Torus knots, the A-polynomial, and SL(2,C) (2405.19197v1)
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 distinguishes torus knots from all other knots, and in particular detects the torus knot T_{a,b} if and only if one of |a| or |b| is $2$ or both are prime powers. 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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