Paolo Ghiggini, Ko Honda
We explain the effect of applying a full Lutz twist along a pre-Lagrangian torus in a contact 3-manifold, on the contact invariant in Heegaard Floer homology with twisted coefficients.
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Legendrian invariants and half Giroux torsion
本文研究了具有非零接触不变量的 Legendrian 链环，证明了零调链环在特定过度扭转接触结构中存在非松散实现，并揭示分离半 Giroux 扭转不必然使不变量消失，最后提出关于链环不变量非零性的猜想刻画。
In this paper we prove a vanishing theorem for the contact Ozsvath--Szabo invariants of certain contact 3--manifolds having positive Giroux torsion. We use this result to establish similar vanishing results for contact structures with underlying 3--manifolds admitting either a torus fibration over the circle or a Seifert fibration over an orientable base. We also show -- using standard techniques from contact topology -- that if a contact 3--manifold (Y,ξ) has positive Giroux torsion then there exists a Stein cobordism from (Y,ξ) to a contact 3--manifold (Y,ξ') such that (Y,ξ) is obtained from…
Algebraic and Giroux torsion in higher-dimensional contact manifolds
本文在任意奇数维构造了具有有限非零代数挠率的接触流形，证明其 tight 且无强辛填充，并证明 Giroux 挠率蕴含代数 1-挠率，从而解决 Massot-Niederkrueger-Wendl 猜想。
Bordered contact invariants and half Giroux torsion
本文利用带边接触不变量和纽结补的最内层接触结构，构造沿分离环面含半 Giroux 挠率但接触不变量非零的无穷多闭接触3维流形，反驳 Ghiggini 猜想并确定 2π 为使不变量消失的最小扭转量。
We prove that the Ozsvath-Szabo contact invariant of a closed contact 3-manifold with positive Giroux torsion vanishes.
Tight contact structures on toroidal plumbed 3-manifolds
该研究针对无坏顶点的配边3-流形，开展紧切触结构计数、Giroux挠率添加条件分析及Stein图构造算法研究，重点关注低顶点价数的情况并拓展至高顶点价数情形。
We extend the Heegaard Floer homological definition of spectral order for closed contact 3-manifolds due to Kutluhan, Matić, Van Horn-Morris, and Wand to contact 3-manifolds with convex boundary. We show that the order of a codimension zero contact submanifold bounds the order of the ambient manifold from above. As the neighborhood of an overtwisted disk has order zero, we obtain that overtwisted contact structures have order zero. We also prove that the order of a small perturbation of a $2π$ Giroux torsion domain has order at most two, hence any contact structure with positive Giroux torsion…
We use contact fiber sums of open book decompositions to define an infinite hierarchy of filling obstructions for contact 3-manifolds, called planar k-torsion for nonnegative integers k, all of which cause the contact invariant in Embedded Contact Homology to vanish. Planar 0-torsion is equivalent to overtwistedness, while every contact manifold with Giroux torsion also has planar 1-torsion, and we give examples of contact manifolds that have planar k-torsion for any $k \ge 2$ but no Giroux torsion, leading to many new examples of nonfillable contact manifolds. We show also that the complement…
We show that if (B,π) is an open book decomposition of a contact 3-manifold (Y,ξ), then the complement of the binding B has no Giroux torsion. We also prove the sutured Heegaard-Floer c-bar invariant of the binding of an open book is non-zero.
We calculate the twisted Reidemeister torsion of the complement of an iterated torus knot associated with a representation of its fundamental group to the complex special linear group of degree two. We also show that the twisted Reidemeister torsions associated with various representations appear in the asymptotic expansion of the colored Jones polynomial.
The twisted torsion of a 3-manifold is well-known to be zero whenever the corresponding twisted Alexander module is non-torsion. Under mild extra assumptions we introduce a new twisted torsion invariant which is always non-zero. We show how this torsion invariant relates to the twisted intersection form of a bounding 4-manifold, generalizing a theorem of Milnor. Using this result, we give new obstructions to 3-manifolds being homology cobordant and to links being concordant. These obstructions are sufficiently strong to detect that the Bing double of the figure eight knot is not slice.
Classification of tight contact structures on some Seifert fibered manifolds: I
本文通过 Legendrian 手术构造接触结构获得下界，并利用凸曲面理论获得上界，对具有四条例外纤维的某些 Seifert 纤维化流形上零 Giroux 挠率的紧接触结构进行了分类。
We define a twisted $L^2$-torsion on the character variety of 3-manifold $M$ and study some of its properties. In the case where $M$ is hyperbolic of finite volume, we prove that the $L^2$-torsion is a real analytic function on a neighborhood of any lift of the holonomy representation.
In this article, we give an explicit formula to compute the non-abelian twisted sign-determined Reidemeister torsion of the exterior of a fibered knot in terms of its monodromy. As an application, we give explicit formulae for the non abelian Reidemeister torsion of torus knots and of the figure eight knot.
A Torres formula for twisted Reidemeister torsion
针对Torres公式的扭曲推广问题，本文利用扭曲Reidemeister挠率给出了Morifuji结果的第二种证明，其形式更贴近Torres原始公式，为该推广结果提供了新的证明途径。
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 the first author and Vidussi.
We use the theory of sutured TQFT to classify contact elements in the sutured Floer homology, with $\Z$ coefficients, of certain sutured manifolds of the form $(Σ\times S^1, F \times S^1)$ where $Σ$ is an annulus or punctured torus. Using this classification, we give a new proof that the contact invariant in sutured Floer homology with $\Z$ coefficients of a contact structure with Giroux torsion vanishes. We also give a new proof of Massot's theorem that the contact invariant vanishes for a contact structure on $(Σ\times S^1, F \times S^1)$ described by an isolating dividing set.
We study a computational method of the hyperbolic Reidemeister torsion (also called in the literature the non-abelian Reidemeister torsion) induced by J. Porti for complete hyperbolic three-dimensional manifolds with cusps. The derivative of the twisted Alexander invariant for a hyperbolic knot exterior gives the hyperbolic torsion. We prove such a derivative formula of the twisted Alexander invariant for hyperbolic link exteriors like the Whitehead link exterior. We provide the framework for the derivative formula to work, which consists of assumptions on the topology of the manifold and on t…
On the positivity of twisted $L^2$-torsion for 3-manifolds
本文研究带空或不可压缩环面边界的紧致可定向不可约3维流形上的扭曲 $L^2$-挠率，证明无限基本群情形下其严格为正，并在上三角表示子簇上连续。
Ozsvath-Szabo contact invariants are a powerful way to prove tightness of contact structures but they are known to vanish in the presence of Giroux torsion. In this paper we construct, on infinitely many manifolds, infinitely many isotopy classes of universally tight torsion free contact structures whose Ozsvath-Szabo invariant vanishes. We also discuss the relation between these invariants and an invariant on T^3 and construct other examples of new phenomena in Heegaard--Floer theory. Along the way, we prove two conjectures of K Honda, W Kazez and G Matic about their contact topological quant…
We give an extension of Fox's formula of the Alexander polynomial for double branched covers over the three-sphere. Our formula provides the Reidemeister torsion of a double branched cover along a knot for a non-trivial one dimensional representation by the product of two factors derived from the knot group. One of the factors is determined by the twisted Alexander polynomial and the other is determined by a rational function on the character variety. As an application, we show that these products distinguish isotopy classes of two-bridge knots up to mirror images.
We generalize the familiar notions of overtwistedness and Giroux torsion in 3-dimensional contact manifolds, defining an infinite hierarchy of local filling obstructions called planar torsion, whose integer-valued order $k \ge 0$ can be interpreted as measuring a gradation in "degrees of tightness" of contact manifolds. We show in particular that any contact manifold with planar torsion admits no contact type embeddings into any closed symplectic 4-manifold, and has vanishing contact invariant in Embedded Contact Homology, and we give examples of contact manifolds that have planar k-torsion fo…
We use Reidemeister torsion to study a twisted Alexander polynomial, as defined by Turaev, for links in the projective space. Using sign-refined torsion we derive a skein relation for a normalized form of this polynomial.
Algebraic properties of twisted Alexander polynomial and Reidemeister torsion of torus knots
本文证明了扭Alexander多项式中与不可约SL_n(C)表示相关的每个系数都是SL_n(C)特征标簇上的A-值局部常函数，其中A是所有复数上的代数整数环。此外，作为Kitano和Nozaki最近结果的推广，我们显示SL_n(C)的Reidemeister扭转对于许多Seifert纤维空间是代数整数。同时，我们讨论了扭结的Reidemeister扭转的幂和，这些扭结的低维不可约表示与TQFT存在神秘关系。
Morifuji computed the twisted Alexander polynomial of twist knots for nonabelian representations. In this paper we compute the twisted Alexander polynomial and the Reidemeister torsion of genus one two-bridge knots, a class of knots which includes twist knots. As an application, we give a formula for the Reidemeister torsion of the 3-manifold obtained by a Dehn surgery on a genus one two-bridge knot.
The Alexander polynomial of a knot has been generalized in three different ways to give twisted invariants. The resulting invariants are usually referred to as twisted Alexander polynomials, higher-order Alexander polynomials and $L^2$-Alexander invariants of knots. We quickly recall the definitions and we summarize and compare some of their properties. We also report on work by the authors on $L^2$-Alexander torsions and we conclude the paper with several conjectures on $L^2$-Alexander torsions.
We calculate the twisted Alexander polynomial with the adjoint action for torus knots and twist knots. As consequences of these calculations, we obtain the formula for the nonabelian Reidemeister torsion of torus knots in \cite{Du} and a formula for the nonabelian Reidemeister torsion of twist knots that is better than the one in \cite{DHY}.
Given a finite simplicial complex, a unimodular representation of its fundamental group and a closed twisted cochain of odd degree, we define a twisted version of the Reidemeister torsion, extending a previous definition of V. Mathai and S. Wu. The main tool is a complex of piecewise smooth currents, defined by J. Dupont.
We extract a nonnegative integer-valued invariant, which we call the "order of algebraic torsion", from the Symplectic Field Theory of a closed contact manifold, and show that its finiteness gives obstructions to the existence of symplectic fillings and exact symplectic cobordisms. A contact manifold has algebraic torsion of order zero if and only if it is algebraically overtwisted (i.e. has trivial contact homology), and any contact 3-manifold with positive Giroux torsion has algebraic torsion of order one (though the converse is not true). We also construct examples for each nonnegative k of…
As a generalization of a classical result on the Alexander polynomial for fibered knots, we show in this paper that the Reidemeister torsion associated to a certain representation detects fiberedness of knots in the three sphere.
This paper attempts to investigate the space of various characteristic classes for smooth manifold bundles with local system on the total space inducing a finite holonomy covering. These classes are known as twisted higher torsion classes. We will give a system of axioms that we require these cohomology classes to satisfy. Higher Franz Reidemeister torsion and twisted versions of the higher Miller-Morita-Mumford classes will satisfy these axioms. We will show that the space oftwisted torsion invariants is two dimensional or one dimensional depending on the torsion degree and spanned by these t…
We prove duality theorems for twisted Reidemeister torsions and twisted Alexander polynomials generalizing the results of Turaev. As a corollary we determine the parity of the degrees of twisted Alexander polynomials of 3-manifolds in many cases.
In this paper we extend Badzioch's, Dorabiala's, and Williams' definition of cohomological higher smooth torsion to a twisted cohomological higher torsion invariant. Additionally, we show that this still satisfies geometric additivity and transfer, and will also satisfy additivity and transfer for coefficients.
Given a homomorphism from a link group to a group, we introduce a $K_1$-class in another way, which is a generalization of the 1-variable Alexander polynomial. We compare the $K_1$-class with $K_1$-classes in \cite{Nos} and with Reidemeister torsions. As a corollary, we show a relation to Reidemeister torsions of finite cyclic covering spaces, and show reciprocity in some senses.
On twisted torsion of compact $3$-manifolds
本文研究具有非空边界的非球形紧致3-流形与另一3-流形去掉3-胞腔后的流形，建立扭曲Reidemeister挠率乘法性质与连通和运算的关系，以理解其拓扑与几何。
In this paper, we prove that the Reidemeister torsion twisted by the adjoint representation, which is considered as a 1-form, on the SU(2)-character variety of a knot exterior is invariant under mutation along a Conway sphere.
Let $G$ be the group of complex points of a real semi-simple Lie group whose fundamental rank is equal to 1, e.g. $G= \SL_2 (\C) \times \SL_2 (\C)$ or $\SL_3 (\C)$. Then the fundamental rank of $G$ is $2,$ and according to the conjecture made in \cite{BV}, lattices in $G$ should have 'little' --- in the very weak sense of 'subexponential in the co-volume' --- torsion homology. Using base change, we exhibit sequences of lattices where the torsion homology grows exponentially with the \emph{square root} of the volume. This is deduced from a general theorem that compares twisted and untwisted $L^…
Adjoint twisted Reidemeister torsion and Gram matrices
本文计算了闭定向及带边双曲3-流形的伴随扭曲Reidemeister挠，基于Dehn填充和理想三角剖分给出了新公式，并在量子不变量的渐近展开研究中发挥重要作用。
Twisted Kuperberg invariants of knots and Reidemeister torsion via twisted Drinfeld doubles
本文通过扭曲Drinfeld双构造纽结的量子不变量，推广了缝合流形的扭曲Kuperberg不变量至非对合情形，并将SL(n,C)扭曲Reidemeister挠率表示为Reshetikhin-Turaev不变量。
We show that the reduced $\mathrm{SL}_2(\mathbb{C})$-twisted Burau representation can be obtained from the quantum group $\mathcal{U}_q(\mathfrak{sl}_2)$ for $q = i$ a fourth root of unity and that representations of $\mathcal{U}_q(\mathfrak{sl}_2)$ satisfy a type of Schur-Weyl duality with the Burau representation. As a consequence, the $\operatorname{SL}_2(\mathbb{C})$-twisted Reidemeister torsion of links can be obtained as a quantum invariant. Our construction is closely related to the quantum holonomy invariant of Blanchet, Geer, Patureau-Mirand, and Reshetikhin, and we interpret their in…
The twisted Ruelle zeta function on compact hyperbolic orbisurfaces and Reidemeister-Turaev torsion
该文研究紧致双曲轨道曲面上扭曲 Ruelle zeta 函数在零点取值与 Reidemeister-Turaev 挠率的关系，推广 Fried 猜想至非酉表示并解决相位符号问题，同时计算消失阶与首项系数。
In this paper we consider the cohomology of a closed arithmetic hyperbolic 3-manifold with coefficients in the local system defined by the even symmetric powers of the standard representation of SL(2,C). The cohomology is defined over the integers and is a finite abelian group. We show that the order of the 2nd cohomology grows exponentially as the local system grows. We also consider the twisted Ruelle zeta function of a closed arithmetic hyperbolic 3-manifold and we express the leading coefficient of its Laurent expansion at the origin in terms of the orders of the torsion subgroups of the c…
Let $Σ_{g,n}$ be a compact oriented surface with genus $g\geq 2$ bordered by $n$ circles. Due to Witten, the twisted Reidemeister torsion coincides with a power of the Atiyah-Bott-Goldman-Narasimhan symplectic form on the space of representations of $π_1(Σ_{g,0})$ in any semi-simple Lie group. In the present paper, we first obtain a multiplicative gluing formula for the twisted Reidemeister torsion of $Σ_{g,0}$ in terms of torsions of $Σ_{2,2},$ $Σ_{2,1},$ and boundary circles $\mathbb{S}^1.$ Then, by using Heusener and Porti's results on $Σ_{g,n},$ we show that the symplectic volume form on t…
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