Narrow Results

It is well known that there is a one-to-one correspondence between signed plane graphs and link diagrams via the medial construction. The component number of the corresponding link diagram is however independent of the signs of the plane graph. Determining this number may be one of the first problems in studying links by using graphs. Some works in this aspect have been done. In this paper, we investigate the component number of links corresponding to lattices. Firstly we provide some general results on component number of links. Then, via these results, we proceed to determine the component number of links corresponding to lattices with free or periodic boundary conditions and periodic lattices with one cap (i.e. spiderweb graphs) or two caps.

We consider the attractor T of injective contractions f₁, …, f_m on R² which satisfy the Open Set Condition. If T is connected, then T's interior T° is either empty or has no holes, and T's boundary ∂T is connected; if further T° is non-empty and connected, then ∂T is a simple closed curve, thus T is homeomorphic to the unit disk {x∈R²: |x|≤1}.