arXiv research

Note on connectedness of primitive disk complex.

For a genus two Heegaard splitting of a lens space, the primitive disk complex is defined to be the full subcomplex of the disk complex for one of the handlebodies of the splitting spanned by all vertices of primitive disks. In this work, we describe the complete combinatorial structure of the primitive disk complex fo…

Disk surgery on primitive disks of genus-3 Heegaard splittings of 3-sphere yields no primitive disks.

Using generalized Riemann maps, normal forms for almost complex domains (D, J) with singular foliations by stationary disks are defined. Such normal forms are used to construct counterexamples and to determine intrinsic conditions, under which the stationary disks are extremal disks for the Kobayashi metric or determin…

Disk complexes show 3-sphere surfaces are topologically minimal.

Disk Embeddings tackle embedding DAGs with exponential growth.

Contractibility of unknot's weak reducing disks shown.

Given a stabilized Heegaard splitting of a $3$-manifold, the primitive disk complex for the splitting is the subcomplex of the disk complex for a handlebody in the splitting spanned by the vertices of the primitive disks. In this work, we study the structure of the primitive disk complex for the genus two Heegaard spli…

New homomorphism proven using immersed curves on disks.

We show that the automorphism group of the disk complex is isomorphic to the handlebody group. Using this, we prove that the outer automorphism group of the handlebody group is trivial.

We first study holomorphic isometries from the Poincaré disk into the product of the unit disk and the complex unit $n$-ball for $n\ge 2$. On the other hand, we observe that there exists a holomorphic isometry from the product of the unit disk and the complex unit $n$-ball into any irreducible bounded symmetric domain …

Study uses knot Floer homology to distinguish slice disks.

This paper classifies planar-Rips complexes and their unit disk graphs up to homotopy.

Kervaire's sphere-link is equivalent to a ribbon sphere-link, simplifying complex 2-complexes.

Classifies geodesics for Carathéodory metric on Teichmüller spaces.

Study of disks in complex projective space with specific fundamental groups.

We prove a relative isoperimetric inequalities for Lagrangian half disks in $\mathbb{C}^2$ with respect to a Lagrangian plane, or a complex plane, or a union of any two of Lagrangian or complex planes that intersect transversally at the origin.

Classifies fake surfaces up to complexity 5.

Alternative proof for 2-bridge knots' bridge positions.

We give a distance estimate for the metric on the disk complex and show that it is Gromov hyperbolic. As another application of our techniques, we find an algorithm which computes the Hempel distance of a Heegaard splitting, up to an error depending only on the genus.

Essential arcs on punctured disks intersect at most twice, proving a size limit.

Corks transform complex curves without changing topology.

Holomorphic immersions blocked in 9D real hypersurface with specific signature.

We simplify proof of the theorem that close to any pseudoholomorphic disk there passes a pseudoholomorphic disk of arbitrary close size with any pre-described sufficiently close direction. We apply these results to the Kobayashi and Hanh pseudodistances. It is shown they coincide in dimensions higher than four. The res…

For a boundary-reducible $3$-manifold $M$ with $\partial M$ a genus $g$ surface, we show that if $M$ admits a genus $g+1$ Heegaard surface $S$, then the disk complex of $S$ is simply connected. Also we consider the connectedness of the complex of reducing spheres. We investigate the intersection of two reducing spheres…

In this paper, we discuss a rigidity property for holomorphic disks in Teichmüller space. In fact, we give a refinement of Tanigawa's rigidity theorem. We will also treat the rigidity property of holomorphic disks for complex manifolds. We observe the rigidity property is valid for bounded strictly pseudoconvex domains…

In this paper we show that if the minimal good resolution graph of a normal surface singularity contains at least two nodes (i.e. vertex with valency at least 3) then the singularity does not admit a smoothing with Milnor fiber having rational homology equal to the rational homology of the 4-disk $D^4$ (called a ration…

Spheres in curve complexes are almost simply connected.

Holomorphic disks on compact Lagrangian surfaces are shown to exist.

Classifies degenerations of complex projective plane with rational singularities.

In this paper we extend the notion of the Kobayashi-Royden pseudonorm for almost complex manifolds. Its basic properties known from the complex analysis are preserved in the nonintegrable case as well. The main theorem on coincidence of the pseudodistance induced by this pseudonorm with the Kobayashi pseudodistance for…

New concept of boundary braids defined for disk configurations.

This paper investigates symmetric ribbon numbers of low-complexity knots.

Given a hyperbolic surface and a simple closed geodesic on it, complex-twists along the curve produce a holomorphic family of deformations in Teichmüller space, degenerating to the Riemann surface where it is pinched. We show there is a corresponding Teichmüller disk such that the two are strongly asymptotic, in the Te…

We realize Stasheff's multiplihedron geometrically as the moduli space of stable quilted disks. This generalizes the geometric realization of the associahedron as the moduli space of stable disks. We show that this moduli space is the non-negative real part of a complex moduli space of stable scaled marked curves.

This paper describes the relationship between the first non-vanishing Milnor invariants of a classical link and the intersection invariant of a twisted Whitney tower. This is a certain 2-complex in the 4-ball, built from immersed disks bounded by the given link in the 3-sphere together with finitely many `layers' of Wh…

The paper finds representations of surface groups in SO(4,1) with specific curvature properties.

In this paper we give a necessary combinatorial condition for a negative--definite plumbing tree to be suitable for rational blow--down, or to be the graph of a complex surface singularity which admits a rational homology disk smoothing. New examples of surface singularities with rational homology disk smoothings are a…

RODMAN improves ML-based disk failure prediction accuracy in cloud environments.

Paper solves long-standing Gaussian curvature conjecture for minimal graphs.

The paper characterizes and studies compact subsets of complex projective space with specific line intersection properties.

Uniform diameter bound for reflection group disk patterns.

The paper develops obstructions for embedding 2D complexes into 4D space.

New method uses Floer homology to create exotic disk pairs.

Given a genus two Heegaard splitting for a non-prime 3-manifold, we define a special subcomplex of the disk complex for one of the handlebodies of the splitting, and then show that it is contractible. As applications, first we show that the complex of Haken spheres for the splitting is contractible, which refines the r…

We prove some rigidity theorems for configurations of closed disks. First, fix two collections $\mathcal{C}$ and $\tilde{\mathcal{C}}$ of closed disks in the Riemann sphere $\hat{\mathbb{C}}$, sharing a contact graph which (mostly-)triangulates $\hat{\mathbb{C}}$, so that for all corresponding pairs of intersecting dis…

Moduli spaces of holomorphic disks in a complex manifold Z, with boundaries constrained to lie in a maximal totally real submanifold P, have recently been found to underlie a number of geometrically rich twistor correspondences. The purpose of this paper is to develop a general Fredholm regularity criterion for holomor…

New definition of skein lasagna module for specific 4-manifolds.