Effective bounds on characterising slopes for all knots

Q: What does this research mean for the field?

For any given knot, there exists an explicitly determinable and optimal bound C(K) such that any slope p/q with |q| > C(K) is a characterising slope that uniquely determines the knot via Dehn surgery. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

Patricia Sorya; Laura Wakelin

doi:10.1090/tran/9673

August 20, 2026Transactions of the American Mathematical SocietyOpen Access

Effective bounds on characterising slopes for all knots

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Key Points

Determine explicit and optimal bounds on surgery slopes to establish when Dehn surgery uniquely characterises a given knot in the 3-sphere.
Combined methods from hyperbolic geometry with topological techniques from prior knot surgery frameworks.
Formulated mathematical criteria to evaluate the orientation-preserving homeomorphism types of manifolds produced by rational Dehn surgery.
Determined an explicit bound C(K) for any knot K such that any slope p/q satisfying |q| > C(K) is a characterising slope.
Identified optimal formulations for the bounding function C(K) across knot complements.

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Implication

Geometric analysis uncovers explicit bounds for characterising slopes across all knots, indicating when surgery manifolds uniquely identify knot complements.

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Sorya et al. (2026) studied this question.

synapsesocial.com/papers/6a895e6bca7ade938187c80d https://doi.org/10.1090/tran/9673

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Effective bounds on characterising slopes for all knots

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Effective bounds on characterising slopes for all knots

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Implication

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