arXiv research

Given a genus-$g$ Heegaard splitting of the $3$-sphere with $g \ge 3$, we show that the primitive disk complex for the splitting is not weakly closed under disk surgery operation. That is, there exist two primitive disks in one of the handlebodies of the splitting such that any disk surgery on one along the other one y…

Note on connectedness of primitive disk complex.

Let $K$ be an unknot in $8$-bridge position in the $3$-sphere. We give an example of a pair of weak reducing disks $D_1$ and $D_2$ for $K$ such that both disks obtained from $D_i$ ($i = 1, 2$) by a surgery along any outermost disk in $D_{3-i}$, cut off by an outermost arc of $D_i \cap D_{3-i}$ in $D_{3-i}$, are not wea…

New knots bound multiple non-isotopic ribbon disks.

We study cobordisms and cobordisms rel boundary of PL locally-flat disk knots $D^{n-2}\into D^n$. Cobordisms of disk knots that do not fix the boundary sphere knots are easily classified by the cobordism properties of these boundaries, and any two even-dimensional disk knots with isotopic boundary knots are cobordant r…

Khovanov homology fails to differentiate certain slice disks.

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…

We study the classification of slice disks of knots up to isotopy and diffeomorphism using an invariant in knot Floer homology. We compute the invariant of a slice disk obtained by deform-spinning, and show that it can be effectively used to distinguish non-isotopic slice disks with diffeomorphic complements. Given a s…

A criterion for Whitney disks connects intersections in 3-manifold homology.

New disks found with similar outer shapes.

We construct an infinite family of slice disks with the same exterior, which gives an affirmative answer to an old question asked by Hitt and Sumners in 1981. Furthermore, we prove that these slice disks are ribbon disks.

Study constructs disks with curved boundaries in a 3D ball.

Paper proposes a method to predict disk failures using multi-layer domain adaptive learning.

New spanning 3-disks found for unlink in 4-sphere.

Even-dimensional simply connected manifolds that are rational homology spheres and double disk bundles are homeomorphic to spheres.

We study the relationship between fibered ribbon 1-knots and fibered ribbon 2-knots by studying fibered slice disks with handlebody fibers. We give a characterization of fibered homotopy-ribbon disks and give analogues of the Stallings twist for fibered disks and 2-knots. As an application, we produce infinite families…

Rare Teichmüller disks converge to small limit sets.

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

Proves existence of non-planar minimal disks in ellipsoids.

A Seifert surface F for a knot K is disk decomposable if there is a taut sutured manifold heirarchy for the complement of F, whose decomposing surfaces are all disks. It follows that F has minimal genus for the knot K, and has handlebody complement, i.e., F is free. We show that these necessary conditions for disk deco…

Disk and sphere graphs embed quasi-isometrically in R^2.

Three configurations of two perpendicular disks in R^3 are examined, the first in which the disks share centers and the other two in which the disks touch at precisely one point. Volume, surface area and mean width calculations dominate the discussion. Integrated mean curvature also appears as an indirect way to comput…

We deform a minimal disk in $\mathbb{R}^4$ with a branch point into symplectic minimally immersed disks with only transverse double points.

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…

We classify the resolution graphs of weighted homogeneous surface singularities which admit rational homology disk smoothings. The nonexistence of rational homology disk smoothings is shown by symplectic geometric methods, while the existence is verified via smoothings of negative weights. In particular, it is shown th…

This paper shows that every totally-geodesic isometry from the unit disk to a finite-dimensional Teichmüller space for the intrinsic Kobayashi metric is either holomorphic or anti-holomorphic; in particular, it is a Teichmüller disk. Additionally, a similar result is proved for a large class of disk-rigid domains, whic…

The paper explores slice disks and their properties using satellite operations and knot Floer homology.

Shorter proof for wave front length in Euclidean disk

New examples show non-locally-flat PL-disk bounds in rational homology balls but not in integer homology balls.

The paper proves a disk's energy minimizer is holomorphic and calculates its Morse index.

Disk and sphere graphs embed quasi-isometrically into Euclidean spaces.

Study AFPP of unions of convex digital disks in 2D.

Quadratic bounds found for graph dimensions.

The study finds at least two short, simple geodesic chords on a disk with convex boundary.

Proves uniqueness of capillary disks in 3D domains.

Can certain shapes be drawn with a pencil and eraser?

Suppose a knot in a $3$-manifold is in $n$-bridge position. We consider a reduction of the knot along a bridge disk $D$ and show that the result is an $(n-1)$-bridge position if and only if there is a bridge disk $E$ such that $(D, E)$ is a cancelling pair. We apply this to an unknot $K$, in $n$-bridge position with re…

Authors prove a contact structure result using branched covers and overtwisted disks.

In this paper we prove a theorem concerning lamination limits of sequences of compact disks $M_n$ embedded in $\mathbb{R}^3$ with constant mean curvature $H_n$, when the boundaries of these disks tend to infinity. This theorem generalizes to the non-zero constant mean curvature case Theorem 0.1 by Colding and Minicozzi…

Paper addresses class imbalance in disk SMART dataset using GANs and genetic algorithms.

The study proves the existence of free boundary minimal disks in convex regions.

This paper proves all endomorphisms of framed little disk operad are automorphisms.

New examples of knots with infinitely many inequivalent slice disks.

4D theorem for disks, generalizing previous work.

We underline some differences between the geometric aspect of Berezin's approach to quantization on homogeneous Kähler manifolds and Bergman's construction for bounded domains in $\mathbb{C}^n$. We construct explicitly the Bergman representative coordinates for the Siegel-Jacobi disk $\mathcal{D}^J_1$, which is a parti…

Holomorphic disks on compact Lagrangian surfaces are shown to exist.

New ribbon disks in 4D space, non-isotopic to each other.

Solves Christoffel problem for disk area measures on spheres.