Linking Structures

As no rings of an iron chain can be separated without cutting, no two triangles in 3-space with one edge piercing through the interior of the other can be separated by any continuous transformation without intersecting their boundaries. Those triangles are called linked (germ: verschlungen). In graph theory it is known, that any straight line embedding of the complete graph of 6 vertices in 3-space contains a pair of linked triangles. Using the technique of Gale diagrams, we can state this more precisely: either one pair or exactly three pairs of linked triangles exist. Clearing the more general case of pairs of simplices in (d+3) vertices in d-space (d odd) and a linking property of (d+4) vertices lead to the axiomatization of (abstract) linking structures including an orientation information, induced by the topological linking number. These structures are geometrically motivated but combinatorial objects and emerged from unpublished concepts of Prof. U. Brehm (TU-Dresden). Oriented matroids and computer-supported proofs allow us to completely classify all geometrical and oriented-matroid realizable linking structures of 6, 7 and 8 points in 3-space as well as 9 points in 5-space.