Adding a suitable unknot to any link equates bridge number and meridional rank
Ryan Blair, Alexandra Kjuchukova, Ella Pfaff · Mathematical Proceedings of the Cambridge Philosophical Society · 2026
Abstract Given any link upper L subset of or equal to upper S cubed $L\subseteq S^3$ L ⊆ S 3 , we show that it is possible to embed an unknot U in its complement so that the link upper L union upper U $L\cup U$ L ∪ U satisfies the Meridional Rank Conjecture (MRC). The bridge numbers in our construction fit into the equality beta left parenthesis upper L union upper U right parenthesis equals 2 beta left parenthesis upper L right parenthesis minus 1 equals rank left parenthesis pi 1 left parenthesis upper S cubed minus left parenthesis upper L union upper U right parenthesis right parenthesis right parenthesis $\beta(L\cup U)=2\beta(L)-1=\text{rank}(\pi_1(S^3\backslash (L\cup U)))$ β ( L ∪ U ) = 2 β ( L ) − 1 = rank ( π 1 ( S 3 ∖ ( L ∪ 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.