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

Ryan Blair; Alexandra Kjuchukova; Ella Pfaff

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

Geometric Topology 2024-11-19 v1

Authors: Ryan Blair , Alexandra Kjuchukova , Ella Pfaff

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.

Keywords

bridge number meridional rank conjecture unknot link complement bridge distance knot theory

Cite

@article{arxiv.2411.10642,
  title  = {Adding a suitable unknot to any link equates bridge number and meridional rank},
  author = {Ryan Blair and Alexandra Kjuchukova and Ella Pfaff},
  journal= {arXiv preprint arXiv:2411.10642},
  year   = {2024}
}

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14 pages, 4 figures, 4 footnotes, 1 stanza

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

Abstract

Keywords

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