We define the {\it 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 the Wirtinger 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 $L$.

中文翻译：

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结组生成器的 Wirtinger 系统

我们定义了一个链接的 {\it Wirtinger number}，一个与子午等级密切相关的不变量。Wirtinger 数是所有子午表示中链接补集的基本群的最小生成器数，其中每个关系都是图中出现的迭代 Wirtinger 关系。我们证明了一条链路的 Wirtinger 数等于它的桥数。这种等式可以看作是建立了 Cappell 和 Shaneson 的子午阶猜想的弱版本，并为这个猜想提出了一种新的方法。我们的结果还导致了一种组合技术，用于获得桥数的强上限。到目前为止，这项技术使我们能够将大约 50,000 个主要节点的桥数添加到节点表中，最多 14 个交叉点。作为另一个应用程序，