Minimum lattice length and ropelength of 2-bridge knots and links

TL;DR

This paper establishes linear upper bounds for the minimum lattice length and ropelength of 2-bridge knots and links, providing new insights into their geometric complexity related to crossing number.

Contribution

It introduces the first linear upper bounds for these quantities specifically for 2-bridge knots and links, advancing understanding of their geometric properties.

Findings

Linear upper bound for minimum lattice length: $ ext{Len}(K) \\leq 8 c(K) + 2$

Linear upper bound for minimum ropelength: $ ext{Rop}(K) \\leq 11.39 c(K) + 12.37$

Focus on 2-bridge knots and links to derive these bounds.

Abstract

Knots are commonly found in molecular chains such as DNA and proteins, and they have been considered to be useful models for structural analysis of these molecules. One interested quantity is the minimum number of monomers necessary to realize a molecular knot. The minimum lattice length $\mbox{Len}(K)$ of a knot $K$ indicates the minimum length necessary to construct $K$ in the cubic lattice. Another important quantity in physical knot theory is the ropelength which is one of knot energies measuring the complexity of knot conformation. The minimum ropelength $\mbox{Rop}(K)$ is the minimum length of an ideally flexible rope necessary to tie a given knot $K$. Much effort has been invested in the research project for finding upper bounds on both quantities in terms of the minimum crossing number $c(K)$ of the knot. It is known that $\mbox{Len}(K)$ and $\mbox{Rop}(K)$ lie between $\mbox{O}(c(K)^{\frac{3}{4}})$ and $\mbox{O}(c(K) [\ln (c(K))]^5)$, but unknown yet whether any family of knots has superlinear growth. In this paper, we focus on 2-bridge knots and links. Linear growth upper bounds on the minimum lattice length and minimum ropelength for nontrivial 2-bridge knots or links are presented: $\mbox{Len}(K) \leq 8 c(K) + 2$. $\mbox{Rop}(K) \leq 11.39 c(K) + 12.37$.