arXiv research

This paper explores twisted Lagrangian tori in C^2 and their Hamiltonian stationarity.

Study shows twist tori equidistribute in moduli space, with other families having singular distributions.

The paper proves properties of branched covers of specific knots and tori.

The paper proves uniqueness of closed timelike geodesics on de-Sitter tori with one singularity.

New invariant shows Dehn twist on connected sum of homology tori is not isotopic to identity.

Integral filling volume of mapping tori grows sublinearly with complexity.

Tight embeddings of 2-tori in 3D space contain short loops.

The paper constructs triangulations for double twist knots using geometric methods.

Smooth tori in S^4 are topologically unknotted.

Proves geodesic connections on 2-torus without invariant tori.

New families of twisted torus knots found with essential surfaces.

Establishes convexity and coercivity of K-energy functional for complex tori.

Study of Veech surfaces and their twist tori on moduli spaces of abelian differentials.

We provide the twisted Alexander polynomials of finite abelian covers over three-dimensional manifolds whose boundary is a finite union of tori. This is a generalization of a well-known formula for the usual Alexander polynomial of knots in finite cyclic branched covers over the three-dimensional sphere.

Geometrically, twist numbers on punctured tori are dense and non-continuous.

Researchers calculate dimensions of skein modules for 2-torus mapping tori.

Study of Dehn twists in free groups generates right-angled Artin groups.

Maps from rational homology solid tori yield rank inequalities in Heegaard Floer homology.

We calculate the Heegaard Floer homologies$HF^+(M,s) for mapping tori M associated to certain surface diffeomorphisms, where s is any Spin^c structure on M whose first Chern class is non-torsion. Let gamma and delta be a pair of geometrically dual nonseparating curves on a genus g Riemann surface Sigma_g, and let sigma…

We provide a systematic approach to describing the Ramond-Ramond (RR) fields as elements in twisted differential K-theory. This builds on a series of constructions by the authors on geometric and computational aspects of twisted differential K-theory, which to a large extent were originally motivated by this problem. I…

The author recently proved the existence of an infinite order cork: a compact, contractible submanifold $C$ of a 4-manifold and an infinite order diffeomorphism $f$ of $\partial C$ such that cutting out $C$ and regluing it by distinct powers of $f$ yields pairwise nondiffeomorphic manifolds. The present paper exhibits …

New findings show the Gilmer-Masbaum map isn't always one-to-one.

We extend the techniques in a previous paper to calculate the Heegaard Floer homology groups for fibered 3-manifolds M whose monodromy is a power of a Dehn twist about a genus-1 separating circle on a surface of genus g > 1. We only consider non-torsion Spin^c-structures on M.

The paper analyzes spectral properties of connection Laplacian on tori, proving convergence to real torus.

We introduce a new family of metrics, called functional metrics, on noncommutative tori and study their spectral geometry. We define a class of Laplace type operators for these metrics and study their spectral invariants obtained from the heat trace asymptotics. A formula for the second density of the heat trace is obt…

Modifying the method of [21], we compute the perturbed $HF^+$ for some special classes of fibered three manifolds in the second highest spin$^c$-structures $S_{g-2}$. The special classes considered in this paper include the mapping tori of Dehn twists along a single non-separating curve and along a transverse pair of c…

The paper explores non-trivial welded knots and ribbon torus-knots, proving their existence.

The twist construction is a geometric model of T-duality that includes constructions of nilmanifolds from tori. This paper shows how one-dimensional foliations on manifolds may be used in a shear construction, which in algebraic form builds certain solvable Lie groups from Abelian ones. We discuss other examples of geo…

Let K \subset Y be a knot in a three manifold which admits a longitude-framed surgery such that the surgered manifold has first Betti number greater than that of Y. We find a formula which computes the twisted Floer homology of the surgered manifold, in terms of twisted knot Floer homology. Using this, we compute the t…

Much work has been done recently towards trying to understand the topological significance of being an L-space. Building on work of Rasmussen and Rasmussen, we give a topological characterisation of Floer simple manifolds such that all non-longitudinal fillings are L-spaces. We use this to partially classify L-space tw…

This paper is the the third part of a series of paper whose aim is to use of the framework of \emph{twisted spectral triples} to study conformal geometry from a noncommutive geometric viewpoint. In this paper we reformulate the inequality of Vafa-Witten \cite{VW:CMP84} in the setting of twisted spectral triples. This i…

The paper characterizes gaps in minimal foliations on tori using energy criteria.

Study the algebraic action of torus on knot complement's skein module.

We consider two families of algebraic varieties $Y_n$ indexed by natural numbers $n$: the configuration space of unordered $n$-tuples of distinct points on $\mathbb{C}$, and the space of unordered $n$-tuples of linearly independent lines in $\mathbb{C}^n$. Let $W_n$ be any sequence of virtual $S_n$-representations give…

This paper is a continuation on the 2012 paper on "Cutting Twisted Solid Tori (TSTs)", in which we considered twisted solid torus links (tst links). We generalize the notion of tst links to "surgerized tst links": recall that when performing $Φ^μ(n(τ), d(τ), M)$ on a tst $\langle τ\rangle$ where $M$ is odd, we obtain t…

The paper tackles mapping tori by proposing a new approach to 3d-3d correspondence.

The paper finds manifolds with many Rarita-Schwinger fields.

Applying logarithmic transformations along 2-tori, we construct a generalized complex structure J_n with n type changing luci for every $n\geq 0$ on genus 1-Lefschetz fibrations with a cusp neighborhood, which include elliptic surfaces with non-zero euler characteristic. Applying a technique of broken Lefschetz fibrati…

After fixing a marking (V, W) of a quasifuchsian punctured torus group G, the complex length l_V and the complex twist tau_V,W parameters define a holomorphic embedding of the quasifuchsian space QF of punctured tori into C^2. It is called the complex Fenchel-Nielsen coordinates of QF. For a complex number c, let Q_gam…

Area-preserving diffeomorphisms of a 2-disc can be regarded as time-1 maps of (non-autonomous) Hamiltonian flows on solid tori, periodic flow-lines of which define braid (conjugacy) classes, up to full twists. We examine the dynamics relative to such braid classes and define a braid Floer homology. This refinement of t…

Extends noncommutative deformations of holomorphic line bundles on complex tori and their mirror partners.

Recently, Ian Agol introduced a class of "veering" ideal triangulations for mapping tori of pseudo-Anosov homeomorphisms of surfaces punctured along the singular points. These triangulations have very special combinatorial properties, and Agol asked if these are "geometric", i.e. realised in the complete hyperbolic met…

Study of exceptional algebroids in type IIA string theory.

The twist construction is a method to build new interesting examples of geometric structures with torus symmetry from well-known ones. In fact it can be used to construct arbitrary nilmanifolds from tori. In our previous paper, we presented a generalization of the twist, a shear construction of rank one, which allowed …

Starting from a higher Courant bracket associated to exceptional generalized geometry, we provide a systematic derivation of all types of fluxes and their Bianchi identities for four-dimensional compactifications of M-theory. We show that these fluxes may be understood as generalized Wess-Zumino terms in certain topolo…

We considered a surgery, called Lagrangian attaching disk surgery, that can be applied to a Lagrangian surface L at the presence of a Lagrangian attaching disk D, to obtain a new Lagrangian surface L' which is always smoothly isotopic to L. We showed that this type of surgery includes all even generalized Dehn twists a…

The curvature of the noncommutative torus $T^2_θ$ ($θ$ irrational) endowed with a noncommutative conformal metric has been the focus of attention of several recent works. Continuing the approach taken in the paper [A. Connes and H. Moscovici, http://arxiv.org/abs/1110.3500] we extend the study of the curvature to twist…

This paper derives finite generating sets for liftable mapping class groups of certain branched covers of tori.