棱柱quandle $P_7$ 在标准四维球面中的纤维实现
Fibred realizations of the prism quandle $P_7$ in the standard four-sphere
- University of Gdańsk（格但斯克大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明由四元数棱柱流形构造的两个同伦四维球面均为标准，并给出棱柱quandle阶五十六的两个不等价纤维实现，解决了阶七情形。
AI中文摘要：
我们证明了由四元数棱柱流形（循环因子阶为七）构造的周期六映射环面上一个截面的两次带框手术产生的两个光滑同伦四维球面都是标准的。它们的带球面在标准四维球面中给出了阶为五十六的棱柱quandle的两个光滑纤维二维纽结实现。这两个纽结具有相同的纽结群和微分同胚的外部，但不等价。结合纤维刚性约化，这给出了同伦四维球面中恰好两个纤维实现，且都是标准的。证明结合了轨道轨道流形的Whitehead链模型、圆作用与手术方法、循环分支覆盖计算，以及次级不变量的自旋配边评估。这解决了经典棱柱族实现问题中阶为七的情形。
英文摘要：
We study smooth fibred two-knots whose fundamental quandle is the prism quandle of order fifty-six. The candidates arise from framed circle surgery along a section of a mapping torus, using the two possible normal-framing classes. We prove that both ambient homotopy four-spheres are diffeomorphic to the standard four-sphere. The candidates have the same knot group and diffeomorphic exteriors, but they are inequivalent even as topological pairs. Every smooth fibred realization of this quandle in a homotopy four-sphere is equivalent, up to knot orientation, to exactly one of these two knots. Standardness follows from a Whitehead-link model for the orbit orbifold together with circle-action recognition. Inequivalence is detected by the third cyclic branched covers, on which a spin-bordism secondary invariant takes different values. Thus the parameter-seven prism quandle admits smooth fibred realizations in the standard four-sphere.
