Helical close packings of ideal ropes

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

The ideal trefoil knot

The most tight conformation of the trefoil knot found by the SONO algorithm is presented. Structure of the set of its self-contact points is analyzed.Comment: 11 pages, 8 figure

Tight open knots

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.Comment: 11 pages, 8 figure

The Shapes of Tight Composite Knots

We present new computations of tight shapes obtained using the constrained gradient descent code RIDGERUNNER for 544 composite knots with 12 and fewer crossings, expanding our dataset to 943 knots and links. We use the new data set to analyze two outstanding conjectures about tight knots, namely that the ropelengths of composite knots are at least 4\pi-4 less than the sums of the prime factors and that the writhes of composite knots are the sums of the writhes of the prime factors.Comment: Summary text file of tight knot lengths and writhing numbers stored in anc/ropelength_data.txt. All other data freely available at http:://www.jasoncantarella.com/ and through Data Conservanc

Curvature and torsion of the tight closed trefoil knot

The curvature and torsion profiles of the tight trefoil knot found with the SONO algorithm are analysed. Their characteristics are discussed. Comparison of curvature and torsion profiles with the tightest parametrically defined knot is performed