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 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.
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 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.
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 \(\mathbb{R}^4\).
Extends knot surgery to exotic four-manifolds.
problem Constructing exotic definite four-manifolds.
method Generalizes RBG construction to n-surgery. result Produces pairs of knots with same n-surgery. 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.
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. 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 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.
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 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. 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.
Extends exotic embeddings of RP^2 to a larger family and produces homotopy spheres.
problem Constructing exotic embeddings of RP^2 and homotopy spheres.
method Using Montesinos knots and roll-spun knots to prove the existence of homotopy spheres.
result An infinite family of homotopy spheres and homotopy CP^2s are produced.
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.
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.
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.
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.
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. 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.
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.
In the previous paper the author defined an infinite order plug (P,φ) which gives rise to infinite Fintushel-Stern's knot-surgeries. Here, we give two 4-dimensional infinitely many exotic families Yn, Zn of exotic enlargements of the plug. The families Yn, Zn have b2=3, 4 and the boundaries are…
We show that, for each integer n, there exist infinitely many pairs of n-framed knots representing homeomorphic but non-diffeomorphic (Stein) 4-manifolds, which are the simplest possible exotic 4-manifolds regarding handlebody structures. To produce these examples, we introduce a new description of cork twists and util…
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…
It is known that there is a unique concordance class in the free homotopy class of S1×pt⊂S1×S2. The constructive proof of this fact is given by the second author. It turns out that all the concordances in this construction are invertible. The knots K⊂S1×S2 with hyperbolic …
We discuss corks, and introduce new objects which we call plugs. Though plugs are fundamentally different objects, they also detect exotic smooth structures in 4-manifolds like corks. We discuss relation between corks, plugs and rational blow-downs. We show how to detect corks and plugs inside of some exotic manifolds.…
It is known that every exotic smooth structure on a simply connected closed 4-manifold is determined by a codimention zero compact contractible Stein submanifold and an involution on its boundary. Such a pair is called a cork. In this paper, we construct infinitely many knotted imbeddings of corks in 4-manifolds such t…
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…
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.
We define an operation on homology B4 which we call an n-twist annulus modification. We give a new construction of smoothly slice knots and exotically slice knots via n-twist annulus modifications. As an application, we present a new example of a smoothly slice knot with non-slice derivatives. Such examples we…
New method connects knot Floer homology with bordered Floer homology.
problem Computing involutive knot Floer homology of satellites.
method Invariant splitting principles for knot Floer complexes and bordered Floer homology.
result Involutive knot Floer homology of satellites can be computed from their companions.
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.
We discuss the relation between Fintushel-Stern knot surgery operation on 4-manifolds and Scharlemann manifolds, and as a corollary show that they all are standard. Along the way we show that the fishtail can exotically knot in the 4-sphere infinitely many ways.
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.
We present various examples of cosmetic bandings on knots and links, that is, bandings on knots and links leaving their types unchanged. As a byproduct, we give a hyperbolic knot which admits exotic chirally cosmetic surgeries yielding hyperbolic manifolds. This gives a counterexample to a conjecture raised by Bleiler,…
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…
This paper is two-fold. At first we will discuss the generation of source terms in the Einstein-Hilbert action by using (topologically complicated) compact 3-manifolds. There is a large class of compact 3-manifolds with boundary: a torus given as the complement of a (thickened) knot admitting a hyperbolic geometry, den…
Smoothly knotted 5RP^2 found in 4-sphere.
problem Finding knotted embeddings in higher dimensions.
method Topological and smooth knotting analysis in 4-sphere.
result Smoothly knotted 5RP^2 found in 4-sphere.
We define a new 4-dimensional symplectic cut and paste operation which is analogous to Fintushel and Stern's rational blow-down. We use this operation to produce multiple constructions of symplectic smoothly exotic complex projective space blown-up eight times, seven times, and six times. We also show how this operatio…
We show that the smooth 4-manifold M obtained by attaching a 2-handle to B4 along a certain knot K⊂∂B4 admits infinitely many absolutely exotic copies Mn, n=0,1,2.., such that each copy Mn is obtained by attaching 2-handle to a fixed compact smooth contractible manifold W along th…
From a handlebody-theoretic perspective, the simplest compact, contractible 4-manifolds, other than the 4-ball, are Mazur manifolds. We produce the first pairs of Mazur manifolds that are homeomorphic but not diffeomorphic. Our diffeomorphism obstruction comes from our proof that the knot Floer homology concordance inv…
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.
Algorithm finds knot friends in 3-manifolds.
problem Identifying knots that share similar 3-manifold complements.
method Developed an algorithm using SnapPy and Regina.
result Constructed a census of simple knots with friends.
The study proves exotic smooth structures and equivalent genus functions in 4-manifolds.
problem Understanding exotic smooth structures and genus functions in 4-manifolds.
method Proving the existence of infinitely many exotic structures and equivalence of genus functions.
result Genus functions of exotic 4-manifolds are pairwise equivalent and stable under certain operations.
The present paper produces examples of Gauss diagram formulae for virtual knot invariants which have no analogue in the classical knot case. These combinatorial formulae contain additional information about how a subdiagram is embedded in a virtual knot diagram. The additional information comes from the second author's…
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.
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.
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.