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.
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.
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.
Studies modules over a category of Jacobi diagrams in handlebodies.
problem Understanding modules over a specific category of Jacobi diagrams.
method Generalizes adjunctions and studies subquotient modules.
result Generalizes adjunctions between modules and Casimir Lie algebra modules.
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.
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.
Study Weinstein structures on toric divisors' complements.
problem Understanding Weinstein structures on toric divisors' complements.
method Define a partially-centered condition on Delzant polytopes, develop an algorithm for Weinstein handlebody diagrams.
result Explicit Weinstein structures for complements of smoothed toric divisors.
Multisections generalize Heegaard splittings and trisections to higher dimensions.
problem Decomposing higher-dimensional manifolds into 1-handlebodies with specific intersection properties.
method Considering multisections of higher-dimensional smooth or PL closed orientable manifolds, and associating diagrams to them.
result Multisection diagrams uniquely determine manifolds in all dimensions up to 6.
The study proves a theorem for alternating knots in handlebodies.
problem Understanding the properties of alternating knots in handlebodies.
method Generalization of the Jones polynomial to handlebodies.
result Any dotted-reduced alternating diagram of a knot in a handlebody realizes the minimal crossing number and has identical writhe to any other diagram of the same knot.
This paper classifies curves in genus two handlebodies.
problem Classifying curves in the boundary of a genus two handlebody.
method Using R-R diagrams and cutting disks, the paper classifies curves as primitive, proper power, or Seifert.
result The paper provides all possible R-R diagrams for these curves.
Complete invariant for surfaces in 3-sphere derived from diagrams of fundamental groups.
problem Constructing a complete invariant for closed surfaces in the three-sphere.
method Diagram of fundamental groups, generalization of Kneser conjecture, extensions of Waldhausen's theorem.
result Proves the diagram of fundamental groups is a complete invariant for closed surfaces in the three-sphere.
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…
This paper classifies knots with simple curves in genus 2 handlebodies.
problem Classifying knots with specific curves in handlebodies.
method Analyzing R-R diagrams and 2-handle additions.
result Classification of knots with proper power curves in genus 2 handlebodies.
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.
Rail knotoids study rail arcs and their isotopy in 3D space.
problem Understanding rail isotopy of arcs in 3D space.
method Introducing rail knotoid diagrams and their equivalence, proving rail isotopy equivalence based on projections.
result Two rail arcs are rail isotopic if and only if their projections onto a plane are equivalent.
New method to compute homology and intersection form of 4-manifolds.
problem Computing homology and intersection form of 4-manifolds.
method Using multisections to define complexes and retraction onto CW-complex.
result Efficient proofs of homology and intersection form from multisection diagrams.
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.
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.
We prove generic fibre of Painlevé moduli spaces are Weinstein handlebodies.
problem Understanding the generic fibre of Painlevé moduli spaces.
method Attach Weinstein handles to Stokes Legendrian.
result Generic fibre of Painlevé moduli spaces are Weinstein handlebodies.
New Brieskorn spheres found to bound rational homology balls but not integral ones.
problem Identifying Brieskorn spheres that bound rational homology balls but not integral ones.
method Using modifications of Fintushel and Stern's argument and handlebody diagrams.
result Found new families of Brieskorn spheres (Σ(2,4n+1,12n+5) and Σ(3,3n+1,12n+5)) that bound rational homology balls but not integral ones. Study on polynomiality and outer nature of functors from Jacobi diagrams to group homomorphisms.
problem Understanding polynomiality and outer nature of functors from Jacobi diagrams to group homomorphisms.
method Analyzing polynomiality and outer nature of functors from Jacobi diagrams to group homomorphisms.
result Results generalize previous work by Katada and study polynomiality and outer nature of these functors.
This research connects 4-manifold trisections to their underlying surfaces and their intersection forms.
problem Understanding the intersection forms of 4-manifolds from their trisection diagrams.
method Expressing the twisted homology and Reidemeister torsion of a 4-manifold in terms of the first homology of its trisection surface and the intersection form.
result The intersection form of a 4-manifold can be derived from the intersection form of its trisection surface.
Extends knotted defect classification to bounded domains using handlebodies.
problem Classifying knotted defects in bounded domains.
method Using continuous maps and monodromies around meridional loops, global defects are described in terms of planar diagrams.
result Classification scheme for defects in handlebodies.
A trisection of a smooth, closed, oriented 4-manifold is a decomposition into three 4-dimensional 1-handlebodies meeting pairwise in 3-dimensional 1-handlebodies, with triple intersection a closed surface. The fundamental groups of the surface, the 3-dimensional handlebodies, the 4-dimensional handlebodies, and the clo…
A new integral for tangles in handlebodies connects to Hopf algebras.
problem Constructing a functor for bottom tangles in handlebodies.
method Extension of Kontsevich integral to handlebodies, functor construction.
result Induces canonical isomorphism between bottom tangles and a specific PROP.
We identify the canonical contact structure on the link of a simple elliptic or cusp singularity by drawing a Legendrian handlebody diagram of one of its Stein fillings. We also show that the canonical contact structure on the link of a numerically Gorenstein surface singularity is trivial considered as a real plane bu…
The paper constructs multisections for m-spun 3-manifolds in higher dimensions.
problem Finding multisections for higher-dimensional manifolds.
method Extending the concept of spun 3-manifolds to higher dimensions, constructing multisections and diagrams.
result Infinitely many examples of non-diffeomorphic multisected manifolds in all dimensions.
The paper studies pseudo links in genus g handlebodies, generalizing knot theory.
problem Modeling DNA knots with missing crossing information.
method Introducing pseudo links as mixed pseudo links in S^3, generalizing Kauffman bracket polynomial and Alexander theorem.
result The theory of pseudo links is closely related to singular links and can be applied to study singular links in genus g handlebodies.
Stein and Weinstein structures are described for disk cotangent bundles of surfaces.
problem Characterizing Stein and Weinstein structures on disk cotangent bundles.
method Using Legendrian handlebody diagrams and symplectic/contact mappings.
result The canonical contact structure on the unit cotangent bundle of S is obtained via surgery.
We present an alternative definition for the Goussarov--Habiro filtration of the Z-module freely generated by oriented integral homology 3-spheres, by means of Lagrangian-preserving homology handlebody replacements (LP-surgeries). Garoufalidis, Goussarov and Polyak proved that the graded space (G_n)_n associated to thi…
We show that if V3 is a handlebody in R3, with curves J1,...,Jg⊂∂V which are the attaching curves for a Heegaard splitting of a homology sphere, then there exists a homeomorphism h:V→V so that each of the curves h(Ji) bounds an orientable surface in R3−int(V). This lead…
The study shows knots from 3-braids cannot be concordant to a specific Legendrian unknot.
problem The concordance of knots from 3-braids to a specific Legendrian unknot.
method Using symplectic handlebody diagrams and Legendrian contact homology, the study derives a contradiction to show the non-concordance.
result The study proves that knots from 3-braids cannot be concordant to a specific Legendrian unknot.
Defines thin position for 4-manifolds and trisections, finding key relations and diagrams.
problem Defining and analyzing thin position for 4-manifolds and their trisections.
method Introduced width of handle decompositions, defined thin position, and studied trisections.
result Described genus 2g+2 and g+2 trisection diagrams for sphere bundles.
Data structure for Heegaard splittings reduces complexity.
problem Representing Heegaard splittings efficiently.
method Data structure for normal curves with respect to triangulations.
result Exponential gains in space complexity for certain families.
New definition of skein lasagna module for specific 4-manifolds.
problem Defining a new homology theory for specific 4-manifolds.
method Elementary definition using diagrams in the 4-manifold.
result Explicit calculations for disk bundles over S^2.
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.
The paper studies symplectic operations on Stein fillings of Brieskorn singularities.
problem Symplectic operations on Stein fillings of Brieskorn singularities.
method Two interpretations: symplectic sum and monodromy substitution in a Lefschetz fibration.
result Generalized chain surgeries and their applications in symplectic geometry.
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.
New algebraic structure for distinguishing twisted virtual handlebody-links.
problem Distinguishing twisted virtual handlebody-links.
method Introducing twisted virtual bikeigebras and using them to define invariants.
result New invariants distinguish some twisted virtual handlebody-links.
We find trisections for 3-manifold bundles with specific Heegaard surfaces.
problem Finding trisections for 3-manifold bundles over S^1.
method Generalized the results of Gay and Kirby to find trisections of genus 3g+1.
result Existence of trisections of genus 3g+1 for bundles with specific Heegaard surfaces.
Formula for signature of handlebody bundles, interpreting cohomology.
problem Signature of handlebody bundles and their monodromy.
method Explicit formula using homological monodromy.
result Cobounding function of Meyer's signature cocycle on handlebody group.
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…
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.
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 …