PreprintCC BY 4.0arXiv
The Crossing Number and Arc Index of Satellite Knots
Abstract
By considering arc presentations and grid diagrams of satellite knots, we prove that if K is a satellite knot with companion C and wrapping number w, then its crossing number satisfies c(K) ≥ 2/5(w2c(C))1/11. We also prove other similar bounds regarding the crossing number and arc index of satellite knots and links. This is partial progress on Problem 1.2 of the K3 Problem List.
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BibTeX
@misc{ketchell2026crossing,
title = {{The Crossing Number and Arc Index of Satellite Knots}},
author = {Sam Ketchell},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20900},
eprint = {2609.20900},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20900},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Crossing Number and Arc Index of Satellite Knots
AU - Sam Ketchell
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.20900
UR - https://arxiv.org/abs/2609.20900
AB - By considering arc presentations and grid diagrams of satellite knots, we prove that if K is a satellite knot with companion C and wrapping number w, then its crossing number satisfies c(K) ≥ 2/5(w^2c(C))^1/11. We also prove other similar bounds regarding the crossing number and arc index of satellite knots and links. This is partial progress on Problem 1.2 of the K3 Problem List.
ER -