PreprintarXiv licencearXiv
An algebraic proof of 1-extension
Abstract
We give a new proof that 1-extension preserves generic local rigidity of bar-joint frameworks in all dimensions. The proof does not use a special position argument, which makes the approach potentially useful for X-replacement in 3-space. Instead, the balancing conditions recently introduced by the first author use directed paths in the line graph, and by modifying the end of these paths, we transfer a self-stress on the 1-extended graph back to the original. Treating the new vertex as variable allows us to extract the balancing condition of the smaller graph as one coefficient of a homogeneous quadratic. A Lean 4 formalization of the main results accompanies the paper.
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BibTeX
@misc{heaton2026algebraic,
title = {{An algebraic proof of 1-extension}},
author = {Alexander Heaton and Martin Winter},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31212},
eprint = {2610.31212},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31212},
}
RIS
TY - UNPB
TI - An algebraic proof of 1-extension
AU - Alexander Heaton
AU - Martin Winter
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31212
UR - https://arxiv.org/abs/2610.31212
AB - We give a new proof that 1-extension preserves generic local rigidity of bar-joint frameworks in all dimensions. The proof does not use a special position argument, which makes the approach potentially useful for X-replacement in 3-space. Instead, the balancing conditions recently introduced by the first author use directed paths in the line graph, and by modifying the end of these paths, we transfer a self-stress on the 1-extended graph back to the original. Treating the new vertex as variable allows us to extract the balancing condition of the smaller graph as one coefficient of a homogeneous quadratic. A Lean 4 formalization of the main results accompanies the paper.
ER -