Narrow Results

A real 3- or 4-manifold has by definition an orientation preserving smooth involution acting on it. We consider Lefschetz fibrations of 4-dimensional manifolds-with-boundary and open book decompositions on their boundary in the existence of a real structure. We prove that there is a real open book which cannot be filled by a real Lefschetz fibration, although it is filled by non-real Lefschetz fibrations.

Let $S_g$ be a closed and oriented surface of genus $g\ge 2$. A closed curve $\gamma$ on $S_g$ is said to fill$S_g$ (or simply be filling), if its complement in the surface is a disjoint union of topological discs. It is assumed that the curve $\gamma$ is always in minimal position. To a filling curve, we associate a number $b$, the number of topological discs in its complement. For $b=1$, such a filling curve is called minimally intersecting. We prove that for every $b\ge 1,$ there exists a filling curve ${\gamma }_b$ on $S_g$ whose complement is a disjoint union of $b$ many topological discs. Furthermore, we provide an upper bound on the number of mapping class group orbits of closed curves which fills $S_g$ minimally.

Carbon Nanotubes (CNTs) filled with metals can be used in capacitors, sensors, rechargeable batteries and so on. In this study, the process of Nickel filling into single wall CNTs was studied by molecular dynamics (MD) simulation. Three models consisting of Nickel atoms and CNTs were established. These models were cooled from 1500 K to 100 K to analyze the factors that influence the filling height, such as temperature, the force between Carbon and Nickel atoms, as well as CNTs diameter. The results showed that filling height increased as the temperature and the force between Carbon–Nickel atoms rised. Filling height reduced with the increasing diameter of CNTs.

Let $F_g$ denote a closed oriented surface of genus $g$. A set of simple closed curves is called a filling of $F_g$ if its complement is a disjoint union of discs. The mapping class group $\text{Mod}(F_g)$ of genus $g$ acts on the set of fillings of $F_g$. The union of the curves in a filling forms a graph on the surface which is a so-called decorated fat graph. It is a fact that two fillings of $F_g$ are in the same $\text{Mod}(F_g)$-orbit if and only if the corresponding fat graphs are isomorphic. We prove that any filling of $F_2$ whose complement is a single disc (i.e. a so-called minimal filling) has either three or four closed curves and in each of these two cases, there is a unique such filling up to the action of $\text{Mod}(F_2)$.

We provide a constructive proof to show that the minimum number of discs in the complement of a filling pair of $F_2$ is two. Finally, given positive integers $g$ and $k$ with $(g,k)\ne (2,1)$, we construct a filling pair of $F_g$ such that the complement is a union of $k$ topological discs.