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PreprintCC BY 4.0arXiv

Lens space surgeries on knots of small genus

  • Misha Schmalian
  • Steven Sivek
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Abstract

We prove that the (−2,3,7) pretzel knot is the only hyperbolic knot of genus at most $5$ with a lens space surgery. Our key technical result asserts that if a genus-g hyperbolic knot K has an elliptic surgery of slope greater than 4g−3, then the invariant foliations of its monodromy have an orientation that is reversed by the monodromy, and so the opposite of the dilatation must be a root of the Alexander polynomial of K. Combined with prior work of Moser, Wu, Bleiler--Litherland, Baker, and Greene, this verifies the Berge conjecture for knots of genus at most $5$.

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Cite

BibTeXRIS
BibTeX
@misc{schmalian2026lens,
  title = {{Lens space surgeries on knots of small genus}},
  author = {Misha Schmalian and Steven Sivek},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.39260},
  eprint = {2609.39260},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.39260},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Lens space surgeries on knots of small genus
AU  - Misha Schmalian
AU  - Steven Sivek
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.39260
UR  - https://arxiv.org/abs/2609.39260
AB  - We prove that the (−2,3,7) pretzel knot is the only hyperbolic knot of genus at most $5$ with a lens space surgery. Our key technical result asserts that if a genus-g hyperbolic knot K has an elliptic surgery of slope greater than 4g−3, then the invariant foliations of its monodromy have an orientation that is reversed by the monodromy, and so the opposite of the dilatation must be a root of the Alexander polynomial of K. Combined with prior work of Moser, Wu, Bleiler--Litherland, Baker, and Greene, this verifies the Berge conjecture for knots of genus at most $5$.
ER  -