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

Ryan Blair; Alexandra Kjuchukova; Ella Pfaff

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arXiv · 2411.10642

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

Ryan Blair·Alexandra Kjuchukova·Ella Pfaff

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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.

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Ryan Blair, Alexandra Kjuchukova, Ella Pfaff. 2024-11-16. Adding a suitable unknot to any link equates bridge number and meridional rank. https://arxiv.org/abs/2411.10642

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