New exotic 4D spaces found using knot slicing techniques.
problem Finding non-diffeomorphic exotic 4D spaces.
method Using RBG links to create slice knots with non-diffeomorphic complements.
result Distinguished new exotic 4D spaces using end Floer homology.
New method uses Floer homology to create exotic disk pairs.
problem Creating exotic disk pairs in 4-manifolds.
method Singular instanton Floer homology with Chern--Simons filtration.
result Constructs exotic pairs of slice disks and knots.
Introduces slice knots and concordance, linking to exotic smooth structures.
problem Understanding slice knots and their equivalence through concordance.
method Explains connections and introduces filtrations, satellite operations.
result Connections between slice knots and exotic smooth structures on \(\mathbb{R}^4\).
Paper explores using 0-surgery to find exotic 4-manifolds.
problem Distinguishing smooth structures on 4-manifolds.
method Uses 0-surgery homeomorphisms to generate exotic pairs.
result Found 5 topologically slice knots potentially leading to exotic spheres.
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 to construct slice knots using annulus modifications.
problem Constructing smoothly and exotically slice knots.
method Defining and applying n-twist annulus modifications. result New examples of slice knots and exotically slice knots.
Study framed links in simply connected 4-manifolds, linking to exotic phenomena.
problem Understanding framed links in simply connected 4-manifolds and their relation to exotic phenomena.
method Analyzing the set of framed smoothly slice links on the boundary of a 1-handlebody in a closed, simply connected, smooth 4-manifold.
result Exotic 4-manifolds are distinguished by links that are slice in the exotic manifold but not in the standard D4. New invariants prove exotic slice disks for knots.
problem Proving exotic slice disks for knots.
method Involutive Khovanov homology and derived invariants.
result Reprove exotic slice disks for knots Jn. Study rules out exotic S4 and #nCP2 construction using zero surgery homeomorphisms.
problem Tackles the possibility of constructing exotic S4 or #nCP2 using zero surgery homeomorphisms. method Uses zero surgery homeomorphisms to relate slice properties of knots stably after a connected sum with a 4-manifold.
result Rules out the possibility of constructing exotic S4 or #nCP2 using zero surgery homeomorphisms. The study distinguishes exotic R^4's using knot Floer homology and Casson handles.
problem Distinguishing exotic R^4's with different knot Floer homologies.
method Using Gadgil's end Floer homology and Casson handles.
result A countably infinite family of pairwise nondiffeomorphic chiral exotic R^4's.
Since the first work on exotic smoothness in physics, it was folklore to assume a direct influence of exotic smoothness to quantum gravity. In the second paper, we calculate the "smoothness structure" part of the path integral in quantum gravity for the exotic R^4 as non-compact manifold. We discuss the influence of th…
Kirby diagrams for exotic R^4's constructed from specific knot complements.
problem Identifying and visualizing exotic R4's using Kirby diagrams. method Provided Kirby diagrams for a family of exotic R4's constructed from specific knot complements. result Generalized Kirby diagrams for a broader family of exotic R4's. Khovanov homology distinguishes exotic 4-manifolds.
problem Existence of exotic compact orientable 4-manifolds.
method Khovanov-Rozansky gl2 skein lasagna module and related invariants. result First analysis-free proof of exotic 4-manifolds.
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.
The linear slice of quasi-Fuchsian once-punctured torus groups is defined by fixing the complex length of some simple closed curve to be a fixed positive real number. It is known that the linear slice is a union of disks, and it always has one standard component containing Fuchsian groups. Komori and Yamashita proved t…
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.
Alternative proof for links not smoothly slice in S2imesS2.
problem Existence of links not smoothly slice in S2imesS2. method Alternative proof using Miyazaki and Yasuhara's result.
result Existence of links not smoothly slice in S2imesS2. 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.
Study knot Floer homology to create concordance invariants and slice genus bounds.
problem Developing concordance invariants using knot Floer homology.
method Using knot Floer homology, define and analyze equivariant concordance invariants.
result Showed a family of strongly invertible slice knots with arbitrarily large equivariant slice genus.
The paper shows links with 2 components are not smoothly slice in a specific 4-manifold.
problem Tackles the smooth sliceness of 2-component links in a specific 4-manifold.
method Uses classical topological and smooth obstructions, along with constructive arguments exploiting symmetries.
result Demonstrates the existence of infinitely many integer homology 3-spheres with specific properties.
In this paper we give a necessary and sufficient condition in which a sequence of Kleinian punctured torus groups converges. This result tells us that every exotically convergent sequence of Kleinian punctured torus groups is obtained by the method due to Anderson and Canary. Thus we obtain a complete description of th…
Smooth tori in S^4 are topologically unknotted.
problem Tackling the topological unknottedness of smooth tori in S^4.
method Analyzing the intersection forms and critical points of tori to prove topological unknottedness.
result Certain smooth tori in S^4 are topologically unknotted.
Paper proves a symplectic inequality using trisections and contact geometry.
problem Proving a symplectic inequality for 4-manifolds.
method Used contact geometry and trisections to reduce to slice-Bennequin inequality, proving with Khovanov homology.
result Gauge-theory-free proofs of landmark 4-manifold topology results.
New 4-manifolds with exotic diffeomorphisms found.
problem Existence of exotic diffeomorphisms in 4-manifolds.
method Proves existence of infinitely many contractible 4-manifolds with exotic diffeomorphisms.
result Infinitely many contractible 4-manifolds with absolutely exotic diffeomorphisms.
Exotic submanifolds in 4-manifolds remain exotic after stabilizations.
problem Constructing exotic codimension-1 submanifolds in 4-manifolds.
method Constructing pairs of exotic codimension-1 submanifolds with diffeomorphic complements and showing they remain exotic after stabilizations.
result Exotic submanifolds remain exotic after any number of stabilizations.
Study exotic Dehn twists in 4-manifolds, producing first known exotic diffeomorphisms.
problem Detecting exotic diffeomorphisms in 4-manifolds.
method 2-parameter families Seiberg-Witten theory over RP2. result Constructed the smallest closed 4-manifold with exotic diffeomorphisms.
New methods distinguish exotic 4-manifolds using Heegaard Floer homology.
problem Distinguishing exotic 4-manifolds from standard ones.
method Using Heegaard Floer homology to define new invariants.
result New invariants remain distinct in covers and can distinguish exotic manifolds.
Constructs Riemannian foliations with exotic tori as leaves.
problem Creating Riemannian foliations with exotic tori.
method Smooth fiber bundles with exotic tori fibers and finite abelian fundamental group total space.
result Examples of Riemannian foliations with exotic tori leaves and finite abelian fundamental group total space.
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.
New actions found on exotic spheres using group theory.
problem Understanding smooth transformations on exotic spheres.
method Recent progress in stable homotopy groups of spheres and group theory.
result Smooth circle and cyclic group actions on exotic spheres produced.
Estimates exotic option prices without a model using market data.
problem Pricing exotic derivatives without a model.
method Uses rough path signatures and implied expected signatures from market prices.
result Prices exotic derivatives using market data and implied expected signatures.
Nonparametric pricing and hedging of exotic derivatives using signature payoffs.
problem Pricing and hedging exotic derivatives accurately and efficiently.
method Introducing signature payoffs and using them to approximate and price exotic derivatives nonparametrically.
result Signature payoffs enable accurate and computationally tractable pricing and hedging of exotic derivatives.
Paper finds exotic diffeomorphisms on specific 4-manifolds.
problem Understanding exotic diffeomorphisms on 4-manifolds with b2+=2. method Comparing winding numbers of parameter families.
result 2\mathbb{C}\mathbb{P}^2 \# 10 (-\mathbb{C}\mathbb{P}^2) admits exotic diffeomorphisms.
Detects exotic embeddings in 4-manifolds with boundary.
problem Detecting exotic diffeomorphisms in 4-manifolds with boundary.
method Defined family versions of Kronheimer-Mrowka invariants and used them to find exotic embeddings.
result First example of exotic 3-spheres in a smooth closed 4-manifold with diffeomorphic complements.
Study exotic phenomena in small 4-manifolds using diffeomorphisms.
problem Understanding exotic smooth structures and embeddings in 4-manifolds.
method Using constraints from Seiberg-Witten theory on diffeomorphisms.
result New methods to detect and bound exotic phenomena.
We show how to construct absolutely exotic smooth structures on compact 4-manifolds with boundary, including contractible manifolds. In particular, we prove that any compact smooth 4-manifold W with boundary that admits a relatively exotic structure contains a pair of codimension-zero submanifolds homotopy equivalent t…
New exotic structures found on 4-manifolds with specific groups.
problem Finding exotic smooth structures on 4-manifolds with certain fundamental groups.
method Constructing new smooth structures using definite intersection forms and fundamental groups.
result Infinitely many exotic smooth structures on 4-manifolds with fundamental group Z/2Z.
4-manifolds can be exotic after connected sum with S^2 x S^2.
problem Existence of exotic contractible 4-manifolds.
method Constructed a cork that remains exotic after connected sum with S^2 x S^2.
result Existence of exotic pair of contractible 4-manifolds.
This paper constructs exotic diffeomorphisms on reducible 4-manifolds with odd b+.
problem Detecting exotic diffeomorphisms on reducible 4-manifolds with odd b+.
method Using a families Bauer-Furuta invariant and pin-cobordism.
result Constructs exotic diffeomorphisms on reducible 4-manifolds with odd b+.
New exotic 4-manifolds found with small trisection genus.
problem Finding exotic 4-manifolds with small trisection genus.
method Constructing genus-3 relative trisections for contractible 4-manifolds.
result Exotic 4-manifolds with boundary have trisection genus of 4.
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.
Extends T-duality to exotic Courant algebroids.
problem Lack of T-duality for exotic Courant algebroids.
method Extends T-duality isomorphism to exotic exact Courant algebroids.
result Exchange of momentum and winding numbers.
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.
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 results on localization of exotic diffeomorphisms in 4-manifolds.
problem Understanding when groups of exotic diffeomorphisms can be localized to smaller embedded submanifolds.
method Analyzing families Seiberg-Witten theory, Dehn twists, and properties of Seifert fibered homology spheres.
result Exotic diffeomorphisms cannot be localized to topologically embedded rational homology balls or homology spheres.
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.
New exotic 4D spaces with nontrivial mappings.
problem Creating exotic copies of R4 with nontrivial mapping class groups. method Modification of diffeomorphism corks to make exteriors simply-connected.
result Construct exotic copies of R4 with mapping class groups of arbitrarily large rank. Construct non-split 2-component links in S4 to produce exotic pairs of 4-manifolds
problem Construct non-split 2-component links in S4 method Construct non-split 2-component links in S4 result Obtain exotic pairs of simply connected, definite 4-manifolds with boundary