Adding a suitable unknot to any link equates bridge number and meridional rank

TL;DR

This paper demonstrates how adding an unknot to any link can satisfy the Meridional Rank Conjecture, establishing a new link between bridge number and meridional rank, and proves the conjecture for new link families.

Contribution

It introduces a method to embed an unknot in a link's complement to satisfy the MRC and extends the conjecture's validity to new infinite link families.

Findings

Bridge number of the augmented link equals 2 times the original minus 1.

The augmented link's meridional rank matches its bridge number.

The MRC is proven for new infinite families of links.

Abstract

Given any link $L\subseteq S^3$, we show that it is possible to embed an unknot $U$ in its complement so that the link $L\cup U$ satisfies the Meridional Rank Conjecture (MRC). The bridge numbers in our construction fit into the equality $\beta(L\cup U)=2\beta(L)-1=\text{rank}(\pi_1(S^3\backslash (L\cup U)))$. In addition, we prove the MRC for new infinite families of links and distinguish them from previously settled cases through an application of bridge distance.