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Sivek–Zentner conjecture (SU(2)-abelian surgeries)

Establish whether torus knots are the only knots in S^3 that admit infinitely many SU(2)-abelian Dehn surgeries.

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Background

The paper references a conjecture of Sivek and Zentner asserting that within S3, torus knots are exactly those with infinitely many SU(2)-abelian surgeries. The authors prove an SL(2,C) analogue (Theorem 1.1), leaving the original SU(2) conjecture as a motivating open direction.

This conjecture connects properties of representation varieties and Dehn surgery behavior to knot classification, paralleling themes in the Property P program and instanton Floer homology.

References

One of our two main results concerns a conjecture by Sivek and Zentner on this theme [SZ], which asserts that torus knots are the only knots in S3 with infinitely many SU(2)-abelian Dehn surgeries.

— Torus knots, the A-polynomial, and SL(2,C) (2405.19197 - Baldwin et al., 29 May 2024) in Section 1 (Introduction), discussion of conjecture