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 shown that all automorphisms of the complex of incompressible surfaces are geometric, that is, induced by a homeomorphism of the handlebody.
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 and one using thin position.
A handlebody link is a union of handlebodies of positive genus embedded in 3-space, which generalizes the notion of links in classical knot theory. In this paper, we consider handlebody links with one genus 2 handlebody and $n-1$ solid tori, $n>1$. Our main result is the complete classification of such handlebody links with six crossings or less, up to ambient isotopy.
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.
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 $F \widetilde{\boldsymbol{\times}} I$, where $F$ is a once-punctured torus. If one of the curves is a proper power of a primitive, the situation is simpler. Either the curves lie on opposite sides of a separating disk in the handlebody, or they bound a nonseparating essential annulus in the 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.
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 of the genus $g$ handlebody group into a genus $h>g$ mapping class group which is not conjugate into a handlebody group.
Let $M$ be a closed orientable 3-manifold with a genus two Heegaard splitting $(V_1, V_2; F)$ and a non-trivial JSJ-decomposition, where all components of the intersection of the JSJ-tori and $V_i$ are not $\partial$-parallel in $V_i$ for $i=1,2$. If $G$ is a finite group of orientation-preserving diffeomorphisms acting on $M$ which preserves each handlebody of the Heegaard splitting and each piece of the JSJ-decomposition of $M$, then $G\cong \mathbb{Z}_2$ or $\mathbb{D}_2$ if $V_j\cap(\cup T_i)$ consists of at most two disks or at most two annuli.
Let $M$ be a closed orientable 3-manifold with a genus two Heegaard splitting $(V_1, V_2; F)$ and a non-trivial JSJ-decomposition, where all components of the intersection of the JSJ-tori and $V_i$ are not $\partial$-parallel in $V_i$ for $i=1,2$. If $G$ is a finite group of orientation-preserving smooth actions on $M$ which preserves each handlebody of the Heegaard splitting and each piece of the JSJ-decomposition of $M$, then $G\cong \mathbb{Z}_2$ or $\mathbb{D}_2$ unless $V_j\cap(\cup T_i)$ has three or more disk components for $j=1$ or 2.
By Thurston's hyperbolization theorem, irreducible handlebody-knots are classified into three classes: hyperbolic, toroidal, and atoroidal cylindrical. It is known that a non-trivial handlebody-knot of genus two has a finite symmetry group if and only if it is atoroidal. The paper investigates the topology of cylindrical handlebody-knots of genus two that admit an unknotting annulus; we show that the symmetry group is trivial if the unknotting annulus is unique and of type $2$.
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 genus two handlebody-knots are finite.
A recent result of Funayoshi-Koda shows that a handlebody-knot of genus two has a finite symmetry group if and only if it is hyperbolic -- the exterior admits a hyperbolic structure with totally geodesic boundary -- or irreducible, atoroidal, cylindrical -- the exterior contains no essential disks or tori but contains an essential annulus. Based on the Koda-Ozawa classification theorem, essential annuli in an irreducible, atoroidal handlebody-knots of genus two are classified into four classes: type $2$, type $3$-$2$, type $3$-$3$ and type $4$-$1$. We show that under mild condition most genus …
A simple closed curve $α$ in the boundary of a genus two handlebody $H$ is primitive if adding a 2-handle to $H$ along $α$ yields a solid torus. If adding a 2-handle to $H$ along $α$ yields a Seifert-fibered space and not a solid torus, the curve is called Seifert. If $α$ is disjoint from an essential separating disk in $H$, does not bound a disk in $H$, and is not primitive in $H$, then it is said to be proper power. As one of the background papers of the classification project of hyperbolic primitive/Seifert knots in $S^3$ whose complete list is given in [BK20], this paper classifies in term…
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 handlebody. We prove that the logarithm of the minimal dilatation (i.e, the minimal entropy) of all pseudo-Anosov elements in the hyperelliptic handlebody group of genus $g$ is comparable to $1/g$. This means that the asymptotic behavior of the minimal pseudo-Anosov dilatation of…
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. We show that for every handlebody-knot pair associated to a genus two "prime bi-knotted" surface, one is irreducible, and the other is reducible. Furthermore, for given two genus two handlebody-knots $H_1$ and $H_2$ satisfying certain conditions, we will construct a "prime bi-knot…
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.
We show that the mapping class group of a handlebody of genus at least 2 (with any number of marked points or spots) is exponentially distorted in the mapping class group of its boundary surface. The same holds true for solid tori with at least two marked points or spots.
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-separating. We also consider the case in which the genus two handlebody is replaced by an orientable 3-manifold with a compressible boundary component of genus greater than or equal to two.
We construct infinitely many linearly independent quasi-homomorphisms on the mapping class group of a Riemann surface with genus at least two which vanish on a handlebody subgroup. As a corollary, we disprove a conjecture of Reznikov on bounded width in Heegaard splittings. Another corollary is that there are infinitely many linearly independent quasi-invariants on the Heegaard splittings of three-manifolds.
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.
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.
Suppose subgroups $A,B < MCG(S)$ in the mapping class group of a closed orientable surface $S$ are given and let $\langle A, B \rangle$ be the subgroup they generate. We discuss a question by Minsky asking when $\langle A, B \rangle \simeq A*_{A \cap B} B$ for handlebody subgroups $A,B$.
In this paper we present two new bases, $B^{\prime}_{H_2}$ and $\mathcal{B}_{H_2}$, for the Kauffman bracket skein module of the handlebody of genus 2 $H_2$, KBSM($H_2$). We start from the well-known Przytycki-basis of KBSM($H_2$), $B_{H_2}$, and using the technique of parting we present elements in $B_{H_2}$ in open braid form. We define an ordering relation on an augmented set $L$ consisting of monomials of all different "loopings" in $H_2$, that contains the sets $B_{H_2}$, $B^{\prime}_{H_2}$ and $\mathcal{B}_{H_2}$ as proper subsets. Using the Kauffman bracket skein relation we relate $B_{…
Let $H_{g}$ be a 3-dimensional handlebody of genus $g$. We determine the twisted first homology group of the mapping class group of $H_{g}$ with coefficients in the first integral homology group of the boundary surface $\partial H_{g}$ for $g\geq 2$.
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 interpretation for the generator of the first cohomology group of the hyperelliptic handlebody group.
We show that if the Hempel distance of a Heegaard splitting is larger than three then the mapping class group of the Heegaard splitting is isomorphic to a subgroup of the mapping class group of the ambient 3-manifold. This implies that given two handlebody sets in the curve complex for a surface that are distance at least four apart, the group of automorphisms of the curve complex that preserve both handlebody sets is finite.
Let $M$ be a $3$--dimensional handlebody of genus $g$. This paper gives examples of hyperbolic knots in $M$ with arbitrarily large genus $g$ bridge number which admit Dehn surgeries which are boundary-reducible manifolds.
The mapping class group $MCG(Σ_g)$ of a surface of genus $g$ has a long-history in topology and group theory. More recently, the mapping class group $MCG(V_g)$ of a handlebody $V_g$ of genus $g$ has become an interesting topic in the study of $3$ manifolds, largely thanks to Heegaard splitting. While $MCG(V_g)$ can be regarded naturally as a sub group of $MCG(Σ_g)$, we could not find any complete proof of this fundamental theorem. It is the purpose of this paper that we give a rigorous proof of embedding of $MCG(V_g)$ into $MCG(V_g)$. The key step is: Any self-homeomorphism $f$ of handlebody $…
For a genus g handlebody H a simplicial complex, with vertices being isotopy classes of certain incompressible surfaces in H, is constructed and several properties are established. In particular, this complex naturally contains, as a subcomplex, the complex of curves of the boundary surface of H. As in the classical theory, the group of automorphisms of this complex is identified with the mapping class group of the handlebody.
The balanced superelliptic handlebody group is the normalizer of the transformation group of the balanced superelliptic covering space in the handlebody group of the total space. We prove that the balanced superelliptic mapping class group is generated by four elements. To prove this, we also proved that the liftable Hilden group is generated by three elements. This generating set for the liftable Hilden group is minimal except for some hyperelliptic cases and the generating set for the balanced superelliptic mapping class group above is also minimal for several cases.
We consider finite group-actions on closed, orientable and nonorientable 3-manifolds; such a finite group-action leaves invariant the two handlebodies of a Heegaard splitting of M of some genus g. The maximal possible order of a finite group-action of an orientable or nonorientable handlebody of genus g > 1 is 24(g-1), and in the present paper we characterize the 3-manifolds M and groups G for which the maximal possible order |G| = 24(g-1) is obtained, for some G-invariant Heegaard splitting of genus g > 1. If M is reducible then it is obtained by doubling an action of maximal possible order 2…
The classification of finite group-actions on closed surfaces of small genus is well-known. In the present paper we are interested in the question of which of these group-actions are bounding (extend to a compact 3-manifold with the surface as its unique boundary component, e.g. to a handlebody) or geometrically bounding (extend to a hyperbolic 3-manifold with totally geodesic boundary), concentrating, as a typical case, on large group-actions on surfaces of genus 3.
