Torus knots, the A-polynomial, and SL(2,ℂ)
File(s)
Accepted version
Author(s)
Baldwin, John A
Sivek, Steven
Type
Journal Article
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.
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.
Date Issued
2026-02-01
Date Acceptance
2025-12-01
Citation
Mathematische Annalen, 2026, 394 (2)
ISSN
0025-5831
Publisher
Springer
Journal / Book Title
Mathematische Annalen
Volume
394
Issue
2
Copyright Statement
Copyright © 2026, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature. This is the author’s accepted manuscript made available under a CC-BY licence in accordance with Imperial’s Research Publications Open Access policy (www.imperial.ac.uk/oa-policy)
License URL
Identifier
http://arxiv.org/abs/2405.19197v1
Subjects
math.GT
math.GT
Publication Status
Published
Article Number
ARTN 26
Date Publish Online
2026-02-13