Links, bridge number, and width trees

TL;DR

This paper introduces width trees as a new combinatorial tool to analyze links in 3-dimensional space, relating them to classical invariants and establishing bounds and uniqueness results based on geometric structures.

Contribution

It defines width trees for links, interprets link invariants through these trees, and proves bounds and uniqueness results linking geometric and topological properties.

Findings

Width trees can bound link invariants from below.

Each width tree corresponds to a knot with certain invariants.

High-distance width trees are uniquely determined by their invariants.

Abstract

To each link $L$ in $S^3$ we associate a collection of certain labelled directed trees, called width trees. We interpret some classical and new topological link invariants in terms of these width trees and show how the geometric structure of the width trees can bound the values of these invariants from below. We also show that each width tree is associated with a knot in $S^3$ and that if it also meets a high enough "distance threshold" it is, up to a certain equivalence, the unique width tree realizing the invariants.