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

TL;DR

This paper demonstrates that a version of the A-polynomial can detect torus knots and characterizes knots with infinitely many SL(2,C)-abelian Dehn surgeries, advancing understanding of knot invariants and surgeries.

Contribution

It proves that a version of the A-polynomial detects torus knots and classifies knots with infinitely many SL(2,C)-abelian Dehn surgeries using instanton Floer homology.

Findings

A-version of the A-polynomial detects torus knots.

Characterization of knots with infinitely many SL(2,C)-abelian Dehn surgeries.

Progress towards a folklore conjecture on boundary slopes.

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.