arXiv research

Enumerated all genus two handlebody-knots with seven crossings.

Complete classification of handlebody links up to six crossings.

Proves certain subgroups of genus 2 handlebody group are convex cocompact.

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 …

Stable subgroups identified in genus two handlebody group.

The JSJ decomposition helps classify genus two handlebody-knots.

Classifies essential annuli in a genus two handlebody exterior.

Correspondence found between Askey-Wilson polynomials and genus-two handlebody skein module.

Classifies essential annuli in genus two handlebody-knots, determining hyperbolicity and constructing obstructions.

The paper studies cylindrical handlebody-knots of genus two with unique unknotting annuli and finds trivial symmetry groups.

Handlebody groups are virtual duality groups in positive genus.

We characterize composite tunnel number one genus two handlebody-knots.

We study the existence of incompressible embeddings of surfaces into the genus two handlebody. We show that for every compact surface with boundary, orientable or not, there is an incompressible embedding of the surface into the genus two handlebody. In the orientable case the embedding can be either separating or non-…

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 consider the hyperelliptic handlebody group on a closed surface of genus $g$. This is the subgroup of the mapping class group on a closed surface of genus $g$ consisting of isotopy classes of homeomorphisms on the surface that commute with some fixed hyperelliptic involution and that extend to homeomorphisms on the …

Study on cylindrical handlebody-knots with symmetry and rigidity properties.

We show that the Dehn function of the handlebody group is exponential in any genus $g\geq 3$. On the other hand, we show that the handlebody group of genus $2$ is cubical, biautomatic, and therefore has a quadratic Dehn function.

Maps from 4D handlebodies to their boundaries have sections.

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.

Study infinite genus surfaces and Schottky groups for uniformization.

We consider the group of isotopy classes of automorphisms of the 3-sphere that preserve a spatial graph or a handlebody-knot embedded in it. We prove that the group is finitely presented for an arbitrary spatial graph or a reducible handlebody-knot of genus two. We also prove that the groups for "most" irreducible genu…

New 3D handlebodies in 4-sphere and 5-ball are not isotopic even with same boundary.

Study reveals uniform difference in stretch factors between genus two handlebody group and outer automorphism group.

Uncountably many fibrations found on genus 2 handlebody.

We provide a classification of the essential surfaces of non-negative Euler characteristic in the exteriors of genus two handlebodies embedded in the 3-sphere.

Using a similar algorithm to Hatcher-Thurston's algorithm for finding a presentation of the mapping class group of a surface, Wajnryb succeeded to find a presentation for the handlebody group. This is long and complicated. In this note I simplify Wajnryb's presentation for the handlebody group of genus g = 2.

In this paper we consider all orientation-preserving $\mathbb{Z}_{4}$-actions on $3$-dimensional handlebodies $V_g$ of genus $g>0$. We study the graph of groups $(Γ($v$),\mathbf{G(v)})$, which determines a handlebody orbifold $V(Γ($v$),{\mathbf{G(v)}})\simeq{V_g}/\mathbb{Z}_{4}$. This algebraic characterization is used…

A specific set of 4g+1 elements is shown to generate the Goeritz group of the genus g+1 Heegaard splitting of a genus g handlebody. These generators are consistent with Powell's proposed generating set for the Goeritz group of the genus g+1 splitting of S^3. There are two proofs: one using purely classical techniques a…

This paper proves that handlebody homeomorphisms are isotopic to identity.

In this paper, we describe the relation between the study of closed connected surfaces embedded in $S^3$ and the theory of handlebody-knots. By Fox's theorem, a pair of handlebody-knots is associated to a closed connected surface embedded in $S^3$ in the sense that their exterior components are pairwise homeomorphic. W…

Researchers study Prym representations for handlebody groups, focusing on cyclic cases.

This paper classifies curves in genus two handlebodies.

Study of large group actions on surfaces, focusing on Hurwitz and handlebody groups.

Suppose a genus two handlebody is removed from a 3-manifold M and then a single meridian of the handlebody is restored. The result is a knot or link complement in M and it is natural to ask whether geometric properties of the link complement say something about the meridian that was restored. Here we consider what the …

We show that the mapping class group of a handlebody of genus at least 2 has a Dehn function of at most exponential growth type.

Embeddings of pairs of disjoint nonparallel primitive simple closed curves in the boundary of a genus two handlebody are classified. Briefly, two disjoint primitives either lie on opposite ends of a product $F \boldsymbol{\times} I$, or they lie on opposite ends of a kind of "twisted" product $F \widetilde{\boldsymbol{…

3-manifolds with hyperbolic handlebody complements are studied.

Proves finiteness for 3-manifolds from handlebodies with 2-handles.

We give a necessary and sufficient condition for a simple closed curve on the boundary of a genus two handlebody to decompose the handlebody into (torus with one boundary component times [0,1]. We use this condition to decide whether a simple closed curve on a genus two Heegaard surface is a GOF-knot (genus one fibered…

We study the orientation preserving involutions of the orientable 3-dimensional handlebody $H_g$, for any genus $g$. A complete classification of such involutions is given in terms of their fixed points.

We prove that the handlebody subgroup of the Torelli group of an orientable surface is generated by genus one BP-maps. As an application, we give a normal generating set for the handlebody subgroup of the level $d$ mapping class group of an orientable surface.

The main theorem of this paper generalizes recent results in Dehn surgery to the case of handlebody attachment. We consider attaching handlebodies and solid tori to the boundary of an irreducible, boundary-irreducible, atoroidal and acylindrical 3-manifold. We show that for a large class of homeomorphisms attaching the…

We study geometric properties of stabilisers in the handlebody group. We find that stabilisers of meridians are undistorted, while stabilisers of primitive curves or annuli are exponentially distorted for large enough genus.

Paper introduces an invariant to distinguish handlebody-knot exteriors.

We construct quantum $\mathcal{U}_q(\mathfrak{sl}_{\,2})$ type invariants for handlebody-knots in the 3-sphere $S^3$. 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 …

Minimal crystallizations bound for 3-manifolds with boundary.

Disk and sphere graphs embed quasi-isometrically in R^2.

For a handlebody of genus $g\geq6$ it is shown that every automorphism of the complex of separating meridians can be extended to an automorphism on the complex of all meridians and, in consequence, it is geometric.