4 ∩ F: k-gon F is divided into (k − 1)-gon F 0 and 5-gon F 00; E1 and E4 are divided into two edges; the number of edges of F1 and F4 increases by 1. 32 / 54 The Endo-Kroto operation as a (2, 6)-truncation
The Endo-Kroto operation is the only (2, k)-truncation that transforms a fullerene into a fullerene. In this case k = 6 and F1 and F4 are pentagons.
33 / 54 Construction of combinatorial fullerenes
Theorem (Buchstaber-Erokhovets, 2015) Any fullerene is combinatorially equivalent to a polytope obtained from the cube by a sequence of edge-truncations and (2, 6)- and (2, 7)-truncations, such that at each step the polytope has facets only 4-, 5-, 6-gons and no more than one 7-gon, and this 7-gon is incident to 4- or 5-gon. Any operation increases the number of facets by one, therefore the number of operations in a sequence is 6 + p6.
34 / 54 Realization of the dodecahedron
(p4, p5, p6) = (3, 6, 0)(p4, p5, p6) = (2, 8, 1)(p4, p5, p6) = (0, 12, 0)
first apply 3 edge-truncations to the cube to obtain the associahedron; then apply 2 edge-truncations of bold edges; at last apply (2, 6)-truncation of two bold edges.
35 / 54 Properties of sequences of operations
Theorem (Buchstaber-Erokhovets, 2015) Consider a sequence of edge-, (2, 6)- and (2, 7)- truncations that give a combinatorial fullerene P from the cube. Then The first and the second operations are edge-truncations; The first 5 operations can be chosen to be edge-truncations; If P is not a dodecahedron, then the first 7 operations can be chosen to be edge-truncations;
The last operation isa (2, 6)- or a (2, 7)-truncation, and F1 and F4 – quadrangle or pentagon. This gives two incident facets: a (k − 1)-gon F 0 and a pentagon F 00. The sequence of operations is not unique. Moreover, any pentagon F 00 and incident facet F 0 of a given fullerene appear as a pair (F 0, F 00) at the last step of some sequence of operations. 36 / 54 Simple partitions of 2-surfaces
A polygonal partition of a compact 2-surface M2 (closed or with boundary) is called simple, if the intersection of any two polygons is either empty or their common edge.
37 / 54 Simple partitions 2-surfaces
Any vertex of a simple partition has valency 3 if it is an interior point of a surface; 2 or 3 if it lies on the boundary.
Let
pk be a number of k-gons in a simple partition,
µi be the number of boundary vertices of valency i = 2, 3.
p4 = 3 p3 = 1, p4 = 2 µ2 − µ3 = 0, χ(M) = 1 µ2 − µ3 = −1, χ(M) = 1. 2 · 3 = 6 · 1 + 03 · 1 + 2 · 2 = 6 · 1 + 1 38 / 54 Simple partitions of 2-surfaces
For any simple partition of M2
2 X 3p3 + 2p4 + p5 = 6χ(M ) − δ + (k − 6)pk , k>7
where δ = µ2 − µ3
There are no hexagonal simple partitions of a closed surface if χ(M2) 6= 0. There exist hexagonal simple partitions of a torus and a Klein bottle.
39 / 54 Simple partitions of a disk into 5- and 6-gons
2 A disk D is a 2-surface homeomorphic to {z ∈ C: |z| 6 1}. For a disk χ(D2) = 1 and the formula has the form
p5 = 6 − δ =⇒ −∞ < δ 6 6
p5 = 0 ⇔ δ = 6; p5 = 6 ⇔ δ = 0. 2 There exist simple partitions of D with arbitrary p5 and p6.
40 / 54 Realization of simple partitions of a disk
p5 = 1, p6 = 5,δ = 5; p5 = 0, p6 = 7,δ = 6; p5 = 7, p6 = 2,δ = −1
41 / 54 Simple partitions of the cylinder into 5- and 6-gons
2 For the cylinder χ(M ) = 0 and p5 = −δ =⇒ δ 6 0. There are simple partitions of the cylinder for any
p5 + p6 > 3 =⇒ p6 > 3 + δ.
p5 = 0, p6 = 5,δ = 0; p5 = 0, p6 = 6,δ = 0; p5 = 5, p6 = 2,δ = −5 For simple partition of the cylinder into hexagons:
µ2 = µ3;
f0 = µ2 + 2p6, f1 = µ2 + 3p6, f2 = p6. 42 / 54 Applications to nanotubes
On each component of the boundary the number of 2-valent vertices is equal to the number of 3-valent.
Corollary Each disk on the end of a capped nanotube has exactly 6 pentagons.
capped (10, 10)-nanotube capped (10, 0)-nanotube 43 / 54 Construction of a nanobud
a) b) c)
a) – the Buckminsterfullerene C60 b) – a disk on a) partitioned into 3 pentagons and 4 hexagons. c) – excision.
44 / 54 Construction of a nanobud
d) e) f )
d) – the graphene sheet; e) – a disk on d) partitioned into 7 hexagons. f ) – excision.
45 / 54 Construction of a nanobud
g) h) i)
g) – C60 and graphene sheet with holes; h) – a cylinder partitioned into 3 heptagons and 3 octagons with boundary components equal to those on g). i) – a nanobud.
46 / 54 Result
We constructed a nanobud N with 9 pentagons; 3 heptagons; 3 octagons; χ(N) = 0. The numbers satisfy the equation
p5 = p7 + 2p8
47 / 54 Moment-angle functor
One of the main objects of the toric topology is the moment-angle functor K → ZK . It assigns to each simplicial complex K with m vertices m a space ZK with an action of a compact torus T , whose orbit m space ZK /T can be identified with the cone CK over K . To any simple partition P of a closed 2-surface into polygons we can associate the nerve-complex KP .
Vertices of KP are facets of P. Vertices of KP form a simplex if and only if the corresponding facets have nonempty common intersection.
Thus to partition P we associate the space ZP := ZKP .
48 / 54 Moment-angle manifolds
Boundary of a simple polytope P with m facets is a polygonal partition of the sphere S2.
In this case KP is the boundary of the dual simplicial polytope P∗.
The moment-angle complex ZP has the structure of a smooth manifold with a smooth action of T m. It is called a moment-angle manifold.
m The orbit space ZP /T can be identified with P itself.
Thus to any fullerene P we associate a smooth manifold ZP m with a smooth actions of the torus T , m = 12 + p6.
49 / 54 Stanley-Reisner ring of a simplicial complex
Let {v1,..., vm} be the set of vertices of a simplicial complex K . Then a Stanley-Reisner ring over Z is defined as
Z[K ] = Z[v1,..., vm]/(vi1 ... vik = 0, if {vi1 ,..., vik } 6∈ K ).
The Stanley-Reisner ring of a flag polytope is quadratic: the relations have only the form vi vj = 0: Fi ∩ Fj = ∅. Two polytopes are combinatorially equivalent if and only if their Stanley-Reisner rings are isomorphic.
Corollary The Stanley-Reisner ring of a fullerene is quadratic.
50 / 54 Multigraded complex
Let
∗ 2 R (K ) = Λ[u1,..., um] ⊗ Z[K ]/(ui vi , vi ), mdegui = (−1, 2{i}), mdegvi = (0, 2{i}), dui = vi , dvi = 0
be a multigraded differential algebra. Theorem (Buchstaber-Panov) We have an isomorphism
∗ ∗,∗ ∗ H[R (K ), d] ' Tor ( [K ], ) ' H (ZK , ) Z[v1,...,vm] Z Z Z
Moreover, this isomorphism defines the structure of a ∗ multigraded algebras in Tor and H (ZK , Z).
51 / 54 Toric topology of fullerenes
Main problems of toric topology of fullerenes: To build combinatorial invariants of fullerenes using methods of toric topology. To find such invariants that distinguish fullerenes from other simple 3-polytopes.
The talk of N. Erokhovets, Fullerenes and combinatorics of 3-dimensional flag polytopes is devoted to the results concerning this problems.
52 / 54 ReferencesI
V.M.Buchstaber, T.E.Panov, Toric Topology AMS Math. Surveys and monogrpaphs. vol. 204, 2015. 518 pp.
V.M.Buchstaber, N. Erokhovets, Truncations of simple polytopes and applications Proc. of Steklov Math Inst, MAIK, Moscow, vol. 289, pp. 104-173, 2015.
X. Lu, Z. Chen,
Curved Pi-Conjugtion, Aromaticity, and the Related Chemistry of Small Fullerenes (C60) and Single-Walled Carbon Nanotubes. Chemical Reviews 105:10(2005), 3643-3696.
M.Deza, M.Dutour Sikiric, M.I.Shtogrin, Fullerenes and disk-fullerenes Russian Math. Surveys, 68:4(2013), 665-720.
V.D.Volodin, Combinatorics of flag simplicial 3-polytopes Russian Math. Surveys, 70:1(2015); arXiv: 1212.4696.
B. Grünbaum, Convex polytopes Graduate texts in Mathematics 221, Springer-Verlag, New York, 2003.
53 / 54 Thank You for the Attention!
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