AI 中文总结
本文证明Lecuona例外族pretzel纽结的Alexander多项式不满足Fox-Milnor条件，从而完成三股pretzel纽结的可切性分类，并移除相关分类中的非例外假设。
AI 中文摘要
我们证明了Lecuona例外族pretzel纽结中每个纽结的Alexander多项式不满足Fox-Milnor条件。因此，这些纽结中没有一个在代数上或拓扑上是可切的。结合Lecuona、Miller和Kim-Lee-Song的工作，这完成了三股pretzel纽结的可切-带形和拓扑可切性分类。这也从Lecuona-Wand关于素纤维带形pretzel纽结（参数重排之前）的分类中移除了非例外假设。
英文摘要
We consider the three-strand pretzel knot $P_a=P(a,-a-2,-(a+1)^2/2)$ and prove that, when $a\equiv1\pmod4$, its Alexander polynomial fails the Fox--Milnor condition. The same remains true after adding any number of opposite odd parameter pairs $q,-q$ and reordering the parameters. Combined with work of Kim--Lee--Song, this gives the same conclusion for every odd $a\ge3$. In particular, no knot in Lecuona's exceptional family is algebraically or topologically slice. Together with earlier work of Lecuona, Miller, and Kim--Lee--Song, this completes the slice--ribbon and topological-sliceness classifications for three-strand pretzel knots and removes the nonexceptional hypothesis from the Lecuona--Wand classification of prime fibered ribbon pretzel knots.
Comments8 pages, 2 figures. Revision: minor changes and a remark
