This note characterizes monohedral tilings of regular polygons with up to three tiles.
problem Characterizing monohedral tilings of regular polygons with up to three tiles.
method Connecting the results for squares and circles to generalize for any regular n-gon. result Characterization of monohedral tilings of any regular n-gon with up to three tiles. New knots found with tough, unsliceable discs.
problem Finding tough knots that can't be sliced smoothly.
method Constructed infinitely many knots with non-approximable slice discs.
result Smoothly sliceable knots have non-approximable slice discs.
Study on the topology of ordered disc configurations, revealing nontrivial homotopy classes.
problem Topology of ordered disc configurations and their homotopy types.
method Analysis of ordered configuration spaces of hard discs, focusing on homotopy types and nontrivial classes.
result Exhibit nontrivial classes in π_{n-3} for all n, and their persistence in deformed ambient discs.
Study minimal discs in metric spaces with a quadratic isoperimetric inequality.
problem Understanding the geometry of minimal discs in metric spaces.
method Associate a compact metric space to each minimal disc, controlling its properties by the isoperimetric inequality.
result The geometry of the associated space can control the shapes of curves and the original space's topology.
Study on non-orientable surfaces for maximal disc packings.
problem Maximizing disc packings in non-orientable surfaces.
method Analyzing compact non-orientable surfaces of genus g≥3 for maximal k-packings. result Characterization of maximal disc packings in non-orientable surfaces.
Modified proof constructs dual spheres for 4-manifolds.
problem Prove dual spheres for 4-manifolds.
method Modified proof of disc embedding theorem, geometric construction.
result Constructs geometrically dual spheres.
Formula derived for mean curvature of a special surface with boundary curve.
problem Understanding the properties of constant mean curvature discs with analytic boundary.
method Analysis of holomorphic quadratic Hopf differential and normal curvature.
result Formula for mean curvature as a weighted average of boundary curve normal curvature.
Modified Engel structures allow complete h-principle for overtwisted discs.
problem Engel structures and their overtwisted discs.
method Engel twist modification and h-principle proof.
result Complete h-principle for overtwisted Engel structures.
We use the famous knot-theoretic consequence of Freedman's disc theorem---knots with trivial Alexander polynomial bound a locally-flat disc in the 4-ball---to prove the following generalization. The degree of the Alexander polynomial of a knot is an upper bound for twice its topological slice genus. We provide examples…
Circular disc can be tiled with up to 3 congruent pieces, showing symmetry.
problem Tiling a circular disc with congruent pieces.
method Proving the existence of a k-fold rotational symmetry for k≤3. result First nontrivial estimate on minimum number of tiles for certain tiling configurations.
Classifies homotopy ribbon discs for certain slice knots.
problem Characterizing homotopy ribbon discs for specific slice knots.
method Classifies Γ-homotopy ribbon slice discs up to topological ambient isotopy. result In the infinite cyclic case, there is a unique equivalence class of such slice discs. For the Baumslag-Solitar group, there are at most two equivalence classes of Γ-homotopy ribbon discs. Classifies 5D stationary black hole domains using plumbing and disc bundles.
problem Classifying the topology of 5D stationary black hole domains.
method Using plumbing constructions and disc bundles over the 2-sphere.
result The domain of outer communication is homeomorphic to S4 or a connected sum of S2imesS2's or CP2. The paper classifies homotopy ribbon discs with specific groups.
problem Classifying homotopy ribbon discs with given fundamental groups.
method Using geometric and algebraic properties of groups, particularly Farrell-Jones conjecture.
result Classification of homotopy ribbon discs for specific knot groups and Baumslag-Solitar groups.
Study on surfaces minimizing elastic energy with boundary constraints.
problem Finding stable configurations of surfaces with elastic boundaries and surface energy.
method Investigation of critical surfaces with mean curvature and spontaneous curvature, coupled to boundary elastic energy.
result Characterization and minimization of surface energy for specific topological shapes.
New real hyperbolic manifolds constructed over surfaces with maximal Euler number ratio.
problem Constructing real hyperbolic manifolds with specific topological properties.
method Applying techniques from [AGG] and using the Gromov-Lawson-Thurston conjecture [GLT].
result New examples achieve the maximal value of $|eM/χS|= rac35$.
Proves uniqueness of Lefschetz fibrations over discs based on singular fibers.
problem Classifying Lefschetz fibrations over the disc based on their singular fibers.
method Inspired by homological mirror symmetry and Karpov--Nogin's theorem, uses topological equivalence and fiber analysis.
result Lefschetz fibrations over discs are uniquely determined by their singular fibers.
Using open book foliations we show that an overtwisted disc in a planar open book can be put in a topologically nice position. As a corollary, we prove that a planar open book whose fractional Dehn twist coefficients grater than one for all the boundary components supports a tight contact structure.
We prove that a Casson tower of height 4 contains a flat embedded disc bounded by the attaching circle, and we prove disc embedding results for height 2 and 3 Casson towers which are embedded into a 4-manifold, with some additional fundamental group assumptions. In the proofs we create a capped grope from a Casson towe…
New spanning tree model connects knot homology, s-invariant, and exotic discs.
problem Understanding exotic discs in the 4-ball for knots.
method Explicitly defined differential in spanning tree complex, described Rasmussen's s-invariant.
result Identified new infinite family of knots bounding exotic discs.
100 years ago exactly, in 1906, Hartogs published a celebrated extension phenomenon (birth of Several Complex Variables), whose global counterpart was stated in full generality later by Osgood (1929): holomorphic functions in a connected neighborhood V(bD) of a connected boundary bD contained in C^n (n >= 2) do extend …
The topological underpinnings are presented for a new algorithm which answers the question: `Is a given knot the unknot?' The algorithm uses the braid foliation technology of Bennequin and of Birman and Menasco. The approach is to consider the knot as a closed braid, and to use the fact that a knot is unknotted if and …
New stabilizing number defined for knots, linking bounds in 4D.
problem Defining a new measure for knot boundaries in 4D.
method Defining stabilizing number sn(K), bounding it by signatures, Casson-Gordon invariants, and 4-genus. result Found examples where stabilizing number is less than 4-genus.
New algebra connects braid group actions to disc curves.
problem Understanding braid group actions on disc curves.
method Constructed a type B zigzag algebra and showed its categorical action on projective modules.
result Type B braid group action on homotopy category of projective modules.
The paper solves an embedding problem for discs with specific curvature properties.
problem Embedding discs with positive Gauss curvature into Euclidean 3-space.
method Analyzes the free boundary isometric embedding problem with specific curvature conditions.
result Discs with specified curvature can be isometrically embedded into R3 orthogonally to the unit sphere. Study the topology of stable vector fields and Lyapunov functions on R^n.
problem Topology of stable vector fields and Lyapunov functions on R^n.
method Differential topology, Lyapunov theory, and results on diffeomorphism groups of discs.
result Path-connected and simply connected spaces of stable vector fields for n≠4,5 and weakly contractible for n≤3.
Using a cell model for the little discs operad in terms of spineless cacti we give a minimal common topological operadic formalism for three a priori disparate algebraic structures: (1) a solution to Deligne's conjecture on the Hochschild complex, (2) the Hopf algebra of Connes and Kreimer, and (3) the string topology …
We construct and analyze minimal disc stackings with bounds on their Morse index.
problem Constructing and analyzing minimal free boundary disc stackings.
method Constructing minimal free boundary disc stackings in a three-dimensional Euclidean unit ball, proving bounds on their Morse index.
result Uniform, linear bounds on the Morse index of all such surfaces.
In previous work with Schoenfeld, we considered a string-type chain complex of curves on surfaces, with differential given by resolving crossings, and computed the homology of this complex for discs. In this paper we consider the corresponding "string homology" of annuli. We find this homology has a rich algebraic stru…
In analogy with the holomorphic case, we compare the topology of Milnor fibrations associated to a meromorphic germ f/g : the local Milnor fibrations given on Milnor tubes over punctured discs around the critical values of f/g, and the Milnor fibration on a sphere.
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…
Minimal discs count as knot invariants in hyperbolic 4-space.
problem Counting minimal discs with a given boundary in hyperbolic 4-space.
method Using J-holomorphic curves in twistor space to count minimal discs, considering singularities at infinity.
result The number of minimal discs with a given boundary is a knot invariant.
New geometric invariant from disc intersections captures all coloured Jones polynomials.
problem Constructing a universal knot invariant from configuration spaces.
method Defining a new local system and Lagrangian submanifolds in the disc.
result The new invariant recovers Habiro's universal invariant and more.
The smoothing theory is revised to generalize to different disc embedding spaces.
problem Generalizing the smoothing theory to various disc embedding spaces.
method Revising the Morlet-Burghelea-Lashof-Kirby-Siebenmann theorem to apply to different versions of disc smooth embedding spaces.
result The delooping of disc embedding spaces is shown to be compatible with the Hatcher and Budney actions.
Topological model for coloured Alexander invariants from quantum group representations.
problem Quantum invariants from Uq(sl(2)) at roots of unity. method Topological model using graded intersection pairings in a covering space.
result Coloured Alexander invariants can be obtained as homology class pairings.
Let Fg denote a closed oriented surface of genus g. A set of simple closed curves is called a filling of Fg if its complement is a disjoint union of discs. The mapping class group Mod(Fg) of genus g acts on the set of fillings of Fg. The union of the curves in a filling forms a graph on the surfa…
We prove that any steady solution to the real analytic Euler equations on a Riemannian 3-sphere must possess a periodic orbit bounding an embedded disc. One key ingredient is an extension of Fomenko's work on the topology of integrable Hamiltonian systems to a degenerate case involving stratified integrals. The result …
Maximizes filling systems on surfaces with given boundary components.
problem Finding the maximum size of filling systems on surfaces with specific boundary conditions.
method Analyzing the structure of filling systems and their complements.
result The maximum size of a filling system on a surface of genus g with 1 ≤ b ≤ 2g-2 boundary components is 2g + b - 1.
Exposes two methods for constructing flat surfaces in 4D spaces.
problem Building locally flat embedded surfaces in 4-manifolds.
method Direct methods and surgery theory.
result Every primitive second homology class in a closed, simply connected 4-manifold is represented by a locally flat embedded torus.
Smoothly isotopic 3-discs in 4-sphere become identical after 5-dimensional push.
problem Smooth isotopy of 3-discs in 4-sphere.
method Pushing 3-discs into 5-dimensional space.
result Isotopic 3-discs in 4-sphere become identical after 5-dimensional push.
The paper proves conditions for the existence of holomorphic discs in Kähler manifolds.
problem Existence of holomorphic discs for higher A∞ operations. method Showing existence of minimal discs with specific properties implies existence of holomorphic discs.
result Minimal discs in Kähler manifolds with certain boundary conditions are holomorphic.
The paper explores the topology of polygonal meshes and their properties.
problem Understanding the topological properties of polygonal meshes.
method Overview of topological concepts, definitions of intrinsic and extrinsic topology, proofs of Euler and Euler-Poincaré formulas, and discussion on cutting meshes.
result Detailed understanding and definitions of polygonal mesh topology, including intrinsic and extrinsic properties.
Motivated by string topology and the arc operad, we introduce the notion of quasi-operads and consider four (quasi)-operads which are different varieties of the operad of cacti. These are cacti without local zeros (or spines) and cacti proper as well as both varieties with fixed constant size one of the constituting lo…
Maximizes the number of curves needed to cover a surface without overlapping.
problem Finding the maximum number of curves needed to cover a surface without overlapping.
method Analyzing the geometric intersection numbers of curves in minimal fillings.
result Maximum size of a filling of a surface is 2g + b - 1.
This manuscript studies manifolds-with-boundary collapsing in the Gromov-Hausdorff topology. The main aim is an understanding of the relationship of the topology and geometry of a limiting sequence of manifolds-with-boundary to that of a limit space, which is presumed to be without geodesic terminals. The main result e…
Classifies ancient flows in a disc with boundary.
problem Ancient convex flows in a disc with boundary.
method Classifies flows using curve shortening.
result Ancient convex flows in a disc are classified.
Study knot groups from disc patterns in 3D space.
problem Understanding groups generated by reflections in planar patterns.
method Construct and study representation spaces of holonomy groups of hyperbolic 3-manifolds.
result Visualizations suggest conjectures and support algebraic calculations.
The paper explores connections between braids, links, and cobordisms using algebraic methods.
problem Investigating functions on manifolds and their connections to braids, links, and cobordisms.
method Algebraic methods including group theory, sheaves, and formal groups.
result Constructs Lazard's one-dimensional universal commutative formal group and applies it to cobordism theory.
Holomorphic discs cover a ball in complex space.
problem Covering a ball in complex space with holomorphic discs.
method Showed a nonsingular holomorphic foliation by complete discs.
result The open unit ball in complex space admits a foliation by complete discs.