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

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.

arXiv comment14 pages, 4 figures, 4 footnotes, 1 stanza