Ribbonlength of folded ribbon unknots in the plane

Folded ribbonlength of 2-bridge knots

A ribbon is a two-dimensional object with one-dimensional properties which is related with geometry, robotics and molecular biology. A folded ribbon structure provides a complex structure through a series of folds. We focus on a folded ribbon with knotted core. The folded ribbonlength $Rib(K)$ of a knot $K$ is the infimum of the quotient of length by width among the ribbons representing a knot type of $K$. This quantity tells how efficiently the folded ribbon is realized. Kusner conjectured that folded ribbonlength is bounded by a linear function of the minimal crossing number $c(K)$. In this …

小交叉数纽结与链环的折带长度上界

Ribbonlength upper bounds for small crossing knots and links

本文提出构造折叠带状纽结的新方法，改进了 $(2,q)$-环面链环、扭转纽结和椒盐卷饼链环的折叠带长上界，并给出八字结为 8 的构造与最优性猜想。

折叠丝带结族的丝带长度

Ribbonlength of families of folded ribbon knots

本文研究折叠丝带结的丝带长度，给出多类结的线性上界，并构造(p,q)环面结的次线性上界，证明三叶结可用丝带长度6实现。

Folded ribbon knots in the plane

This survey reviews Kauffman's model of folded ribbon knots: knots made of a thin strip of paper folded flat in the plane. The ribbonlength is the length to width ratio of such a ribbon, and the ribbonlength problem asks to minimize the ribbonlength for a given knot type. We give a summary of known results. For the most part, these are upper bounds of ribbonlength of twist knots and certain families of torus knots. We discuss result of G. Tian, which give upper bounds of ribbonlength in terms of crossing number. In addition, it turns out the choice of fold affects the ribbonlength. We end with…

Ribbonlength and crossing number for folded ribbon knots

We study Kauffman's model of folded ribbon knots: knots made of a thin strip of paper folded flat in the plane. The ribbonlength is the length to width ratio of such a folded ribbon knot. We show for any knot or link type that there exist constants $c_1, c_2>0$ such that the ribbonlength is bounded above by $c_1\cdot Cr(K)^2$, and also by $c_2\cdot Cr(K)^{3/2}$. We use a different method for each bound. The constant $c_1$ is quite small in comparison to $c_2$, and the first bound is lower than the second for knots and links with $Cr(K)\leq$ 12,748.

结带长度界限与不超过9个交叉的结与结带链

Ribbonlength bounds for pretzel links and knots with $\leq 9$ crossings

本文研究了不超过9个交叉的结与结带链的结带长度界限，证明了结带链的最小折叠结带长度存在统一上界，并提供了相关数据表。