The 'simplest' entanglements of the graph of edges of the cube are enumerated, forming two-cell {6, 3} (hexagonal mesh) complexes on the genus-one two-dimensional torus. Five chiral pairs of knotted graphs are found. The examples contain non-trivial knotted and/or linked subgraphs [(2, 2), (2, 4) torus links and (3, 2), (4, 3) torus knots].

There has been renewed interest in recent years in McKinsey and Tarski’s interpretation of modal logic in topological spaces and their proof that S4 is the logic of any separable dense-in-itself metric space. Here we extend this work to the modal mu-calculus and to a logic of tangled closure operators that was developed by Fernández-Duque after these two languages had been shown by Dawar and Ot...

On the Shapes of Tangled Curves

In the first picture in Figure 1 there are three mutually tangent circles packed in a large circle on the outside, with four curvilinear triangles in-between. Such a configuration of circles is called a Descartes configuration and is the starting point for a bounded Apollonian circle packing (ACP). One may also consider an unbounded Apollonian circle packing in which the original Descartes conf...

The paper deals with the problem of optimal packing of two sets of circles (2-D spheres) into a simply connected container. The number of circles is given. The radii of these circles are equal within each set, but, generally speaking, they differ between sets. There are two different statements of a such problem. The simplest one is when the circles of a larger radius are located first, and the...