arXiv research

Twists agrarian and $\ell^2$-Betti numbers for locally indicable groups.

We introduce twisted Alexander norms of a compact connected orientable 3-manifold with first Betti number bigger 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 …

We introduce a norm on the space of test configurations, which we call the minimum norm. We conjecture that uniform K-stability with respect to this norm is equivalent to the existence of a constant scalar curvature Kähler metric. This notion of uniform K-stability is analogous to coercivity of the Mabuchi functional. …

Using recent results of Agol, Przytycki-Wise and Wise we show that twisted Alexander polynomials detect the Thurston norm of any irreducible 3-manifold which is not a closed graph manifold.

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 polynomial…

We prove that the twisted Reidemeister torsion of a 3-manifold corresponding to a fibered class is monic and we show that it gives lower bounds on the Thurston norm. The former fixes a flawed proof in [FV10], the latter gives a quick alternative argument for the main theorem of [FK06].

We show that the Thurston norm of any irreducible 3-manifold can be detected using twisted Reidemeister torsions corresponding to integral representations and also corresponding to representations over finite fields. In particular our result holds for all graph manifolds, these are not covered by the earlier work of th…

Study spectral distribution of twisted Laplacian on high genus hyperbolic surfaces.

We study when the Thurston norm is detected by twisted Alexander polynomials associated to representations of the 3-manifold group to SL(2, C). Specifically, we show that the hyperbolic torsion polynomial determines the genus for a large class of hyperbolic knots in the 3-sphere which includes all special arborescent k…

We introduce a norm on the real 1-cohomology of finite 2-complexes determined by the Euler characteristics of graphs on these complexes. We also introduce twisted Alexander-Fox polynomials of groups and show that they give rise to norms on the real 1-cohomology of groups. Our main theorem states that for a finite 2-com…

We continue the study of the twisted Novikov homology, introduced in our joint paper with H.Goda (arXiv:math.DG/0312374), and its generalizations. The main applications of the developed algebraic techniques are to the topology of 3-manifolds. We show in particular that the twisted Novikov homology of a 3-manifold M of …

We study incompressible surfaces constructed by Culler-Shalen theory in the context of twisted Alexander polynomials. For a $1$st cohomology class of a $3$-manifold the coefficients of twisted Alexander polynomials induce regular functions on the $SL_2(\mathbb{C})$-character variety. We prove that if an ideal point giv…

We assign to a finite $CW$-complex and an element in its first cohomology group a twisted version of the $L^2$-Euler characteristic and study its main properties. In the case of an irreducible orientable $3$-manifold with empty or toroidal boundary and infinite fundamental group we identify it with the Thurston norm. W…

We consider knots whose diagrams have a high amount of twisting of multiple strands. By encircling twists on multiple strands with unknotted curves, we obtain a link called a generalized augmented link. Dehn filling this link gives the original knot. We classify those generalized augmented links that are Seifert fibere…

Recently twisted and higher order Alexander polynomials were used by Cochran, Harvey, Friedl--Kim and Turaev to give lower bounds on the Thurston norm. We first show how Reidemeister torsion relates to these Alexander polynomials. We then give lower bounds on the Thurston norm in terms of the Reidemeister torsion which…

Let $E$ be a flat complex vector bundle over a closed oriented odd dimensional manifold $M$ endowed with a flat connection $\nabla$. The refined analytic torsion for $(M,E)$ was defined and studied by Braverman and Kappeler. Recently Mathai and Wu defined and studied the analytic torsion for the twisted de Rham complex…

For a compact, orientable, irreducible 3-manifold with toroidal boundary that is not the product of a torus and an interval or a cable space, each boundary torus has a finite set of slopes such that, if avoided, the Thurston norm of a Dehn filling behaves predictably. More precisely, for all but finitely many slopes, t…

Let N be a closed, oriented 3-manifold. A folklore conjecture states that $S^{1} \times N$ admits a symplectic structure if and only if $N$ admits a fibration over the circle. We will prove this conjecture in the case when N is irreducible and its fundamental group satisfies appropriate subgroup separability conditions…

We derive a simple closed formula for the SL(2,C) Casson invariant for Seifert fibered homology 3-spheres using the correspondence between SL(2,C) character varieties and moduli spaces of parabolic Higgs bundles of rank two. These results are then used to deduce the invariant for Dehn surgeries on twist knots by combin…

The paper introduces new invariants to refine Alexander polynomials and bounds BNSR Σ-invariants.

The taut polynomial equals a twisted Alexander polynomial.

A classical result in knot theory says that the Alexander polynomial of a fibered knot is monic and that its 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 Alexa…

Finite quotients of fibered hyperbolic 3-manifold groups detect taut polynomials.

We prove that twisted correction terms in Heegaard Floer homology provide lower bounds on the Thurston norm of certain cohomology classes determined by the strong concordance class of a 2-component link $L$ in $S^3$. We then specialise this procedure to knots in $S^2\times S^1$, and obtain a lower bound on their geomet…

Let M be an oriented irreducible 3-manifold with infinite fundamental group and empty or toroidal boundary. Consider any element φin the first cohomology of M with integral coefficients. Then one can define the φ-twisted L^2-torsion function of the universal covering which is a function from the set of positive real nu…

We prove that the Nielsen zeta function is a rational function or a radical of a rational function for orientation preserving homeomorphisms on closed orientable 3-dimensional manifolds which are special Haken or Seifert manifolds. In the case of pseudo-Anosov homeomorphism of surface we compute an asymptotic for the n…

Construct minimal Lagrangian surfaces in complex projective plane via loop group method.

The paper studies flow lines on Higgs bundle moduli spaces, classifying them via secant varieties.

Let $M^3$ be an oriented 3-manifold. We investigate when one of the fibers or a combination of fiber components, $F_{best}$, of a \emph{harmonic} map $f: M^3 \to S^1$ with Morse-type singularities delivers the Thurston norm $χ_-([F_{best}])$ of its homology class $[F_{best}] \in H_2(M^3; \Z)$. In particular, for a map …

The paper proves vanishing theorems for complex line bundles using a new adiabatic limit approach.

In this article we study the Quillen norm on the determinant line bundle associated with a family of complex curves with cusps, which admit singular fibers. More precisely, we fix a family of complex curves $π: X \to S$, which admit at most double-point singularities. Let $(ξ, h^ξ)$ be a holomorphic Hermitian vector bu…

Let $\overline{M}$ be a compact Riemann surface and let $g^{TM}$ be a metric over $\overline{M} \setminus D_M$, where $D_M \subset \overline{M}$ is a finite set of points. We suppose that $g^{TM}$ is equal to the Poincaré metric over a punctured disks around the points of $D_M$. The metric $g^{TM}$ endows the twisted c…

Study extends complex sections on non-holomorphic objects on Kähler manifolds.

We study the topology of the boundary manifold of a line arrangement in CP^2, with emphasis on the fundamental group G and associated invariants. We determine the Alexander polynomial Delta(G), and more generally, the twisted Alexander polynomial associated to the abelianization of G and an arbitrary complex representa…

We introduce the concept of twisted contact groupoids, as an extension either of contact groupoids or of twisted symplectic ones, and we discuss the integration of twisted Jacobi manifolds by twisted contact groupoids. We also investigate the very close relationships which link homogeneous twisted Poisson manifolds wit…

New infinite families of twisted torus knots found.

The study finds hyperbolic twisted torus links for certain twists.

The paper introduces a method to deform submanifolds in Euclidean space using Drinfel'd twists.

Paper constructs Chern character for higher twists and shows isomorphism between K-theory and cohomology.

The paper studies the twisted Calabi flow on Kähler manifolds.

Construct noncommutative deformations of algebraic submanifolds in R^n.

Paper proves any twisted link can be described as a unique twisted braid.

Study on singular twisted links and virtual braids, extending knot theory concepts.

In this paper, we develop differential twisted K-theory and define a twisted Chern character on twisted K-theory which depends on a choice of connection and curving on the twisting gerbe. We also establish the general Riemann-Roch theorem in twisted K-theory and find some applications in the study of twisted K-theory o…

Proves existence of Kähler-Einstein metrics with certain twisting forms.

The paper extends knot theory to twisted virtual braids and links.

Solves infinite family of cubic polynomial problems.

The paper introduces colorings and invariants for twisted links and shows how double coverings can be equivalent.