Title: Mutually touching infinite cylinders in the 3D world of lines,
Abstract: Recently we gave arguments that only two unique topologically different
configurations of 7 equal all mutually touching round cylinders (the
configurations being mirror reflections of each other) are possible in 3D,
although a whole world of configurations is possible already for round
cylinders of arbitrary radii. It was found that as many as 9 round cylinders
(all mutually touching) are possible in 3D while the upper bound for arbitrary
cylinders was estimated to be not more than 14 under plausible arguments. Now
by using the chirality and Ring matrices that we introduced earlier for the
topological classification of line configurations, we have given arguments that
the maximal number of mutually touching straight infinite cylinders of
arbitrary cross-section (provided that its boundary is a smooth curve) in 3D
cannot exceed 10. We generated numerically several configurations of 10
cylinders, restricting ourselves with elliptic cylinders. Configurations of 8
and 9 equal elliptic cylinders (all in mutually touching) are generated
numerically as well. A possibility and restriction of continuous
transformations from elliptic into round cylinder configurations are discussed.
Some curious results concerning the properties of the chirality matrix (which
coincides with Seidel's adjacency matrix important for the Graph theory) are
presented.