Torus knots, the A-polynomial, and SL(2,C)
arXiv:2405.19197 · doi:10.1007/s00208-026-03363-2
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.
22 pages; v2: accepted version
References in corpus (10)
- Knot homology groups from instantons
- Witten's conjecture and Property P
- Non-triviality of the A-polynomial for knots in S^3
- Instanton Floer homology and the Alexander polynomial
- Integer homology 3-spheres admit irreducible representations in SL(2,C)
- Instantons and L-space surgeries
- Detection of knots and a cabling formula for A-polynomials
- Stein fillings and SU(2) representations
- SU(2)-cyclic surgeries and the pillowcase
- Small Dehn surgery and SU(2)