The most tight conformations of prime knots are found with the use of the SONO algorithm. Their curvature and torsion profiles are calculated. Symmetry of the knots is analysed. Connections with the physics of polymers are discussed.

Closely packed conformations of helices formed on the ideal rope are considered. The pitch versus radius relations which define a closely packed helix are determined. The relations stem from the turn-to-turn distance and curvature limiting conditions. Plots of the relations are shown to cross each other. The physical sense of the crossing point is discussed.Comment: 14 pages, 10 figure

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The physical sense of tight knots provided by the SONO algorithm is discussed. A method allowing one to predict their length is presented. An upper bound for the minimum length of a smooth trefoil knot is determined.

A set of self-contact points of the most tight, parametrically tied trefoil knot is determined. The knot is subjected to further tightening procedure based on the shrink-on-no-overlaps algorithm. Changes in the structure of the set of the self-contact points are monitored and the final form of the set is determined.