Fibred realizations of the prism quandle P7 in the standard four-sphere
Abstract
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.
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BibTeX
@misc{jablonowski2026fibred,
title = {{Fibred realizations of the prism quandle $P_7$ in the standard four-sphere}},
author = {Michal Jablonowski},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.29839},
eprint = {2609.29839},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29839},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Fibred realizations of the prism quandle P_7 in the standard four-sphere
AU - Michal Jablonowski
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2609.29839
UR - https://arxiv.org/abs/2609.29839
AB - 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.
ER -