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arXiv · 2609.39260

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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Misha Schmalian, Steven Sivek. 2026-09-30. Lens space surgeries on knots of small genus. https://arxiv.org/abs/2609.39260

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