AAG Seminar: A-Polynomials and Zero Loci of Regulator Maps

Abstract: The A-polynomial is an invariant which captures information about families of hyperbolic structures on a compact orientable 3-manifold with toric boundary, such as knot exteriors. I will discuss its definition and a condition in algebraic K-theory precisely characterizing the irreducible Laurent polynomials which are factors of some A-polynomial. This condition can be framed in terms of zero loci of regulator maps, for which our finiteness result (joint with Matt Kerr and RJ Acuna) resolves a conjecture of Guilloux and Marché regarding the finiteness of the A-factor locus. We also define a measure of complexity on A-factors called "overgenus" and show that there are finitely many A-facotrs of bounded positive overgenus.

Host: Matt Kerr