  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.

Metric Geometry