Twisted Alexander norms give lower bounds on the Thurston norm

Abstract

Abstract. We introduce twisted Alexander norms of a compact connected orientable 3-manifold with first Betti number greater than one, generalizing norms of McMullen and Turaev. We show that twisted Alexander norms give lower bounds on the Thurston norm of a 3-manifold. Using these we completely determine the Thurston norm of many 3-manifolds which are not determined by norms of McMullen and Turaev.

Non-commutative multivariable Reidemester torsion and the Thurston norm

“…As a final remark we point out that the results in this paper completely generalize the results in [6]. Furthermore, the results can easily be extended to studying 2-complexes together with the Turaev norm which is modeled on the definition of the Thurston norm of a 3-manifold.…”

“…The general commutative case is the main result in Friedl-Kim [6]. The proof we give here is different in its nature from the proofs in [11] and [6].…”

Twisted Alexander Polynomials and Symplectic Structures

“…Several results (particularly for the b 1 (N ) = 1 case) have a more elegant presentation when formulated in terms of torsion. Second, some of the results concerning the fiberability of a 3-manifold that are presented here appear (although in different form or generality) in two papers of the first author and Taehee Kim (see [FK05] and [FK06]) as well as, for the case of knots, in previous work of Cha (see [Ch03]) and Goda, Kitano and Morifuji (see [GKM05]). However, we have decided for their inclusion both in light of Conjectures 1.1 and 4.2 (as they specify the actual evidence to it) and because the proofs presented are different in nature and have often emerged independently.…”

“…For these Alexander polynomials, a McMullen-type inequality (similar to Lemma 3.9) is one of the main results of [FK05] and [FK06]:…”

Twisted Alexander Polynomials and Ideal Points Giving Seifert Surfaces

“…We take an epimorphism α : π 1 E → t , where t is the infinite cyclic group generated by the indeterminate t. For a representation ρ : π 1 E → GL n (F), we define a representation α ⊗ ρ : π 1 E → GL n (F(t)) by α ⊗ ρ(γ ) = α(γ )ρ(γ ) for γ ∈ π 1 E. If H α⊗ρ * (E; F(t) n ) = 0, then the Reidemeister torsion τ α⊗ρ (E) is defined and is known by Kirk and Livingston [13] and Kitano [14] to be essentially equal to the twisted Alexander polynomial associated to α and ρ. Friedl and Kim [4, Theorem 1.1] showed that deg τ α⊗ρ (E) ≤ n(2g(K) − 1) (see also [5]). It is known by Cha [1], Friedl and Kim [4], and Goda et al [10] that if K is a fibered knot, then deg τ α⊗ρ (E) = n(2g(K) − 1) and τ α⊗ρ (E) is represented by a fraction of monic polynomials in F[t, t −1 ].…”