PreprintCC BY 4.0arXiv
Computation of unknotting numbers: which knot breaks the Bernhard-Jablan Conjecture
Abstract
We determine $2,540$ unknotting numbers of prime knots with at most $13$ crossings, that are unknown in the KnotInfo snapshot of 9 September 2026. The lower-bound calculations use Heegaard Floer correction-term obstructions and Greene's spanning-tree model. We also give explicit crossing-change constructions for upper bounds. For $959$ alternating knots with range [2,3], we verify that no crossing change in a minimal diagram gives a knot of unknotting number one. We determine the unknotting numbers of all four knots in Brittenham and Hermiller's construction and thereby identify 13n3370 as an explicit counterexample to the original Bernhard-Jablan conjecture.
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BibTeX
@misc{lee2026computation,
title = {{Computation of unknotting numbers: which knot breaks the Bernhard-Jablan Conjecture}},
author = {Seong-Jin Lee},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09861},
eprint = {2609.09861},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09861},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Computation of unknotting numbers: which knot breaks the Bernhard-Jablan Conjecture
AU - Seong-Jin Lee
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.09861
UR - https://arxiv.org/abs/2609.09861
AB - We determine $2,540$ unknotting numbers of prime knots with at most $13$ crossings, that are unknown in the KnotInfo snapshot of 9 September 2026. The lower-bound calculations use Heegaard Floer correction-term obstructions and Greene's spanning-tree model. We also give explicit crossing-change constructions for upper bounds. For $959$ alternating knots with range [2,3], we verify that no crossing change in a minimal diagram gives a knot of unknotting number one. We determine the unknotting numbers of all four knots in Brittenham and Hermiller's construction and thereby identify 13n3370 as an explicit counterexample to the original Bernhard-Jablan conjecture.
ER -