Seven, eight, and nine mutually touching infinitely long straight round cylinders: Entanglement in Euclidean space

Abstract: It has been a challenge to make seven straight round cylinders mutually touch before our now 10-year old discovery [Phys. Rev. Lett. 93, 015505 (2004)] of configurations of seven mutually touching infinitely long round cylinders (then coined 7-knots). Because of the current interest in string-like objects and entanglement which occur in many fields of Physics it is useful to find a simple way to treat ensembles of straight infinite cylinders. Here we propose a treatment with a chirality matrix. By comparing 7-knot with variable radii with the one where all cylinders are of equal radii (here 7*-knot, which for the first time appeared in [phys. stat. solidi, b 246, 2098 (2009)]), we show that the reduction of 7-knot with a set of non-equal cylinder radii to 7*-knot of equal radii is possible only for one topologically unique configuration, all other 7-knots being of different topology. We found novel configurations for mutually touching infinitely long round cylinders when their numbers are eight and ultimately nine (here coined 8-knots and 9-knots). Unlike the case of 7-knot, where one angular parameter (for a given set of fixed radii) may change by sweeping a scissor angle between two chosen cylinders, in case of 8- and 9-knots their degrees of freedom are completely exhausted by mutual touching so that their configurations are "frozen" for each given set of radii. For 8-knot the radii of any six cylinders may be changeable (for example, all taken equal) while two remaining are uniquely determined by the others. We show that 9-knot makes the ultimate configuration where only three cylinders can have changeable radii and the remaining six are determined by the three. Possible generalizations and connection with Physics are mentioned.