Piecewise-linear embeddings of decussate extended θ graphs and tetrahedra

An n theta graph is an n-valent graph with two vertices. From symmetry considerations, it has vertex-edge transitivity 1 1. Here, they are considered extended with divalent vertices added to the edges to explore the simplest piecewise-linear tangled embeddings with straight, non-intersecting edges (sticks). The simplest tangles found are those with 3n sticks, transitivity 2 2, and with 2 L(n - 1)/2] ambient-anisotopic tangles. The simplest finite and 1-, 2-and 3-periodic decussate structures (links and tangles) are described. These include finite cubic and icosahedral and 1- and 3-periodic links, all with minimal transitivity. The paper also presents the simplest tangles of extended tetrahedra and their linkages to form periodic polycatenanes. A vertex- and edge-transitive embedding of a tangled srs net with tangled and polycatenated theta graphs and vertex-transitive tangled diamond (dia) nets are described.