Bridge numbers and meridional ranks of knotted surfaces and welded knots

TL;DR

This paper explores the meridional rank conjecture for knotted surfaces in four-dimensional space, establishing cases where bridge number equals meridional rank and examining their properties through constructions, twist-spinning, and welded knot analysis.

Contribution

It introduces a construction for classical knots with specific meridian quotients, proves the equality of bridge number and meridional rank for certain knotted spheres, and relates welded knot bridge numbers to ribbon tori.

Findings

Bridge number equals meridional rank for certain knotted spheres.

Meridional rank is not additive under connected sum for knotted spheres.

Relationship established between welded knot bridge numbers and ribbon tori.

Abstract

The Meridional Rank Conjecture asks whether the bridge number of a knot in $S^3$ is equal to the minimal number of meridians needed to generate the fundamental group of its complement. In this paper we investigate the analogous conjecture for knotted surfaces in $S^4$. Towards this end, we give a construction to produce classical knots with quotients sending meridians to elements of any finite order and which detect their meridional ranks. We establish the equality of bridge number and meridional rank for these knots and knotted spheres obtained from them by twist-spinning. On the other hand, we show that the meridional rank of knotted spheres is not additive under connected sum, so that either bridge number also collapses, or meridional rank is not equal to bridge number for knotted spheres. We also show a relationship between the bridge numbers of welded knots and ribbon tori using the Tube map, and give applications to bridge trisections of knotted surfaces.