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.
We introduce the notion of a G-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.
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 J4 detects non-deformation diffeomorphisms when H∗ is not semisimple and H is not factorizable. Given a Heegaard splitting of a closed 3-manifold, the skein modules of the two handlebodies are modules over the skein algebra of their common boundary surface. The zeroth Hochschild homology of the skein algebra of a surface with coefficients in the tensor product of the skein modules of two handlebodies is interpret…
New invariants detect exotic smooth structures in 4-manifolds.
problem Detect exotic smooth structures in 4-manifolds using invariants of 2-handlebodies.
method Investigates invariants of 4-dimensional 2-handlebodies from the Temperley-Lieb category in positive characteristic.
result Height n=2 invariant vanishes on CP2, CP2, and S2imesS2 for p>3. 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.
Paper defines numerical criteria to test handlebody link irreducibility.
problem Determining the irreducibility of handlebody links.
method A set of numerical criteria to test handlebody links for irreducibility.
result Effective method to recognize irreducibility of handlebody knots and most handlebody links.
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 k-handlebodies. result High-dimensional handlebodies are products of simpler shapes.
A handlebody-link is a disjoint union of embeddings of handlebodies in S3 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 G-families of biquandles, partially multiplicative biquandles and group decomposable biquandles. These structures can be used to color the semiarcs in Y-oriented spatial trivalent graph diagrams representing S1-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 …
We generalize unoriented handlebody-links to the twisted virtual case, obtaining Reidemeister moves for handlebody-links in ambient spaces of the form Σ×[0,1] for Σ 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 3-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 T of handlebody-tangles and present it by generators and relations. The result tells us that every functor on T that gives rise to inv…
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.
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 g handlebody group into the genus g 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 G-family of quandles colorings. The above …
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 2. 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.
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 H=H1∪AH2 of two handlebodies H1 and H2 is a handlebody if and only if the core curve of A 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 α-rebuilding property to handlebody group homology growth. Introduces algorithm for Weinstein handlebodies of certain divisors.
problem Constructing Weinstein handlebodies for complements of smoothed toric divisors.
method Explicit coordinates and simple example for algorithm.
result Produces Weinstein Kirby diagrams for complements of smoothed toric divisors.
We construct quantum Uq(sl2) type invariants for handlebody-knots in the 3-sphere S3. A handlebody-knot is an embedding of a handlebody in a 3-manifold. These invariants are linear sums of Yokota's invariants for colored spatial graphs which are defined by using the Kauffman bracket. We …
The paper disproves the existence of certain subgroups with nontrivial rational abelianization.
problem The existence of finite-index subgroups with nontrivial rational abelianization in handlebody groups.
method Proved that meridian multitwists vanish in H1(Γ;Q) and showed that H1(Γ;Q)=0 for specific subgroups. result No finite-index subgroups of the handlebody group contain nontrivial rational abelianization.
It is known that the order of a finite group of diffeomorphisms of a 3-dimensional handlebody of genus g > 1 is bounded by the linear polynomial 12(g-1), and that the order of a finite group of diffeomorphisms of a 4-dimensional handlebody (or equivalently, of its boundary 3-manifold), faithful on the fundamental group…
Study compact PL 4-manifolds with special handle decompositions.
problem Existence of special handlebody decompositions for simply-connected closed PL 4-manifolds.
method Investigate colored triangulations inducing handle decompositions without 1-handles or 1- and 3-handles.
result Detect a class of compact simply-connected PL 4-manifolds with empty or connected boundary that admit such decompositions.
In this paper, we describe the relation between the study of closed connected surfaces embedded in S3 and the theory of handlebody-knots. By Fox's theorem, a pair of handlebody-knots is associated to a closed connected surface embedded in S3 in the sense that their exterior components are pairwise homeomorphic. W…
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 BHomeo(Hg) to BHomeo(Ug). Characterizes amalgamations and self-amalgamations of handlebodies.
problem Characterizing amalgamations and self-amalgamations of handlebodies.
method Analyzes the properties of amalgamations and self-amalgamations of handlebodies.
result Characterizes amalgamations and self-amalgamations of handlebodies.