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Counterexamples to Two Conjectures on Tangles in Matroid Minors
Abstract
We give counterexamples to two conjectures of Brettell, Jowett, Oxley, Semple, and Whittle concerning tangles in matroid minors. We construct a matroid P0 with a tangle 𝒯 of order seven. For each i∈{4,5,6,7}, let 𝒯i denote its order-i truncation. No proper minor N of P0 admits, for every such i, a tangle of order i whose breadth equals that of 𝒯i, and P0 is not weakly $4$-connected. We also construct a matroid P1 with a tangle of order four and breadth six, but no internally $4$-connected minor of P1 admits a tangle of order four.
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BibTeX
@misc{chen2026counterexamples,
title = {{Counterexamples to Two Conjectures on Tangles in Matroid Minors}},
author = {Zhen Chen and Zhuo Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.23059},
eprint = {2610.23059},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.23059},
}
RIS
TY - UNPB
TI - Counterexamples to Two Conjectures on Tangles in Matroid Minors
AU - Zhen Chen
AU - Zhuo Li
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.23059
UR - https://arxiv.org/abs/2610.23059
AB - We give counterexamples to two conjectures of Brettell, Jowett, Oxley, Semple, and Whittle concerning tangles in matroid minors. We construct a matroid P_0 with a tangle 𝒯 of order seven. For each i∈{4,5,6,7}, let 𝒯_i denote its order-i truncation. No proper minor N of P_0 admits, for every such i, a tangle of order i whose breadth equals that of 𝒯_i, and P_0 is not weakly $4$-connected. We also construct a matroid P_1 with a tangle of order four and breadth six, but no internally $4$-connected minor of P_1 admits a tangle of order four.
ER -