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A counterexample to a conjecture of Plotnick on knot exteriors
Abstract
We resolve a conjecture of S. Plotnick from 1983 concerning the homotopy type of four-dimensional knot complements. We show that the exteriors of two branched twist spins of the trefoil, which were shown by Plotnick not to be homotopy equivalent as boundary pairs and hence not homeomorphic, are in fact simple homotopy equivalent. The main ingredient of the proof is the construction of an explicit element of augmentation $49$ in the integral group ring of SL2(𝔽5) which is a unit modulo the group norm and commutes with the element inducing the monodromy on the fundamental group.
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BibTeX
@misc{hofmann2026counterexample,
title = {{A counterexample to a conjecture of Plotnick on knot exteriors}},
author = {Tommy Hofmann and John Nicholson},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21051},
eprint = {2610.21051},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21051},
}
RIS
TY - UNPB
TI - A counterexample to a conjecture of Plotnick on knot exteriors
AU - Tommy Hofmann
AU - John Nicholson
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21051
UR - https://arxiv.org/abs/2610.21051
AB - We resolve a conjecture of S. Plotnick from 1983 concerning the homotopy type of four-dimensional knot complements. We show that the exteriors of two branched twist spins of the trefoil, which were shown by Plotnick not to be homotopy equivalent as boundary pairs and hence not homeomorphic, are in fact simple homotopy equivalent. The main ingredient of the proof is the construction of an explicit element of augmentation $49$ in the integral group ring of SL_2(𝔽_5) which is a unit modulo the group norm and commutes with the element inducing the monodromy on the fundamental group.
ER -