Christopher Scaduto
Crowley and Nordström introduced an invariant of $G_2$-structures on the tangent bundle of a closed 7-manifold, taking values in the integers modulo 48. Using the spectral description of this invariant due to Crowley, Goette and Nordström, we compute it for many of the closed torsion-free $G_2$-manifolds defined by Joyce's generalized Kummer construction.
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Previously the two of the authors defined a notion of dual Calabi-Yau manifolds in a G_2 manifold, and described a process to obtain them. Here we apply this process to a compact G_2 manifold, constructed by Joyce, and as a result we obtain a pair of Borcea-Voisin Calabi-Yau manifolds, which are known to be mirror duals of each other.
Spectral invariants of Joyce orbifolds
本文提出两种新的谱不变量，用于区分Joyce流形，并发现其与扭曲的Epsteinζ函数存在联系。
The study of fibrations of the target manifolds of string/M/F-theories has provided many insights to the dualities among these theories or even as a tool to build up dualities since the work of Strominger, Yau, and Zaslow on the Calabi-Yau case. For M-theory compactified on a Joyce manifold $M^7$, the fact that $M^7$ is constructed via a generalized Kummer construction on a 7-torus ${\smallBbb T}^7$ with a torsion-free $G_2$-structure $ϕ$ suggests that there are natural fibrations of $M^7$ by ${\smallBbb T}^3$, ${\smallBbb T}^4$, and K3 surfaces in a way governed by $ϕ$. The local picture of s…
The present paper deals with mirror symmetry aspects of compact ``barely'' $G_2$ manifolds, that is, $G_2$ manifolds of the form (CY$\times S^1)/\mathbb{Z}_2$. We propose that the mirror of any barely $G_2$ manifold is another barely one and which is constructed as a fibration of the \emph{mirror} of the CY base. Also, we describe the Joyce manifolds of the first kind as ``barely'' and we show that the underlying CY of all the family is self-mirror with $h^{1,1}=h^{2,1}=19$. We thus propose that the mirror of a Joyce space of the first kind will be another Joyce space of the first kind.We also…
Entropy-Minimizing Diffeomorphisms on a $G_2$-Manifold
本文研究了 $G_2$ 流形上的微分同胚，构造了实现拓扑熵下界的无限多个例子，并探讨了其与代数表面动力学的类比及局限性。
We define a Z/48-valued homotopy invariant nu of a G_2-structure on the tangent bundle of a closed 7-manifold in terms of the signature and Euler characteristic of a coboundary with a Spin(7)-structure. For manifolds of holonomy G_2 obtained by the twisted connected sum construction, the associated torsion-free G_2-structure always has nu = 24. Some holonomy G_2 examples constructed by Joyce by desingularising orbifolds have odd nu. We define a further homotopy invariant xi of G_2-structures such that if M is 2-connected then the pair (nu, xi) determines a G_2-structure up to homotopy and diff…
On Counting Flat Connections over $G_2$-Orbifolds
本文研究无挠 $G_2$-轨道空间上平坦丛的 $G_2$-瞬子模空间，通过引入和乐扰动证明其紧致与光滑性，定义了在 $C^0$-形变下不变的整数值不变量，并计算了 Joyce 构造中特定轨道空间的该不变量。
We solve some computational problems for triangulated closed three-dimensional manifolds using groups of simplicial homology and cohomology modulo 2. Two efficient algorithms for computing the intersection numbers of 1- and 2-dimensional cycles are developed. By means of these algorithms it is possible to construct a basis of cohomology group from the homology group of two cycles of complementary dimensions.
Algorithms that decompose a manifold into simple pieces reveal the geometric and topological structure of the manifold, showing how complicated structures are constructed from simple building blocks. This note describes a way to algorithmically construct a trisection, which describes a $4$-dimensional manifold as a union of three $4$-dimensional handlebodies. The complexity of the $4$-manifold is captured in a collection of curves on a surface, which guide the gluing of the handelbodies. The algorithm begins with a description of a manifold as a union of pentachora, or $4$-dimensional simplice…
Computing closed essential surfaces in 3-manifolds
本文提出一种基于Jaco-Oertel框架的实用算法，用于测试三角剖分3-流形是否包含闭本质曲面，并在大量流形上验证得到新结果，方法亦适用于纽结与3-流形识别等计算难题。
The notion of Gem-Matveev complexity has been introduced within crystallization theory, as a combinatorial method to estimate Matveev's complexity of closed 3-manifolds; it yielded upper bounds for interesting classes of such manifolds. In this paper we extend the definition to the case of non-empty boundary and prove that for each compact irreducible and boundary-irreducible 3-manifold it coincides with the modified Heegaard complexity introduced by Cattabriga, Mulazzani and Vesnin. Moreover, via Gem-Matveev complexity, we obtain an estimation of Matveev's complexity for all Seifert 3-manifol…
A fundamental way to study 3-manifolds is through the geometric lens, one of the most prominent geometries being the hyperbolic one. We focus on the computation of a complete hyperbolic structure on a connected orientable hyperbolic 3-manifold with torus boundaries. This family of 3-manifolds includes the knot complements. This computation of a hyperbolic structure requires the resolution of gluing equations on a triangulation of the space, but not all triangulations admit a solution to the equations. In this paper, we propose a new method to find a triangulation that admits a solution to the …
In this paper, we give an algorithm to compute the hat version of the Heegaard Floer homology of a closed oriented three-manifold. This method also allows us to compute the filtrations coming from a null-homologous link in a three-manifold.
A single qubit may be represented on the Bloch sphere or similarly on the $3$-sphere $S^3$. Our goal is to dress this correspondence by converting the language of universal quantum computing (UQC) to that of $3$-manifolds. A magic state and the Pauli group acting on it define a model of UQC as a positive operator-valued measure (POVM) that one recognizes to be a $3$-manifold $M^3$. More precisely, the $d$-dimensional POVMs defined from subgroups of finite index of the modular group $PSL(2,\mathbb{Z})$ correspond to $d$-fold $M^3$- coverings over the trefoil knot. In this paper, one also invest…
Bordered Heegaard Floer homology is an invariant for three-manifolds with boundary. In particular, this invariant associates to a handle decomposition of a surface F a differential graded algebra, and to an arc slide between two handle decompositions, a bimodule over the two algebras. In this paper, we describe these bimodules for arc slides explicitly, and then use them to give a combinatorial description of HF^ of a closed three-manifold, as well as the bordered Floer homology of any 3-manifold with boundary.
In this paper, we develop general techniques for computing the G-index of a closed, spin, hyperbolic 2- or 4-manifold, and apply these techniques to compute the G-index of the fully symmetric spin structure of the Davis hyperbolic 4-manifold.
Computing Heegaard Floer invariants of closed contact 3-manifolds from open books
本文提出两个改进的SageMath程序，从抽象开卷计算闭接触3-流形的hat版本Heegaard Floer同调，并用于计算Ozsváth--Szabó接触不变量及检测谱阶数有限性这一更强的Stein可填充性障碍。
I give a formula for computing the number of regular $Γ$-coverings of closed orientable Seifert 3-manifolds, for a given finite group $Γ$. The number is computed using a 3d TQFT with finite gauge group, through a cut-and-glue process.
By means of a slight modification of the notion of GM-complexity, the present paper performs a graph-theoretical approach to the computation of (Matveev's) complexity for closed orientable 3-manifolds. In particular, the existing crystallization catalogue C^{28}, due to Lins, is used to obtain upper bounds for the complexity of closed orientable 3-manifolds triangulated by at most 28 tetrahedra. The experimental results actually coincide with the exact values of complexity, for all but three elements. Moreover, in the case of at most 26 tetrahedra, the exact value of the complexity is shown to…
We perform two explicit computations of bordered Heegaard Floer invariants. The first is the type D trimodule associated to the trivial S^1 bundle over the pair of pants P. The second is a bimodule that is necessary for self-gluing, when two torus boundary components of a bordered manifold are glued to each other. Using the results of these two computations, we describe an algorithm for computing HF-hat of any graph manifold.
Triple-cup product forms of 3-manifolds and Heegaard diagrams
本文基于Heegaard图给出3-流形三杯积形式的显式计算公式，并展示其可通过Turaev的Heegaard曲面同伦交积形式还原。
For a compact 3-manifold $N$ with non-empty boundary, Zickert gave a combinatorial formula for computing the volume and Chern-Simons invariant of a boundary parabolic representation $π_1(N)\rightarrow \mathrm{PSL}(2,\mathbb{C})$. In this paper, we introduce a notion of deformed Ptolemy varieties and extend the formula of Zickert to a representation that is not necessarily boundary parabolic. This allows us to compute the volume and Chern-Simons invariant of a $\mathrm{PSL}(2,\mathbb{C})$-representation of a closed 3-manifold.
Hardness of computation of quantum invariants on 3-manifolds with restricted topology
本文证明了在Reshetikhin-Turaev模型中，存在确定性多项式时间算法可将任意闭3-流形转换为具有相同量子不变量但拓扑受限的流形，表明计算量子不变量的难度在严格限制输入拓扑与组合结构时依然存在。
For a closed 3-manifold $M$ in a certain class, we give a presentation of the cellular chain complex of the universal cover of $M$. The class includes all surface bundles, some surgeries of knots in $S^3$, some cyclic branched cover of $S^3$, and some Seifert manifolds. In application, we establish a formula for calculating the linking form of a cyclic branched cover of $S^3$, and develop procedures of computing some Dijkgraaf-Witten invariants.
The Teichmueller polynomial of a fibered 3-manifold plays a useful role in the construction of mapping class having small stretch factor. We provide an algorithm that computes this polynomial of the fibered face associated to a pseudo-Anosov mapping class of a disc homeomorphism. As a byproduct, our algorithm allows us to derive all the relevant informations on the topology of the different fibers that belong to the fibered face.
We propose a way of computing 4-manifold invariants, old and new, as chiral correlation functions in half-twisted 2d $\mathcal{N}=(0,2)$ theories that arise from compactification of fivebranes. Such formulation gives a new interpretation of some known statements about Seiberg-Witten invariants, such as the basic class condition, and gives a prediction for structural properties of the multi-monopole invariants and their non-abelian generalizations.
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