Meridional rank and bridge number for a class of links

TL;DR

This paper establishes a relationship between meridional rank and bridge number for certain links, showing that specific conditions imply the links are 3-bridge, and also provides a lower bound for satellite knots.

Contribution

It proves that links with meridional rank 3 and certain 2-fold branched covers are 3-bridge links, and shows the meridional rank of satellite knots is at least 4.

Findings

Links with meridional rank 3 and graph manifold covers are 3-bridge links

Meridional rank of satellite knots is at least 4

Partial answer to the relation between bridge number and meridional rank

Abstract

We prove that links with meridional rank 3 whose 2-fold branched covers are graph manifolds are 3-bridge links. This gives a partial answer to a question by S. Cappell and J. Shaneson on the relation between the bridge numbers and meridional ranks of links. To prove this, we also show that the meridional rank of any satellite knot is at least 4.