Torus knots, the A-polynomial, and SL(2,C)

John A. Baldwin; Steven Sivek

doi:10.1007/s00208-026-03363-2

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arXiv · 2405.19197

Torus knots, the A-polynomial, and SL(2,C)

John A. Baldwin·Steven Sivek

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Abstract

The A-polynomial of a knot is defined in terms of SL(2,C) representations of the knot group, and encodes information about essential surfaces in the knot complement. In 2005, Dunfield-Garoufalidis and Boyer-Zhang proved that it detects the unknot using Kronheimer-Mrowka's work on the Property P conjecture. Here we use more recent results from instanton Floer homology to prove that a version of the A-polynomial detects whether a knot is a torus knot. We moreover completely determine which individual torus knots are detected by this A-polynomial. These results enable progress towards a folklore conjecture about boundary slopes of non-torus knots. Finally, we use similar ideas to prove that a knot in the 3-sphere admits infinitely many SL(2,C)-abelian Dehn surgeries if and only if it is a torus knot, affirming a variant of a conjecture due to Sivek-Zentner.

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BibTeX RIS

John A. Baldwin, Steven Sivek. 2024-05-29. Torus knots, the A-polynomial, and SL(2,C). https://doi.org/10.1007/s00208-026-03363-2

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