Communications in Analysis and Geometry

Volume 28 (2020)

Number 2

Wirtinger systems of generators of knot groups

Pages: 243 – 262

DOI: https://dx.doi.org/10.4310/CAG.2020.v28.n2.a2

Authors

R. Blair (Department of Mathematics, California State University, Long Beach, Cal., U.S.A.)

A. Kjuchukova (Max Planck Institute for Mathematics, Bonn, Germany)

R. Velazquez

P. Villanueva (Department of Agricultural and Biosystems Engineering, Iowa State University, Ames, Ia., U.S.A.)

Abstract

We define the Wirtinger number of a link, an invariant closely related to the meridional rank. The Wirtinger number is the minimum number of generators of the fundamental group of the link complement over all meridional presentations in which every relation is an iterated Wirtinger relation arising in a diagram. We prove that the Wirtinger number of a link equals its bridge number. This equality can be viewed as establishing a weak version of Cappell and Shaneson’s Meridional Rank Conjecture, and suggests a new approach to this conjecture. Our result also leads to a combinatorial technique for obtaining strong upper bounds on bridge numbers. This technique has so far allowed us to add the bridge numbers of approximately 50,000 prime knots of up to 14 crossings to the knot table. As another application, we use theWirtinger number to show there exists a universal constant $C$ with the property that the hyperbolic volume of a prime alternating link $L$ is bounded below by $C$ times the bridge number of $L$.

Received 4 May 2017

Accepted 30 October 2017

Published 6 May 2020