Folded ribbonlength of 2-bridge knots

TL;DR

This paper proves that the folded ribbonlength of 2-bridge knots is linearly bounded by twice their minimal crossing number plus two, confirming a conjecture about the efficiency of representing such knots with folded ribbons.

Contribution

It establishes an upper bound of 2c(K)+2 for the folded ribbonlength of 2-bridge knots, advancing understanding of knot complexity and ribbon efficiency.

Findings

Folded ribbonlength of 2-bridge knots is bounded by 2c(K)+2.

Confirms Kusner's conjecture for 2-bridge knots.

Provides a linear bound relating knot complexity to ribbon length.

Abstract

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 paper, we confirm that the folded ribbonlength of a 2-bridge knot $K$ is bounded above by $2c(K)+2$.