The plain sphere number of a link

Ryan Blair, Alexandra Kjuchukova and Ella Pfaff

Vol. 342 (2026), No. 2, 217–233

DOI: 10.2140/pjm.2026.342.217

Abstract

Let $L$ be a link in $S^3$. We consider a natural class of meridional presentations for ${\pi }_1(S^3\setminus L)$ in which the relations are witnessed by certain embedded two-spheres easily observed in a fixed diagram of $L$. The Wirtinger presentation is a special case. We prove that the bridge number of $L$ equals the smallest number of generators of ${\pi }_1(S^3\setminus L)$ over all such presentations.

Keywords

knot, link, bridge number, meridian, Wirtinger presentation, fundamental group, link exterior

Mathematical Subject Classification

Primary: 57K10, 57K31, 57M05

Milestones

Received: 24 June 2025

Revised: 20 February 2026

Accepted: 9 March 2026

Published: 23 April 2026