# Satellite constructions and geometric classification of Brunnian links

## TL;DR

This paper introduces new satellite constructions called the satellite sum and tie, enabling the generation of infinitely many Brunnian links and providing a unique decomposition theorem based on a labelled tree structure.

## Contribution

It presents novel satellite constructions for Brunnian links and a decomposition theorem that uniquely characterizes their structure using a labelled tree.

## Key findings

- Constructed two new satellite operations for Brunnian links.
- Proved the ability to generate infinitely many new Brunnian links.
- Established a unique tree-arrow structure for classifying Brunnian links.

## Abstract

In this paper, we construct two families of satellite constructions for Brunnian links, called the satellite sum and the satellite tie. An interesting fact is that by applying the satellite sum and the satellite tie constructions, we can build infinitely many new Brunnian links from any given Brunnian links. With the helps of the satellite sum and the satellite tie, we give a new decomposition theorem for Brunnian links. We prove that every Brunnian link determines a unique labelled "tree-arrow structure" such that each vertex of the tree represents a generalized Hopf link, a hyperbolic Brunnian link, or a hyperbolic Brunnian link in an unlink-complement.