The bridge number of arborescent links with many twigs

TL;DR

This paper proves the meridional rank conjecture for a class of arborescent links with specific branching properties, establishing an upper bound on their bridge number related to the structure of their underlying trees.

Contribution

It introduces a new proof for the meridional rank conjecture for arborescent links with certain branching conditions, and provides an upper bound on the bridge number based on the tree components.

Findings

Proved the meridional rank conjecture for a class of arborescent links.

Established an upper bound on the bridge number in terms of the tree components.

Applicable to all arborescent links with specified branching properties.

Abstract

We prove the meridional rank conjecture for arborescent links associated to plane trees with the following property: all branching points carry a straight branch to at least three leaves. The proof involves an upper bound on the bridge number in terms of the maximal number of link components of the underlying tree, valid for all arborescent links.