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PreprintCC BY 4.0arXiv

Negative-Definite Plumbings and New Unknotting Numbers

  • Seong-Jin Lee
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Abstract

We use Neumann's normal form to decide whether the double branched cover of a knot bounds a negative-definite plumbing tree in either orientation, starting from the recognised Seifert or graph-manifold structure of the cover. For many non-alternating knots with at most thirteen crossings whose unknotting numbers were unknown, this identifies the cases in which the Heegaard Floer homology of the cover can be computed from such a plumbing, and shows for most of the others that it cannot. We determine 112 new unknotting numbers. For 43 of them, the lower bound rests on Heegaard Floer homology computed from the plumbing; for the other 69, it comes from linking pairings and from a theorem of McCoy and Park on special alternating knots. Finally, we show that the knot 13n1587 is a further explicit counterexample to the Bernhard--Jablan conjecture, after 13n3370.

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Cite

BibTeXRIS
BibTeX
@misc{lee2026negativedefinite,
  title = {{Negative-Definite Plumbings and New Unknotting Numbers}},
  author = {Seong-Jin Lee},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.25430},
  eprint = {2610.25430},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.25430},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Negative-Definite Plumbings and New Unknotting Numbers
AU  - Seong-Jin Lee
PY  - 2026
DA  - 2026/10/18
PB  - arXiv
DO  - 10.48550/arxiv.2610.25430
UR  - https://arxiv.org/abs/2610.25430
AB  - We use Neumann's normal form to decide whether the double branched cover of a knot bounds a negative-definite plumbing tree in either orientation, starting from the recognised Seifert or graph-manifold structure of the cover. For many non-alternating knots with at most thirteen crossings whose unknotting numbers were unknown, this identifies the cases in which the Heegaard Floer homology of the cover can be computed from such a plumbing, and shows for most of the others that it cannot. We determine 112 new unknotting numbers. For 43 of them, the lower bound rests on Heegaard Floer homology computed from the plumbing; for the other 69, it comes from linking pairings and from a theorem of McCoy and Park on special alternating knots. Finally, we show that the knot 13n1587 is a further explicit counterexample to the Bernhard--Jablan conjecture, after 13n3370.
ER  -