See previous submission from 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. A 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 7.83<i>Q</i>^3/2, where <i>Q</i> is the total number of helices or the minor index of the torus knot, and the prefactor is exact and a 75% reduction from the previous work. Links formed by extending these helices and bending them into a <i>T</i>(3<i>Q</i>, <i>Q</i>) 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.

交替环面纽结或环链（alternating torus knot or link）可通过将重复双螺旋（double helix）的两端相连构造得到。带有额外螺旋的结构可闭合为非交替环面纽结或环链（non-alternating torus knot or link）。此前已有研究通过优化双螺旋的尺寸与螺距，推导得到交替环面纽结绳长（ropelength）的上界，但非交替纽结尚未得到广泛研究，且已知其绳长更为紧凑。
本文以同心螺旋（concentric helices）作为非交替环面纽结的结构单元，探讨了最小化其轮廓长度的相关考量。通过优化螺旋的几何构型与组合方式，我们得到了包含3至39个螺旋的系统的高效构型。基于上述案例得到的启示，我们进一步提出了适用于更大规模系统的高效构造方法，并证明：分布于多层壳层中的同心螺旋，其优化后的绳长约为7.83<i>Q</i>^3/2，其中<i>Q</i>为螺旋总数量或环面纽结的副指标（minor index），该前因子（prefactor）为精确值，相较于此前研究降低了75%。
将这些螺旋延伸并弯折为<i>T</i>(3<i>Q</i>, <i>Q</i>)型环面链所形成的环链，其绳长约为交叉数（crossing number）的四分之三次方的12倍。上述结果将非交替环面纽结绳长的上下界之比从29缩小至1.4至3.8之间。