Ropelength-minimizing concentric helices and non-alternating torus knots

An alternating torus knot or link may be constructed from a repeating double helix after connecting its two ends. A structure with additional helices may be closed to form a non-alternating torus knot or link. Previous work has optimized the dimensions and pitch of double helices to derive upper bounds on the ropelength of alternating torus knots, but non-alternating knots have not been studied extensively and are known to be tighter. Here, we examine concentric helices as units of non-alternating torus knots and discuss considerations for minimizing their contour length. By optimizing both the geometry and combinatorics of the helices, we find efficient configurations for systems with between 3 and 39 helices. By using insights from those cases, we develop an efficient construction for larger systems and show that concentric helices distributed between many shells have an optimized ropelength of approximately <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>7.83</mml:mn> <mml:msup> <mml:mi>Q</mml:mi> <mml:mrow> <mml:mn>3</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> </mml:math> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Q</mml:mi> </mml:math> is the total number of helices or the minor index of the torus knot, and the prefactor is exact and a 75% reduction from previous work. Links formed by extending these helices and bending them into a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>T</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3</mml:mn> <mml:mi>Q</mml:mi> <mml:mo>,</mml:mo> <mml:mi>Q</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> torus link have a ropelength that is approximately 12 times the three-quarter power of the crossing number. These results reduce the ratio between the upper and lower bounds of the ropelength of non-alternating torus knots from 29 to between 1.4 and 3.8.