In the 1950s Milnor defined a family of higher-order invariants generalizing the linking number. Even the first of these new invariants, the triple linking number, has received fruitful study since its inception. In the case that a link [Formula: see text] has vanishing pairwise linking numbers, this triple linking number gives an integer-valued invariant. When the linking numbers fail to vanish, the triple linking number is only well-defined modulo their greatest common divisor. In recent work Davis–Nagel–Orson–Powell produce a single invariant called the total triple linking number refining the triple linking number and taking values in an abelian group called the total Milnor quotient. They present examples for which this quotient is nontrivial even though each of the individual triple linking numbers take values in the trivial group, [Formula: see text], and so carry no information. As a consequence, the total triple linking number carries more information than do the classical triple linking numbers. The goal of this paper is to compute this group and show that when [Formula: see text] is a link of at least six components it is nontrivial. Thus, this total triple linking number carries information for every [Formula: see text]-component link, even though the classical triple linking numbers often carry no information.

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关于米尔诺三联数的不确定性

在 1950 年代，Milnor 定义了一个泛化链接数的高阶不变量族。即使是这些新不变量中的第一个，三联数，从一开始就得到了卓有成效的研究。在链接[公式：见正文]具有消失的成对链接数的情况下，这个三重链接数给出整数值不变量。当连接数不能消失时，三重连接数只是明确定义的模它们的最大公约数。在最近的工作中，Davis-Nagel-Orson-Powell 产生了一个称为总三联数的不变量，改进了三联数并在称为总 Milnor 商的阿贝尔群中取值。他们提出了这个商不平凡的例子，即使每个单独的三联数都取平凡组中的值，[公式：见文本]，因此不携带任何信息。因此，总的三联数比经典的三联数携带更多的信息。本文的目标是计算该组，并表明当 [公式：见文本] 是至少六个组件的链接时，它是不平凡的。因此，这个总的三重链接数包含每个 [公式：见文本] 组件链接的信息，即使经典的三重链接数通常不包含任何信息。