The A-polynomial And Holonomy Perturbations

TL;DR

This paper uses holonomy perturbations to prove the non-triviality of the A-polynomial for certain knots in 3-manifolds and provides constraints on the A-polynomial of knots in the 3-sphere.

Contribution

It introduces holonomy perturbations as a method to establish the non-triviality of the A-polynomial for null-homotopic knots in irreducible 3-manifolds, extending previous results.

Findings

Proves non-triviality of A-polynomial for null-homotopic knots in irreducible 3-manifolds.

Provides constraints on the A-polynomial of knots in the 3-sphere.

Extends understanding of A-polynomial properties using holonomy perturbations.

Abstract

Dunfield-Garoufalidis and Boyer-Zhang proved that the A-polynomial of a nontrivial knot in $S^{3}$ is nontrivial. In this paper, we use holonomy perturbations to prove the non-triviality of the A-polynomial for a nontrivial, null-homotopic knot in an irreducible 3-manifold. Also, we give a strong constraint on the A-polynomial of a knot in the 3-sphere.