Gluck twisting certain knots results in standard 4-spheres.
problem Understanding when Gluck twists yield standard 4-spheres.
method Analyzing smooth homotopy 4-spheres and diffeomorphisms.
result Infinite collection of twisted doubles of corks are standard.
Standard trisection diagrams found for a specific type of knot.
problem Finding standard trisection diagrams for Gluck twists.
method Using a specific method to obtain trisection diagrams for spun (p+1,p)-torus knots. result Standard trisection diagrams were found for the spun (p+1,p)-torus knots. We give a simple criterion when a Gluck twisting an odd smooth 4-manifold along a 2-sphere S⊂X does not change its diffeomorphism type. We obtain this by handlebody techniques and plug twisting operation, getting a slightly stronger version of the known fact that Gluck twisting of a 2-sphere S⊂X of a …
The Gluck twist preserves the diffeomorphism type of certain satellite 2-knots.
problem Preserving the diffeomorphism type of satellite 2-knots under the Gluck twist.
method Using new descriptions of satellite 2-knots, the paper shows that the Gluck twist does not change the diffeomorphism type of certain satellite 2-knots in three ways.
result The Gluck twist preserves the diffeomorphism type of certain satellite 2-knots.
The union of singular orbits of an effective locally smooth circle action on the 4-sphere consists of two 2-knots, K and K′, intersecting at two points transversely. Each of K and K′ is called a branched twist spin. A twist spun knot is an example of a branched twist spin. The Gluck twists along…
New examples of Schoenflies balls are produced using a 5D approach.
problem Identifying Schoenflies balls that are not standard.
method Using a 5-dimensional perspective, algebraic and geometric handle cancellation.
result New examples of Schoenflies balls not known to be standard are produced.
We show that by performing the Gluck twist along the 2-knot Kpq2 derived from two ribbon presentations of the ribbon 1-knot K(p,q) we get the standard 4-sphere S4. In the proof we apply Kirby calculus.
Heegaard diagrams for 5-manifolds help in understanding their structure.
problem Understanding the structure of 5-dimensional manifolds.
method Introducing a version of Heegaard diagrams for 5-dimensional cobordisms and manifolds, showing diffeomorphisms through specific moves.
result Every smooth 5-manifold can be represented by a Heegaard diagram, and diagrams representing diffeomorphic manifolds are related by certain moves.
Gluck twists on spheres yield equivalent 4-manifolds under certain conditions.
problem Understanding when Gluck twists on spheres result in homeomorphic or simple homotopy equivalent manifolds.
method Analyzing the Gluck twist operation on 4-manifolds with spheres of different relationships (concordant, homotopic) and examining their resulting manifolds.
result Gluck twists on concordant or homotopic spheres yield equivalent 4-manifolds under specific conditions.
The paper examines trisection diagrams of spun knots and shows they are standard for certain cases.
problem Whether trisection diagrams induced by the Gluck surgery on specific knots are standard.
method Explicit depiction and analysis of trisection diagrams for spun (2n+1,−2)-torus knots. result Trisection diagrams are standard for spun (2n+1,−2)-torus knots when n=1 and homologically standard for all n. A theorem of Katanaga, Saeki, Teragaito, and Yamada relates Gluck and Price twists of 4-manifolds. Using trisection diagrams, we give a purely diagrammatic proof of this theorem, and answer a question of Kim and Miller.
Computes Vafa-Witten invariants of 3-manifolds.
problem Computing invariants of 3-manifolds.
method Explicit computations using Vafa-Witten theory.
result Explicit invariants computed for specific 3-manifolds.
Knots in 3-manifolds are equivalent if isotopic, except in special cases.
problem Understanding when knots in 3-manifolds are equivalent and isotopic.
method Analyzing prime, closed, oriented 3-manifolds and irreducible manifolds, and considering orientation-preserving mapping class groups and homeomorphisms.
result Knots in prime, closed, oriented 3-manifolds are isotopic if and only if the orientation preserving mapping class group is trivial.
The paper extends symplectic techniques to generalized complex geometry.
problem Creating stable generalized complex structures on high-dimensional manifolds.
method Introducing generalized Luttinger surgery and generalized Gluck twist.
result Produced stable generalized complex structures with non-homotopy-equivalent components.
Study shows how any group can be fundamental of a non-orientable 4-manifold with exotic structure.
problem Realizing any finitely presented group as the fundamental group of a non-orientable 4-manifold with exotic structure.
method Performing a Gluck twist to obtain an exotic smooth structure, and using 2-covers to stabilize the structure.
result Any finitely presented group can be realized as the fundamental group of a non-orientable 4-manifold with exotic structure.
Study of contractible manifolds and their twists to determine if they are S4.
problem Determine if a specific manifold is S4 using twists and handlebody descriptions. method Analyze two contractible manifolds, one Stein and one non-Stein, with non-trivial boundary mapping classes. Use a specific homotopy sphere to test if it is S4. result Show that a specific manifold is indeed S4. New non-orientable 4-manifolds created via knotting operations.
problem Creating non-orientable 4-manifolds with specific properties.
method Twisting operations on embedded spheres and projective planes.
result Existence of non-diffeomorphic manifolds homeomorphic to Y. 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.
For any positive integer n we give a Zn-cork with a Zn-effective embedding in a 4-manifold being homeomorphic to E(n). This means that a cork gives a subset Zn in the differential structures on E(n). Further, we describe handle decompositions of the twisted doubles (homotopy…
Extends Rasmussen's invariant to new surfaces in four-manifolds.
problem Computing genus bounds for surfaces in specific four-manifolds.
method Extending Khovanov-Lee homology and using Hochschild homology.
result Proves inequalities relating the invariant to surface genus.
We give a new characterization of symplectic surfaces in CP^2 via bridge trisections. Specifically, a minimal genus surface in CP^2 is smoothly isotopic to a symplectic surface if and only if it is smoothly isotopic to a surface in transverse bridge position. We discuss several potential applications, including the cla…
We show the homotopy spheres Σn=−W⌣fnW, formed by doubling the infinite order loose-cork (W,f) by iterates of the cork diffeomorphism f:∂W→∂W is S4. To do this we first show that Σn are obtained by Gluck twistings of S4; then from this we show how to cancel 3-han…
In this paper, we investigate existence of inequivalent smooth structures on closed smooth non-orientable 4-manifolds building upon results of Akbulut, Cappell-Shaneson, Fintushel-Stern, Gompf, and Stolz. We add to the number of known constructions and provide new examples of exotic manifolds that are obtained as an ap…
Akbulut has recently shown that an infinite family of Cappell-Shaneson homotopy 4-spheres is diffeomorphic to the standard 4-sphere. In the present paper, a strictly larger family is shown to be standard by a simpler method. This new approach uses no Kirby calculus except through the relatively simple 1979 paper of Akb…
The paper extends Gluck and Warner's result on fibrations of spheres by great subspheres.
problem Understanding when two Hopf fibrations of S2n−1 agree on a fiber. method Characterizing the conditions for two Hopf fibrations of S2n−1 to agree on a fiber. result A complete characterization of the conditions for two Hopf fibrations of S2n−1 to agree on a fiber. The paper constructs homotopy 4-spheres using pochette surgery.
problem Creating homotopy 4-spheres from pochette surgeries.
method Pochette surgery generalizes Gluck surgery to construct embeddings of pochettes into the 4-sphere and proves homotopy 4-spheres are diffeomorphic to the 4-sphere.
result Homotopy 4-spheres obtained from pochette surgeries are all diffeomorphic to the 4-sphere.
In this paper, we study surfaces embedded in 4-manifolds. We give a complete set of moves relating banded unlink diagrams of isotopic surfaces in an arbitrary 4-manifold. This extends work of Swenton and Kearton-Kurlin in S4. As an application, we show that bridge trisections of isotopic surfaces in a trisected …
New modules derived from Khovanov homology for links.
problem Constructing new link homology modules from Khovanov homology.
method Functoriality proof for cobordisms in 4D relative 1-handlebody complements.
result Functoriality of Rozansky-Willis's homology for cobordisms.
Study pochette surgery on 4-manifolds, focusing on 4-spheres.
problem Understanding pochette surgery on 4-spheres and its effects.
method Using linking number of pochette embeddings, compute homology and analyze surgeries.
result Pochette surgery on any homology 4-sphere can be computed via homology, and trivial cord surgeries do not change diffeomorphism type.
Semisimple 4D field theories can't distinguish smooth 4-manifolds.
problem Detecting exotic smooth structures in 4-manifolds.
method Proving field theories lead to stable invariants, distinguishing only homeomorphic and homotopy equivalent manifolds.
result Semisimple 4D field theories can't distinguish homotopy equivalent 4-manifolds.
We study embedded spheres in 4-manifolds (2-knots) via doubly pointed trisection diagrams, showing that such descriptions are unique up to stabilization and handleslides, and we describe how to obtain trisection diagrams for certain cut-and-paste operations along 2-knots directly from doubly pointed trisection diagrams…
We study generalized Killing spinors on the standard sphere S3, which turn out to be related to Lagrangian embeddings in the nearly Kähler manifold S3×S3 and to great circle flows on S3. Using our methods we generalize a well known result of Gluck and Gu concerning divergence-free geod…
The volume of a k-dimensional foliation F in a Riemannian manifold Mn is defined as the mass of image of the Gauss map, which is a map from M to the Grassmann bundle of k-planes in the tangent bundle. Generalizing a construction by Gluck and Ziller, "singular" foliations by 3-spheres are constructed on…
A remarkable and elementary fact that a locally compact set F of Euclidean space is a smooth manifold if and only if the lower and upper paratangent cones to F coincide at every point, is proved. The celebrated von Neumann's result (1929) that a locally compact subgroup of the general linear group is a smooth manifold,…
New constraints rule out some optimal domains for helicity maximisation.
problem Finding a smooth domain of fixed volume that maximizes helicity.
method Established additional geometric constraints on optimal domains.
result Ruled out the optimality of a broad class of solid tori.
Solves isoperimetric problem for curl operator on compact 3-manifolds.
problem Isoperimetric problem for the curl operator on compact 3-manifolds.
method Analyzes eigenvalues and optimal domains of the curl operator.
result Optimal lower bounds for eigenvalues of curl operator are always attained.
We introduce the concept of twisted contact groupoids, as an extension either of contact groupoids or of twisted symplectic ones, and we discuss the integration of twisted Jacobi manifolds by twisted contact groupoids. We also investigate the very close relationships which link homogeneous twisted Poisson manifolds wit…
As shown by H. Gluck in 1962, the diffeotopy group of S^1 \times S^2 is isomorphic to Z_2 + Z_2 + Z_2. Here an alternative proof of this result is given, relying on contact topology. We then discuss two applications to contact topology: (i) it is shown that the fundamental group of the space of contact structures on S^…
New infinite families of twisted torus knots found.
problem Identifying new types of twisted torus knots.
method Finding new infinite families of twisted torus knots with a single negative twist.
result Eight new infinite families of twisted torus knots are discovered.
The study finds hyperbolic twisted torus links for certain twists.
problem Identifying hyperbolic twisted torus links.
method Analyzing (p,q)-torus links with r strands twisted s times. result All hyperbolic twisted torus links are identified for ∣s∣>3. The paper introduces a method to deform submanifolds in Euclidean space using Drinfel'd twists.
problem Constructing noncommutative deformations of submanifolds in Euclidean space.
method Using Drinfel'd twist deformation of differential geometry, the paper proposes a general procedure to construct noncommutative deformations of an embedded submanifold.
result The method allows for consistent projection of connections and can be applied to various submanifolds like cylinders and hyperboloids.
Paper constructs Chern character for higher twists and shows isomorphism between K-theory and cohomology.
problem Mapping higher twisted K-theory to higher twisted cohomology.
method Constructing Chern character for higher twists and showing isomorphism.
result Chern character gives isomorphism between higher twisted K-theory and higher twisted cohomology.
The classical Sturm-Hurwitz-Kellogg theorem asserts that a function, orthogonal to an n-dimensional Chebyshev system on a circle, has at least n+1 sign changes. We prove the converse: given an n-dimensional Chebyshev system on a circle and a function with at least n+1 sign changes, there exists an orientation preservin…
The paper studies the twisted Calabi flow on Kähler manifolds.
problem Analyzing the behavior of the twisted Calabi flow on compact Kähler manifolds.
method Establishing convexity, proving short-time existence, and demonstrating stability of the flow.
result The stability of the twisted Calabi flow near twisted constant scalar curvature Kähler metrics.
Construct noncommutative deformations of algebraic submanifolds in R^n.
problem Deforming algebraic submanifolds in noncommutative geometry.
method Using twisted differential geometry and Drinfel'd twists, constructing noncommutative deformations of algebraic submanifolds.
result Explicitly worked out deformations of quadrics in R^3.
Paper proves any twisted link can be described as a unique twisted braid.
problem Understanding twisted links and braids.
method Proved using Alexander and Markov Theorems for classical braids and links.
result Any twisted link can be described as the closure of a unique twisted braid.
Study on singular twisted links and virtual braids, extending knot theory concepts.
problem Extending knot theory concepts to singular twisted links and virtual braids.
method Definition and analysis of singular twisted virtual braids and their monoid structure.
result Presentation of monoid and reduced monoid for singular twisted virtual braids.
In this paper, we develop differential twisted K-theory and define a twisted Chern character on twisted K-theory which depends on a choice of connection and curving on the twisting gerbe. We also establish the general Riemann-Roch theorem in twisted K-theory and find some applications in the study of twisted K-theory o…