Twisted Alexander polynomials detect fibered 3-manifolds

Abstract

A classical result in knot theory says that for a fibered knot the Alexander polynomial is monic and that the degree equals twice the genus of the knot. This result has been generalized by various authors to twisted Alexander polynomials and fibered 3-manifolds. In this paper we show that the conditions on twisted Alexander polynomials are not only necessary but also sufficient for a 3-manifold to be fibered. By previous work of the authors this result implies that if a manifold of the form S 1 ×N 3 admits a symplectic structure, then N fibers over S 1 . In fact we will completely determine the symplectic cone of S 1 × N in terms of the fibered faces of the Thurston norm ball of N .

Virtually symplectic fibered 4-manifolds

“…The 'if' direction of the theorem was proved in [5,10,15,31] and the 'only if' direction was proved in [16], extending earlier work in [26,8,9,6,12,13]. This theorem was recently extended to 4-manifolds with a fixed point free S 1 -action by Bowden [7].…”

“…The recent dramatic increase in our understanding of 3-manifolds due to the work of Agol [1,2], Przytycki-Wise [27] and Wise [35,36] now makes it possible to settle Q1 and Q2 for many such 4-manifolds. For example Q1 has been completely answered for 4-manifolds which fiber over a 3-manifold in [12,16] and for those which fiber over a surface in [31,18,33]. To the best of our knowledge, the case of 4-manifolds which fiber over S 1 has not been dealt with in any detail.…”

“…not every 4-manifold that is finitely covered by a symplectic 4-manifold is necessarily symplectic itself. For example, if N is a non-fibered hyperbolic 3-manifold, then by work of Agol [2] N admits a finite cover N which is fibered, and it follows from [31] that S 1 ×Ñ is symplectic whereas by [12] the manifold S 1 × N is not symplectic. (Also see Proposition 6 below.)…”

Taut sutured manifolds and twisted homology

“…(2) In [FV11,FV12b] it is shown that twisted Alexander polynomials corresponding to representations of π 1 (N ) can detect whether R is a fiber or not.…”

Twisted Alexander polynomials of 2-bridge knots for parabolic representations

“…A theory of twisted Alexander polynomials was rapidly developed during the past ten years. Among them, one of the most important progress is the determination of fiberedness [FV2] and genus (the Thurston norm) [FV3] of knots by the collection of the twisted Alexander polynomials corresponding to all finite-dimensional representations. For literature on other applications and related topics, we refer to the survey paper by Friedl and Vidussi [FV1].…”