New Hopf algebras help classify 4D shapes.
problem Classifying 4D shapes up to deformations.
method Developed non-factorizable ribbon Hopf algebras.
result Some derived invariants are boundary-dependent.
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48 results for “4D Handlebodies”
Kirby color defined in Khovanov homology for 4D handlebodies.
problem Invariance of 4D handlebodies under Kirby moves.
method Functoriality and cabling properties of Khovanov homology, handle slide isomorphism.
result Kirby-colored Khovanov homology is invariant under handle slide moves.
Maps from 4D handlebodies to their boundaries have sections.
problem Understanding mappings between 4D handlebodies and their boundaries.
method Induced map analysis and section construction.
result Existence of sections for maps from to .
Quantum invariants for surfaces in 4D 2-handlebodies.
problem Quantum invariants of ribbon surfaces in 4D 2-handlebodies.
method Unimodular ribbon categories, labeled Kirby graphs, and modified traces.
result Recovery and generalization of existing invariants.
Algorithm constructs Kirby diagrams for 4D open books.
problem Constructing Kirby diagrams for 4D open books.
method Algorithm using Heegaard diagrams of pages.
result Diffeomorphic open books constructed with different pages and monodromies.
New proof of Laudenbach and Poénaru's theorem on 4D 1-handlebodies.
problem Proving diffeomorphisms extend to 4D 1-handlebodies and compression bodies.
method Classification of Heegaard splittings and Cerf's theorem on 3-ball diffeomorphisms.
result Every diffeomorphism of the positive boundary extends to the whole compression body.
Constructs a TQFT for 3-manifolds and extends it to 4-dimensional 2-handlebodies.
problem Building a TQFT for 3-manifolds and extending it to 4D.
method Using a Frobenius algebra and skein relations, constructing a TQFT for 3-manifolds and extending it to 4D using an inductive state-sum construction.
result Extends a TQFT to 4-dimensional 2-handlebodies.
New algebraic presentations for 3D cobordisms and 4D 2-handlebodies.
problem Finite algebraic presentations for cobordism and handlebody categories.
method Direct proof using a new functor construction.
result Algebraic presentations of 3Cob and 4HB categories.
Functor connects 4D 2-handlebodies to ribbon categories, detecting non-deformation diffeomorphisms.
problem Detecting non-deformation diffeomorphisms in 4D 2-handlebodies.
method Constructs a braided monoidal functor from 4D 2-handlebodies to unimodular ribbon categories.
result Functor detects non-deformation diffeomorphisms when is not semisimple and is not factorizable.
Refined invariants for 4D 2-handlebodies, linking quantum groups and cohomology.
problem Constructing and studying new invariants for 4D 2-handlebodies.
method Defining invariants for pairs , using unimodular ribbon Hopf coalgebras.
result Decomposition formulas for original invariants in terms of refined ones.
The paper proves unique factorization of knotted handlebodies and examines handlebody-knot symmetry.
problem Uniqueness of factorization of knotted handlebodies along decomposing 2-spheres.
method Analyzes factorization of knotted handlebodies in the 3-sphere, proving uniqueness for specific cases.
result Determines chirality of 6_{10} handlebody-knot and constructs an infinite family of hyperbolic handlebody-knots.
Complete classification of handlebody links up to six crossings.
problem Classifying handlebody links with specific properties.
method Complete enumeration and ambient isotopy consideration.
result Complete classification of handlebody links with six crossings or less.
High-dimensional handlebodies are shown to be products of simpler shapes.
problem Understanding the structure of high-dimensional handlebodies.
method Introducing Kirby diagrams to simplify the structure of -handlebodies.
result High-dimensional handlebodies are products of simpler shapes.
A handlebody-link is a disjoint union of embeddings of handlebodies in and an HL-homotopy is an equivalence relation on handlebody-links generated by self-crossing changes. The second author and Ryo Nikkuni classified the set of HL-homotopy classes of 2-component handlebody-links completely using the linking numb…
The JSJ decomposition helps classify genus two handlebody-knots.
problem Classifying genus two handlebody-knots.
method JSJ decomposition and annulus classification applied to handlebody-knot exteriors.
result Genus two handlebody-knots classified using annulus diagrams.
We give examples of knots in a genus 2 handlebody which have nontrivial Dehn surgeries yielding handlebodies and show that these knots are not 1--bridge.
Finite presentation for a specific group in 3D handlebody topology.
problem Finding a finite presentation for a specific group in 3D handlebody topology.
method Constructed a finite presentation for the liftable Hilden group to derive the presentation for the balanced superelliptic handlebody group.
result A finite presentation was given for the balanced superelliptic handlebody group.
Study on finiteness properties of handlebody mapping class groups.
problem Understanding finiteness properties of asymptotically rigid handlebody groups.
method Introduced asymptotically rigid mapping class groups and determined their finiteness properties based on the space of ends of handlebodies.
result Homology of these groups coincides with stable homology of handlebody groups in some cases.
Bridge positions of handlebody-knots are equivalent when stable.
problem Equivalence of bridge positions in handlebody-knots.
method Demonstrated stability of bridge positions.
result Bridge positions of handlebody-knots are stably equivalent.
We introduce several algebraic structures related to handlebody-knots, including -families of biquandles, partially multiplicative biquandles and group decomposable biquandles. These structures can be used to color the semiarcs in -oriented spatial trivalent graph diagrams representing -oriented handlebody-k…
A complex of incompressible surfaces in a handlebody is constructed so that it contains, as a subcomplex, the complex of curves of the boundary of the handlebody. For genus 2 handlebodies, the group of automorphisms of this complex is used to characterize the mapping class group of the handlebody. In particular, it is …
In this paper we define a set of numerical criteria for a handlebody link to be irreducible. It provides an effective, easy-to-implement method to determine the irreducibility of handlebody links; particularly, it recognizes the irreducibility of all handlebody knots in the Ishii-Kishimoto-Moriuchi-Suzuki knot table an…
We generalize unoriented handlebody-links to the twisted virtual case, obtaining Reidemeister moves for handlebody-links in ambient spaces of the form for a compact closed 2-manifold up to stable equivalence. We introduce a related algebraic structure known as twisted virtual bikeigebras whose axiom…
Paper introduces an invariant to distinguish handlebody-knot exteriors.
problem Challenges in distinguishing handlebody-knots with homeomorphic exteriors.
method Defined an invariant (annulus diagram) using characteristic submanifold theory and Koda-Ozawa classification for essential annuli.
result The annulus diagram can differentiate handlebody-knot families.
Homological stability proved for handlebody mapping class groups.
problem Homological stability for handlebody mapping class groups.
method Categorical framework developed by Randal-Williams and Wahl, allowing for any number of marked discs and boundary points.
result Homology of handlebody groups stabilizes with respect to genus and number of marked discs for all finite degree coefficient systems.
Defines annulus complex of handlebodies and proves its connectivity.
problem None explicitly stated; focus on definition and proof.
method Combinatorial methods to prove connectivity.
result Connectivity of annulus complex of handlebodies proven.
Enumerated all genus two handlebody-knots with seven crossings.
problem Counting genus two handlebody-knots with specific crossings.
method Extending an existing table of genus two handlebody-knots.
result Enumerated all genus two handlebody-knots with seven crossings.
We give an explicit formula for the signature of handlebody bundles over the circle in terms of the homological monodromy. This gives a cobounding function of Meyer's signature cocycle on the mapping class group of a -dimensional handlebody, i.e., the handlebody group. As an application, we give a topological interp…
Proves certain subgroups of genus 2 handlebody group are convex cocompact.
problem Characterizing subgroups of genus 2 handlebody group.
method Proving convex cocompactness of purely pseudo-Anosov subgroups.
result Finitely generated, purely pseudo-Anosov subgroups are convex cocompact.
A handlebody-knot is a handlebody embedded in the 3-sphere. We establish a uniform method to construct invariants for handlebody-links. We introduce the category of handlebody-tangles and present it by generators and relations. The result tells us that every functor on that gives rise to inv…
We introduce the notion of a -family of quandles which is an algebraic system whose axioms are motivated by handlebody-knot theory, and use it to construct invariants for handlebody-knots. Our invariant can detect the chiralities of some handlebody-knots including unknown ones.
3-manifolds with hyperbolic handlebody complements are studied.
problem Understanding the geometry of 3-manifolds with hyperbolic handlebody complements.
method Capping off spherical and torus boundaries, constructing handlebodies, and applying octahedral decomposition.
result Bounds on volume for some handlebody complements are derived.
New invariant detects more elements in 4D diffeomorphism group.
problem Detecting more elements in the mapping class group of 4D handlebodies.
method Defined and computed a new invariant for .
result Infinitely many non-isotopic separating 3-balls found.
Proves finiteness for 3-manifolds from handlebodies with 2-handles.
problem Finiteness of skein modules for 3-manifolds.
method Using handlebodies and 2-handles, formulate and prove the conjecture.
result Proved finiteness for 3-manifolds with handlebodies of genus 2 and 3.
Study Torelli subgroups of handlebody groups and their cohomology.
problem Understanding the cohomology of handlebody Torelli groups.
method Introduce Torelli subgroups, use Johnson homomorphisms, and symplectic representations.
result Describe cup products in the first rational cohomology groups of handlebody Torelli groups.
We show that finite index subgroups of the handlebody group are rigid in their ambient mapping class group: any injective map of a finite index subgroup of the genus handlebody group into the genus mapping class group is conjugation by a mapping class group element. On the other hand, we construct an injection …
We give lower bounds for the tunnel number of knots and handlebody-knots. We also give a lower bound for the cutting number, which is a "dual" notion to the tunnel number in the handlebody-knot theory. We provide necessary conditions for constituent handlebody-knots by using -family of quandles colorings. The above …
Handlebody diagram algebrasmath.QA
The paper explores handlebody versions of various diagram algebras.
problem None explicitly stated, but related to algebraic structures.
method Study of handlebody versions of classical diagram algebras and reformulation of cellular algebras.
result All mentioned algebras are part of the reformulated cellular algebra theory.
Automorphisms of handlebodies arise naturally in the a classification of automorphisms of three-manifolds. Among automorphisms of handlebodies, there are certain automorphisms called irreducible (or generic), which are analogues of pseudo-Anosov automorphisms of surfaces. We show that irreducible automorphisms of handl…
Handlebody groups are virtual duality groups in positive genus.
problem Understanding the structure of handlebody groups.
method Analyzing mapping class groups and their dualizing modules.
result Description of dualizing modules for handlebody groups.
The paper studies cylindrical handlebody-knots of genus two with unique unknotting annuli and finds trivial symmetry groups.
problem Understanding the topology and symmetry of cylindrical handlebody-knots of genus two.
method Analysis of Thurston's hyperbolization theorem and investigation of unknotting annuli.
result The symmetry group is trivial if the unknotting annulus is unique and of type .
Classifies essential annuli in genus two handlebody-knots, determining hyperbolicity and constructing obstructions.
problem Classifying essential annuli in genus two handlebody-knots.
method Introducing τ- and ρ-tangles and good rectangles, classifying these structures.
result Categorization of atoroidal 3-decomposable genus two handlebody-knots based on essential annuli.
Researchers study Prym representations for handlebody groups, focusing on cyclic cases.
problem Understanding Prym representations for handlebody groups and their images.
method Restrict Prym representations to handlebody groups and twist groups, focusing on cyclic cases.
result Determined the image of Prym representations in the cyclic case for handlebody groups.
Primitive curves in handlebodies form a connected complex.
problem Understanding the structure of curves in handlebodies.
method Defining and analyzing primitive curves and constructing sequences between them.
result The primitive curve complex for a handlebody is connected.
Injective map from top cohomology of moduli spaces to handlebodies.
problem Understanding cohomology of moduli spaces of surfaces and handlebodies.
method Constructing a classifying space for handlebody mapping class group.
result Top weight cohomology of moduli spaces maps injectively into handlebodies.
Detects handlebodies and mapping class extensions using bordered Floer homology.
problem Detecting handlebodies and mapping class extensions over compression bodies.
method Combining bordered Floer homology with ideas from Casson-Long and train tracks.
result Algorithm to detect mapping class extensions over any compression body.
The main results of the paper is that we give a characteristics for an annulus sum and a once-punctured torus sum of two handlebodies to be a handlebody as follows: 1. The annulus sum of two handlebodies and is a handlebody if and only if the core curve of is a longitude for either $H_…
Survey on handlebody groups and their properties.
problem Understanding handlebody groups and their properties.
method Survey and recent trends in geometric group theory.
result Application of the cheap -rebuilding property to handlebody group homology growth.