Efficiency of non-identical double helix patterns in minimizing ropelength of torus knots

Abstract

The ropelength of a knotted string with volume is defined as the ratio of the length of its central curve to the radius of its sectional disc. In a physical context, achieving minimal ropelength corresponds to a state of minimal potential energy, and geometrically, it signifies a tightly-packed conformation. The quest to establish a connection between the topological complexity of knotted strings and their minimal ropelength has persisted into recent years. In this paper, we introduce a new upper bound on the minimal ropelength of (2, n)-torus knots and links: Rop(T(2, n)) <= 7.3163 Cr(T(2, n)) + 17.1657. This upper bound is derived from a torus knot conformation constructed based on a tightened pattern of double helix with non-identical radii of winding. A comparative analysis with conformations generated from a superhelix and a circular helix underscores the efficiency of the non-identical double helix pattern, particularly when it appears as a long repeated motif in knotted strings.