The fundamental group and Galois coverings of hexagonal systems in 3-space

We consider hexagonal systems embedded into the 3-dimensional space R3. We define the fundamental group π1(G) of such a system G and show that in case G is a finite hexagonal system with boundary, then π1(G) is a (non-abelian) free group. In this case, the rank of π1(G) equals m(G) − h(G) − n(G) + 1 where n(G) (resp. m(G), h(G)) denotes the number of vertices (resp. edges, hexagons) in G. A hexagonal system is a finite connected graph with all edges lying in regular hexagons. Hexagonal systems have been extensively studied, as the natural representations of hydrocarbon molecules, analyzing various physico-chemical properties of the molecules represented. Spectral graph theory is often employed for this purpose because the algebraic invariants associated to the graphs contain relevant information about the molecular structure, see [2, 9]. Additionally, the application of topological techniques has become increasingly important in the field over the past years, see [6, 8]. The consideration of benzenoid hydrocarbons as planar hexagonal systems is a well developed theory [2, 9]. As first discussed by Wasserman [14], one may conceive of classes of hydrocarbons, including knotted rings and linked rings (catananes), with interesting topological properties. On this trend, in [13] it was reported the synthesis of the first molecular Möbius strip and in [5], the synthesis of a molecular trefoil knot. The development of nanotechnology makes relevant the consideration of nanotubes and other related hexagonal systems in 3-space, see [10]. In this work we consider hexagonal systems, embedded into the 3-dimensional space R. Given (a possibly infinite) hexagonal system G stabilized by the action of a group Γ which acts freely on the vertices of G, the natural quotient π: G → G/Γ is said to be a Galois covering defined by the action of Γ. For a finite hexagonal system G, we build a universal Galois covering π: G̃ → G, (that is, for any Galois covering π′: G′ → G, there is a unique morphism π̄: G̃ → G′ with π = π′π̄). The map π is defined by the action of the fundamental group π1(G). This group π1(G) is also the fundamental group of the CW -realization cw(G) of the hexagonal system G. We say that an edge e in G is on the boundary if e belongs only to one hexagon in G. In case G has no boundary and cw(G) is orientable, a well-known argument implies that cw(G)