The paper constructs exotic knotted surfaces and curves in 4-manifolds.
problem Understanding exotic knotted surfaces and curves in 4-manifolds.
method Local knotting and construction of surfaces and curves.
result First examples of exotically knotted complex curves and symplectic 2-spheres.
Constructs exotic proper 2-knots from open 2-handles.
problem Proving topological equivalence of exotic proper knotted surfaces.
method Flexible construction of exotic open 2-handles, distinguishing through genus functions and end Floer homology.
result Constructs infinite families of irreducible exotic proper knotted surfaces.
New module constructs exotic surfaces in 4-manifolds.
problem Constructing exotic surfaces in 4-manifolds.
method Using Bar-Natan Khovanov homology and skein lasagna gluing maps.
result Exotic surfaces require at least one internal stabilization.
Exotic diffeomorphisms found on complex surfaces and 4-manifolds.
problem Finding exotic diffeomorphisms on 4-manifolds.
method Minimal complex surfaces and spin 4-manifolds with S3 boundary. result First known instances of exotic diffeomorphisms of irreducible 4-manifolds.
The paper constructs exotic complex projective surfaces using rational blowdowns.
problem Finding exotic complex projective surfaces with specific invariants.
method Rational blowdowns and construction of Kollár--Shepherd-Barron--Alexeev surfaces.
result First exotic surfaces constructed, including minimal and symplectic ones.
Single stabilization is not always enough to make exotic surfaces isotopic.
problem Determining if a single stabilization is sufficient to make exotic surfaces isotopic.
method Study of stabilization distance with satellite operations using Floer theoretic techniques.
result Found examples of exotic disks in the four-ball with arbitrarily large stabilization distance.
The paper presents a new method to create exotic 4-manifolds and surfaces that remain exotic after stabilization.
problem Stabilization of exotic 4-dimensional phenomena and knotted surfaces.
method Elementary approach to constructing exotic 4-manifolds and surfaces, including examples in closed, simply connected 4-manifolds.
result The construction yields exotic surfaces in the 4-ball that remain exotic after stabilization, detected by Khovanov homology.
The paper constructs exotic surface links in 4-ball, proving their Brunnian nature.
problem Investigating exotic surface links in 4-ball.
method Two constructions of Brunnian exotic surface links, using satellite operations and covering links.
result Provided constructions of exotic Brunnian surface links with varied properties.
Study of torus surgeries on knot traces, finding exotic surfaces and traces.
problem Understanding exotic surfaces and traces through torus surgeries.
method Realizing annulus twisting as torus surgery, using key technical insight.
result Exotic elliptic surfaces and traces discovered, improving known geography.
New techniques in Khovanov homology help distinguish exotic surfaces in 4-ball.
problem Distinguishing exotic surfaces in the 4-ball that are not diffeomorphic.
method Developed new techniques for distinguishing cobordism maps on Khovanov homology using knot symmetries and braid factorizations.
result Distinguishes smooth surfaces in the 4-ball that are exotically knotted.
On all compact complex surfaces (modulo finite unramified coverings), we classify all of the locally homogeneous geometric structures which are locally isomorphic to the exotic homogeneous surfaces of Lie.
A study on the relation between the smooth structure of a symplectic homotopy K3 surface and its symplectic symmetries is initiated. A measurement of exoticness of a symplectic homotopy K3 surface is introduced, and the influence of an effective action of a K3 group via symplectic symmetries is investigated. It is show…
New findings show stabilization isn't enough for certain knotted surfaces.
problem The necessity of stabilization for exotic knotted surfaces.
method Examined smooth 4-manifolds with nonempty boundaries containing exotically knotted 2-spheres.
result The 'one is enough' theorem does not apply to closed surfaces with characteristic homology classes.
New stabilization methods for exotic surfaces in 4-manifolds.
problem Understanding and stabilizing exotic surfaces in 4-manifolds.
method Relating internal and external stabilization, proving results using this relation.
result Many exotic surfaces become smoothly isotopic after external stabilization.
Study on exotic smooth embeddings of surfaces in 4-manifolds, revealing different properties and complexities.
problem Understanding exotic smooth embeddings of surfaces in 4-manifolds.
method Analyzing smooth, proper embeddings of noncompact surfaces in 4-manifolds, focusing on exotic planes and annuli.
result Exotic planes and annuli exhibit radically different properties, with one class being simple enough to draw explicit level diagrams.
Detects exotic surfaces without smooth invariants, providing first example of knotted RP².
problem Detecting exotic surfaces without smooth invariants.
method Two versions of families (Seiberg-Witten) generalizations of Donaldson's diagonalization theorem.
result First example of exotically knotted RP² with diffeomorphic complements.
We prove a general fusion theorem for complete orientable minimal surfaces in R3 with finite total curvature. As a consequence, complete orientable minimal surfaces of weak finite total curvature with exotic geometry are produced. More specifically, universal surfaces (i.e., surfaces from which all minimal …
Authors prove existence of exotic surfaces and invariants not detecting self-diffeomorphisms.
problem Existence and properties of exotic surfaces and diffeomorphisms.
method Vanishing theorem of family Bauer--Furuta invariant for diffeomorphisms on spin 4-manifolds.
result Family Bauer--Furuta invariants do not detect exotic self-diffeomorphisms on S4 or S2imesS2. Starting with the Dolgachev surface E(1)_{2,3} we construct an infinite family of distinct exotic copies of the rational surface E(1), each of which admits a handlebody decomposition without 1- and 3- handles, and we draw these handlebodies.
Paper defines end Khovanov homology to detect exotic planes.
problem Detecting exotic Lagrangian and symplectic planes in 4-space.
method Define and apply end Khovanov homology to surfaces in 4-space.
result First known examples of exotic Lagrangian and symplectic planes found.
We study the relationship between exotic R^4's and Stein surfaces as it applies to smoothing theory on more general open 4-manifolds. In particular, we construct the first known examples of large exotic R^4's that embed in Stein surfaces. This relies on an extension of Casson's Embedding Theorem for locating Casson han…
Surface corks modify 4-manifold structures without changing their homeomorphism type.
problem Understanding how to change 4-manifold structures using surface corks.
method Introducing surface corks as compact, contractible submanifolds intersecting smoothly embedded surfaces in 4-manifolds.
result Explicit construction of a transverse surface cork that is diffeomorphic to a 4-ball.
Paper extends Miyazawa's construction to more 4-manifolds, finding exotic involutions and embeddings.
problem Finding exotic involutions and embeddings in 4-manifolds.
method Real Seiberg-Witten theory.
result Many infinite families of exotic involutions and embeddings.
In this paper, we study stable equivalence of exotically knotted surfaces in 4-manifolds, surfaces that are topologically isotopic but not smoothly isotopic. We prove that any pair of embedded surfaces in the same homology class become smoothly isotopic after stabilizing them by handle additions in the ambient 4-manifo…
New method preserves option structure while using neural networks for volatility.
problem Inconsistent exotic option prices with model calibration.
method Volatility Feature Approach (VFA) using neural networks.
result VFA outperforms model calibration approach for practical volatility surfaces.
Characterizes quasiconformally homogeneous ladder surfaces.
problem Characterize quasiconformally homogeneous Riemann surfaces.
method Breaks the problem into four cases and proves one case.
result Every quasiconformally homogeneous ladder surface is quasiconformally equivalent to a regular cover of a closed surface.
We give new rational blowdown constructions of exotic CP^2#n(-CP^2) (5\leq n\leq 9) without using elliptic fibrations. We also show that our 4-manifolds admit handle decompositions without 1- and 3-handles, for 7\leq n\leq 9. A strategy for rational blowdown constructions of exotic CP^2#n(-CP^2) (1\leq n\leq 4) is also…
Algorithm counts intersections of normal curves efficiently.
problem Efficiently solving word problems in mapping class groups of punctured surfaces.
method Fast algorithm for counting intersections of normal curves on triangulated surfaces.
result Efficient solution of the word problem for mapping class groups of punctured surfaces.
Software simplifies triangulations of 4-manifolds, revealing exotic structures.
problem Understanding smooth 4-manifolds from discrete and algorithmic perspectives.
method New software tools, including an algorithm for triangulations from Kirby diagrams and a heuristic for simplification.
result Presented new triangulations of exotic pairs, corks, and plugs, including the smallest known K3 surface.
Since the first work on exotic smoothness in physics, it was folklore to assume a direct influence of exotic smoothness to quantum gravity. Thus, the negative result of Duston (arXiv:0911.4068) was a surprise. A closer look into the semi-classical approach uncovered the implicit assumption of a close connection between…
For any finitely presentable group G, we show the existence of an isolated complex surface singularity link which admits infinitely many exotic Stein fillings such that the fundamental group of each filling is isomorphic to G. We also provide an infinite family of closed exotic smooth four-manifolds with the fundam…
The paper explores slice disks and their properties using satellite operations and knot Floer homology.
problem Tackles the conjecture about slice disks and their positive Whitehead doubles.
method Uses techniques from knot Floer homology, Seiberg-Witten theory, and Khovanov homology.
result Provides evidence for the conjecture and constructs exotic disks.
New method calibrates LV surfaces for exotic derivatives with smoother, more stable Greeks.
problem Challenges in LV calibration leading to spiky surfaces and unstable Greeks.
method Automatic local regression to pre-process market observables and smooth LV surfaces.
result Significantly smoother LV surfaces and greatly improved Greek stability with negligible additional cost.
Overlay framework simplifies exotic derivative pricing.
problem Inconsistent pricing models for exotics.
method Combines path reweighting and conic optimisation.
result Practically model-independent price bands for exotics.
New 2-links created in 4D spaces, topologically unknotted but not smoothly.
problem Creating exotic 2-links in 4D spaces.
method A construction procedure to generate infinite sets of 2-links.
result First examples of 2-links that are topologically unknotted but not smoothly.
The study creates symplectic Lefschetz fibrations and rational blowdowns for new 4-manifolds.
problem Creating symplectic Lefschetz fibrations and rational blowdowns for new 4-manifolds.
method Producing simply connected, minimal, symplectic Lefschetz fibrations and rationally blowing down Lefschetz fibrations with clustered nodal fibers.
result New constructions of small symplectic exotic 4-manifolds.
We first construct a genus zero positive allowable Lefschetz fibration over the disk (a genus zero PALF for short) on the Akbulut cork and describe the monodromy as a positive factorization in the mapping class group of a surface of genus zero with five boundary components. We then construct genus zero PALFs on infinit…
Classifies geodesic planes outside convex core of geometrically finite 3-manifolds.
problem Classifying geodesic planes in geometrically finite 3-manifolds.
method Constructive proof involving exotic rays and roofs.
result Existence of exotic roofs depends on the existence of exotic rays and bending lamination properties.
Convex-cocompact groups in infinite hyperbolic space are deformable.
problem Understanding deformability of convex-cocompact groups in infinite hyperbolic spaces.
method Proving convex-cocompact representations form an open set and using bending to deform them.
result Deformable convex-cocompact representations of surface groups not conjugate to exotic PSL(2,R) representations.
We present an infinite sequence of smooth embeddings of a connected sum of 6 projective planes in the 4-sphere, which are all ambient homeomorphic, but pairwise ambient non-diffeomorphic. The double covers of the 4-sphere ramified along these surfaces form a family of the exotic $\Bbb CP^2#5\bar{\Bbb CP^2}$ constructed…
The paper bounds the genus of surfaces in four-manifolds with indefinite forms.
problem Bounding the genus of surfaces in four-manifolds with indefinite intersection forms.
method Using adjunction inequalities and the 10/8 + 4 theorem, the paper provides methods to bound the genus of surfaces.
result The set of H-slice knots can detect exotic smooth structures on closed 4-manifolds.
New invariant distinguishes non-orientable surfaces.
problem Distinguishing non-orientable surfaces bounded by the same knot.
method Mixed invariant from Lee and Bar-Natan deformations of Khovanov homology.
result Distinguishes exotic non-orientable surfaces.
We introduce a new operation, double point surgery, on immersed surfaces in a 4-manifold, and use it to construct knotted configurations of surfaces in many 4-manifolds. Taking branched covers, we produce smoothly exotic actions of Z/m x Z/n on simply connected 4-manifolds with complicated fixed-point sets.
Study on four-dimensional Dehn twists and Milnor fibrations, revealing new phenomena.
problem Understanding the monodromy of Milnor fibrations of surface singularities.
method Using Seiberg-Witten invariant and monopole Floer homology.
result Infinite order non-triviality results for boundary Dehn twists.
Paper extends cobordism maps in Khovanov and instanton homologies.
problem Define and prove compatibility of cobordism maps in Khovanov and instanton homologies.
method Extend embedded cobordism map to immersed cobordisms and prove compatibility.
result Induced map on Khovanov homology is injective for certain concordances.
In a recent paper of Akhmedov, Etnyre, Mark and Smith, it was shown that there exist infinitely many contact Seifert fibered 3-manifolds each of which admits infinitely many exotic (homeomorphic but pairwise non-diffeomorphic) simply-connected Stein fillings. Here we extend this result to a larger set of contact Seifer…
In this note we study logarithmic transformations in the sense of differential topology on two fibers of the Hopf surface. It is known that such transformations are susceptible to yield exotic smooth structures on four-manifolds. We will show here that this is not the case for the Hopf surface, all integer homology Hop…
Classifies 4-manifolds with specific properties and fundamental group.
problem Classifying 4-manifolds with fundamental group Z and boundary. method Topological and smooth classification methods, including Hermitian forms and 2-handlebody constructions. result Every Hermitian form over Z[t±1] arises as the equivariant intersection form of exotic smooth 4-manifolds.