# Braid Index Bounds Ropelength From Below

## TL;DR

This paper establishes a lower bound on the ropelength of any link based on its absolute braid index, linking geometric and topological properties of links.

## Contribution

It introduces the concept of absolute braid index and proves it provides a universal lower bound for the ropelength of links.

## Key findings

- Ropelength is bounded below by a constant times the absolute braid index.
- Defines the absolute braid index as the maximum braid index over all orientations.
- Establishes a fundamental link between geometric length and topological complexity.

## Abstract

For an un-oriented link $\mathcal{K}$, let $L(\mathcal{K})$ be the ropelength of $\mathcal{K}$. It is known that when $\mathcal{K}$ has more than one component, different orientations of the components of $\mathcal{K}$ may result in different braid index. We define the largest braid index among all braid indices corresponding to all possible orientation assignments of $\mathcal{K}$ the {\em absolute braid index} of $\mathcal{K}$ and denote it by $\textbf{B}(\mathcal{K})$. In this paper, we show that there exists a constant $a>0$ such that $L(\mathcal{K})\ge a \textbf{B}(\mathcal{K}) $ for any $\mathcal{K}$, {\em i.e.}, the ropelength of any link is bounded below by its absolute braid index (up to a constant factor).