Twisted Alexander Polynomials and Ptolemy Varieties of twist knots and surface bundles

Michael Marinelli (CUNY)

Abstract: The twisted Alexander polynomial (TAP) of a knot is a variation of the classical Alexander polynomial twisted by a linear representation of the knot group. In 2011, Dunfield-Friedl-Jackson conjectured that the twisted Alexander polynomial associated to the holonomy representation of a hyperbolic knot determines both genus and fibering. In 2013, Morifuji-Tran proved this conjecture for double-twist knots using the Riley polynomial and proved that the statement holds for TAPs computed using parabolic representations of the knot group. In this talk, we will give an alternate method for proving this result for twist knots using the Ptolemy variety. Our method gives an elegant formula for twisted Alexander polynomials and can be extended to other families of knots and 3-manifolds. We will outline an extension of our methods to a family of punctured torus bundles.