Wirtinger systems of generators of knot groups

摘要

We define the {it Wirtinger number} of a link, an invariant closely relatedto the meridional rank. The Wirtinger number is the minimum number ofgenerators of the fundamental group of the link complement over all meridionalpresentations in which every relation is an iterated Wirtinger relation arisingin a diagram. We prove that the Wirtinger number of a link equals its bridgenumber. This equality can be viewed as establishing a weak version of Cappelland Shaneson's Meridional Rank Conjecture, and suggests a new approach to thisconjecture. Our result also leads to a combinatorial technique for obtainingstrong upper bounds on bridge numbers. This technique has so far allowed us toadd the bridge numbers of approximately 50,000 prime knots of up to 14crossings to the knot table. As another application, we use the Wirtingernumber to show there exists a universal constant $C$ with the property that thehyperbolic volume of a prime alternating link $L$ is bounded below by $C$ timesthe bridge number of $L$.