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We define a torsion invariant T for every balanced sutured manifold (M, γ), and show that it agrees with the Euler characteristic of sutured Floer homology (SFH). The invariant T is easily computed using Fox calculus. With the help of T, we prove that if (M, γ) is complementary to a Seifert surface of an alternating knot, then SFH(M, γ) is either 0 or ℤ in every Spinc structure. The torsion invariant T can also be used to show that a sutured manifold is not disc decomposable, and to distinguish between Seifert surfaces. The support of SFH gives rise to a norm z on H2(M, ∂ M; ℝ). The invariant T gives a lower bound on the norm z, which in turn is at most the sutured Thurston norm xs. For closed 3‐manifolds, it is well known that Floer homology determines the Thurston norm, but we show that z

Every element in the first cohomology group of a 3-manifold is dual to embedded surfaces. The Thurston norm measures the minimal 'complexity' of such surfaces. For instance the Thurston norm of a knot complement determines the genus of the knot in the 3-sphere. We show that the degrees of twisted Alexander polynomials give lower bounds on the Thurston norm, generalizing work of McMullen and Turaev. Our bounds attain their most concise form when interpreted as the degrees of the Reidemeister torsion of a certain twisted chain complex. We show that these lower bounds give the correct genus bounds for all knots with 12 crossings or less, including the Conway knot and the Kinoshita-Terasaka knot which have trivial Alexander polynomial.We also give obstructions to fibering 3-manifolds using twisted Alexander polynomials and detect all knots with 12 crossings or less that are not fibered. For some of these it was unknown whether or not they are fibered. Our work in particular extends the fibering obstructions of Cha to the case of closed manifolds.

Abstract. We give a short introduction to the theory of twisted Alexander polynomials of a 3-manifold associated to a representation of its fundamental group. We summarize their formal properties and we explain their relationship to twisted Reidemeister torsion. We then give a survey of the many applications of twisted invariants to the study of topological problems. We conclude with a short summary of the theory of higher order Alexander polynomials.

We give a useful classification of the metabelian unitary representations of π 1 (M K ), where M K is the result of zero-surgery along a knot K ⊂ S 3 . We show that certain eta invariants associated to metabelian representations π 1 (M K ) → U (k) vanish for slice knots and that even more eta invariants vanish for ribbon knots and doubly slice knots. We show that our vanishing results contain the Casson-Gordon sliceness obstruction. In many cases eta invariants can be easily computed for satellite knots. We use this to study the relation between the eta invariant sliceness obstruction, the eta-invariant ribbonness obstruction, and the L 2 -eta invariant sliceness obstruction recently introduced by Cochran, Orr and Teichner. In particular we give an example of a knot which has zero eta invariant and zero metabelian L 2 -eta invariant sliceness obstruction but which is not ribbon.

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 .

Given an L2‐acyclic connected finite CW‐complex, we define its universal L2‐torsion in terms of the chain complex of its universal covering. It takes values in the weak Whitehead group prefixWhwfalse(Gfalse). We study its main properties such as homotopy invariance, sum formula, product formula and Poincaré duality. Under certain assumptions, we can specify certain homomorphisms from the weak Whitehead group prefixWhwfalse(Gfalse) to abelian groups such as the real numbers or the Grothendieck group of integral polytopes, and the image of the universal L2‐torsion can be identified with many invariants such as the L2‐torsion, the L2‐torsion function, twisted L2‐Euler characteristics and, in the case of a 3‐manifold, the dual Thurston norm polytope.

We study a twisted Alexander polynomial naturally associated to a hyperbolic knot in an integer homology 3-sphere via a lift of the holonomy representation to SL(2,C). It is an unambiguous symmetric Laurent polynomial whose coefficients lie in the trace field of the knot. It contains information about genus, fibering, and chirality, and moreover is powerful enough to sometimes detect mutation.We calculated this invariant numerically for all 313,209 hyperbolic knots in S 3 with at most 15 crossings, and found that in all cases it gave a sharp bound on the genus of the knot and determined both fibering and chirality.We also study how such twisted Alexander polynomials vary as one moves around in an irreducible component X 0 of the SL(2,C)-character variety of the knot group. We show how to understand all of these polynomials at once in terms of a polynomial whose coefficients lie in the function field of X 0 . We use this to help explain some of the patterns observed for knots in S 3 , and explore a potential relationship between this universal polynomial and the Culler-Shalen theory of surfaces associated to ideal points.