arXiv research

If M is a manifold with compressible boundary, we analyze essential disks in M, as well as incompressible, but not necessarily boundary incompressible, surfaces in M. We are most interested in the case where M is a handlebody or compression body. The analysis depends on a new normal surface theory. We hope the normal s…

This paper studies properties of weak reducing pairs in critical Heegaard splittings.

We define a laminar branched surface to be a branched surface satisfying the following conditions: (1) Its horizontal boundary is incompressible; (2) there is no monogon; (3) there is no Reeb component; (4) there is no sink disk (after eliminating trivial bubbles in the branched surface). The first three conditions are…

A manifold M is simple if it contains no essential disk, sphere, annulus or torus. If M is simple and two Dehn fillings M(r_1), M(r_2) are nonsimple, then there is an upper bound on Δ(r_1,r_2), the geometric intersection number between r_1 and r_2. There are 10 possibilities, depending on the types of M(r_i). In this p…

Let $D_n$ be the $n$-punctured disk. We prove that a family of essential simple arcs starting and ending at the boundary and pairwise intersecting at most twice is of size at most $\binom{n+1}{3}$. On the way, we also show that any nontrivial square complex homeomorphic to a disk whose hyperplanes are simple arcs inter…

Study on cylindrical handlebody-knots with symmetry and rigidity properties.

In this paper we show that for a given 3-manifold and a given Heegaard splitting there are finitely many preferred decomposing systems of $3g - 3$ disjoint essential disks. These are characterized by a combinatorial criterion which is a slight strengthening of Casson-Gordon's rectangle condition. This is in contrast to…

We consider a Heegaard splitting M=H_1 \cup_S H_2 of a 3-manifold M having an essential disk D in H_1 and an essential surface F in H_2 with |D \cap F|=1. (We require that boundary of F is in S when H_2 is a compressionbody with non-empty "minus" boundary.) Let F be a genus g surface with n boundary components. From S,…

The study embeds graphs on translation surfaces, proving essential-systolic embeddings and estimating surface genera.

Wu has shown that if a link or a knot $L$ in $S^3$ in thin position has thin spheres, then the thin sphere of lowest width is an essential surface in the link complement. In this paper we show that if we further assume that $L \subset S^3$ is prime, then the thin sphere of lowest width also does not have any vertical c…

We extend the notion of thin multiple Heegaard splittings of a link in a 3-manifold to take into consideration not only compressing disks but also cut-disks for the Heegaard surfaces. We prove that if H is a c-strongly compressible bridge surface for a link K contained in a closed orientable irreducible 3-manifold M th…

Suppose N is a compressible boundary component of a compact orientable irreducible 3-manifold M and Q is an orientable properly embedded essential surface in M in which each component is incident to N and no component is a disk. Let VN and QN denote respectively the sets of vertices in the curve complex for N represent…

We show that for any extreme curve in a 3-manifold M, there exist a canonical mean convex hull containing all least area disks spanning the curve. Similar result is true for asymptotic case in hyperbolic 3-space such that for any asymptotic curve, there is a canonical mean convex hull containing all minimal planes span…

Factorization homology theories of topological manifolds, after Beilinson, Drinfeld and Lurie, are homology-type theories for topological $n$-manifolds whose coefficient systems are $n$-disk algebras or $n$-disk stacks. In this work we prove a precise formulation of this idea, giving an axiomatic characterization of fa…

It is a well-known procedure for constructing a torus knot or link that first we prepare an unknotted torus and meridian disks in the complementary solid tori of it, and second smooth the intersections of the boundary of meridian disks uniformly. Then we obtain a torus knot or link on the unknotted torus and its Seifer…

We consider the Dirichlet Laplacian in a two-dimensional strip composed of segments translated along a straight line with respect to a rotation angle with velocity diverging at infinity. We show that this model exhibits a "raise of dimension" at infinity leading to an essential spectrum determined by an asymptotic thre…

We bound the hyperbolic volumes of a large class of knots and links, called homogeneously adequate knots and links, in terms of their diagrams. To do so, we use the decomposition of these links into ideal polyhedra, developed by Futer, Kalfagianni, and Purcell. We identify essential product disks in these polyhedra.

We use an accessibility result of Delzant and Potyagailo to prove Swarup's Strong Accessibility Conjecture for Gromov hyperbolic groups with no 2-torsion. It follows that, if M is an irreducible, orientable, compact 3-manifold with hyperbolic fundamental group, then any hierarchy in which M is decomposed alternately al…

A Heegaard splitting of a closed, orientable three-manifold satisfies the disjoint curve property if the splitting surface contains an essential simple closed curve and each handlebody contains an essential disk disjoint from this curve [Thompson, 1999]. A splitting is full if it does not have the disjoint curve proper…

It is shown that given any link-manifold, there is an algorithm to decide if the manifold contains an embedded, essential planar surface; if it does, the algorithm will construct one. If a slope on the boundary of the link-manifold is given, there is an algorithm to determine if the slope bounds an embedded punctured-d…

We construct a pairing, which we call factorization homology, between framed manifolds and higher categories. The essential geometric notion is that of a vari-framing of a stratified manifold, which is a framing on each stratum together with a coherent system of compatibilities of framings along links between strata. O…

New Grunsky operator for disk maps to complex plane.

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

In this paper we study the relation between two diagrammatic representations of links in lens spaces: the disk diagram and the grid diagram and we find how to pass from one to the other. We also investigate whether the HOMFLY-PT invariant and the Link Floer Homology are essential invariants, that is, we try to understa…

This paper classifies curves in genus two handlebodies.

Minimal surfaces with uniform curvature (or area) bounds have been well understood and the regularity theory is complete, yet essentially nothing was known without such bounds. We discuss here the theory of embedded (i.e., without self-intersections) minimal surfaces in Euclidean 3-space without a priori bounds. The st…

Let $S_{g,p}$ denote the genus $g$ orientable surface with $p$ punctures. We show that nested train track sequences constitute $O((g,p)^{2})$-quasiconvex subsets of the curve graph, effectivizing a theorem of Masur and Minsky. As a consequence, the genus $g$ disk set is $O(g^{2})$-quasiconvex. We also show that splitti…

Study pseudo-holomorphic disks in real analytic hypersurfaces using exterior differential systems.

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…

Many open problems and important theorems in low-dimensional topology have been formulated as statements about certain 2--complexes called gropes. This paper describes a precise correspondence between embedded gropes in 4--manifolds and the failure of the Whitney move in terms of iterated `towers' of Whitney disks. The…

New knots bound multiple non-isotopic ribbon disks.

In this paper the properties of the Kauffman bracket skein module of $L(p,q)$ are investigated. Links in lens spaces are represented both through band and disk diagrams. The possibility to transform between the diagrams enables us to compute the Kauffman bracket skein module on an interesting class of examples consisti…

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.

We show that bordered Heegaard Floer homology detects incompressible surfaces and bordered-sutured Floer homology detects partly boundary parallel tangles and bridges, in natural ways. For example, there is a bimodule Lambda so that the tensor product of CFD(Y) and Lambda is Hom-orthogonal to CFD(Y) if and only if the …

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…

Embeddings of pairs of disjoint nonparallel primitive simple closed curves in the boundary of a genus two handlebody are classified. Briefly, two disjoint primitives either lie on opposite ends of a product $F \boldsymbol{\times} I$, or they lie on opposite ends of a kind of "twisted" product $F \widetilde{\boldsymbol{…

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.

Primitive curves in handlebodies form a connected complex.

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.