Adding a suitable unknot to any link equates bridge number and meridional rank
Abstract
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.
Fields of Study & Subjects
Computational Geometry and Mesh Generation
Advanced Graph Theory Research
Mathematics and Applications
Publication details
- Journal / source
- Mathematical Proceedings of the Cambridge Philosophical Society
- Publisher
- Cambridge University Press (CUP)
- Volume
- 181
- Issue
- 3
- Pages
- 1–17
- Publication date
- 29 July 2026
- Language
- EN