2013/12/21 by Peter V. Pikhitsa, Pikhitsa, Peter V., Mansoo Choi +1
Engineering · Computer Science · #Advanced Materials and Mechanics #Music Technology and Sound Studies
paper · pdf · doi:10.48550/arxiv.1312.6207
It has been a challenge to make seven straight round cylinders mutually touch\nbefore our now 10-year old discovery [Phys. Rev. Lett. 93, 015505 (2004)] of\nconfigurations of seven mutually touching infinitely long round cylinders (then\ncoined 7-knots). Because of the current interest in string-like objects and\nentanglement which occur in many fields of Physics it is useful to find a\nsimple way to treat ensembles of straight infinite cylinders. Here we propose a\ntreatment with a chirality matrix. By comparing 7-knot with variable radii with\nthe one where all cylinders are of equal radii (here 7*-knot, which for the\nfirst time appeared in [phys. stat. solidi, b 246, 2098 (2009)]), we show that\nthe reduction of 7-knot with a set of non-equal cylinder radii to 7*-knot of\nequal radii is possible only for one topologically unique configuration, all\nother 7-knots being of different topology. We found novel configurations for\nmutually touching infinitely long round cylinders when their numbers are eight\nand ultimately nine (here coined 8-knots and 9-knots). Unlike the case of\n7-knot, where one angular parameter (for a given set of fixed radii) may change\nby sweeping a scissor angle between two chosen cylinders, in case of 8- and\n9-knots their degrees of freedom are completely exhausted by mutual touching so\nthat their configurations are "frozen" for each given set of radii. For 8-knot\nthe radii of any six cylinders may be changeable (for example, all taken equal)\nwhile two remaining are uniquely determined by the others. We show that 9-knot\nmakes the ultimate configuration where only three cylinders can have changeable\nradii and the remaining six are determined by the three. Possible\ngeneralizations and connection with Physics are mentioned.\n