arXiv subjects

Makoto Sakuma

Publications and source records attributed to Makoto Sakuma.

At least 19 recordsLinked to original sources

arXiv2024

The equivariant genera of marked strongly invertible knots associated with $2$-bridge knots

A marked strongly invertible knot is a triple $(K,h,δ)$ of a knot $K$ in $S^3$, a strong inversion $h$ of $K$, and a subarc $δ\subset \operatorname{Fix}(h)\cong S^1$ bounded by $\operatorname{Fix}(h)\cap K\cong S^0$. An invariant Seifert surface for $(K,h,δ)$ is an $h$-invariant Seifert surface for $K$ that intersects $\operatorname{Fix}(h)$ in the arc $δ$. In this paper, we completely determine the equivariant genus (the minimum of the genera of invariant Seifert surfaces for $(K,h,δ)$) of every marked strongly invertible knot $(K,h,δ)$ with $K$ a $2$-bridge knot.

math.GT↗

arXiv2023

Combinatorial local convexity implies convexity in finite dimensional CAT(0) cubed complexes

We give a proof to the following theorem, which is well-known among experts: A connected subcomplex $W$ of a finite dimensional CAT(0) cubed complex $X$ is convex if and only if Lk$(v, W)$ is a full subcomplex of Lk$(v, X)$ for every vertex $v$ of $W$.

math.GT↗

arXiv2023

Two-parabolic-generator subgroups of hyperbolic 3-manifold groups

We give a detailed account of Agol's theorem and his proof concerning two-meridional-generator subgroups of hyperbolic 2-bridge link groups, which is included in the slide of his talk at the Bolyai conference 2001. We also give a generalization of the theorem to two-parabolic-generator subgroups of hyperbolic 3-manifold groups, which gives a refinement of a result due to Boileau-Weidmann.

math.GT↗

arXiv2022

Invariant Seifert surfaces for strongly invertible knots

We study invariant Seifert surfaces for strongly invertible knots, and prove that the gap between the equivariant genus (the minimum of the genera of invariant Seifert surfaces) of a strongly invertible knot and the (usual) genus of the underlying knot can be arbitrary large. This forms a sharp contrast with Edmonds' theorem that every periodic knot admits an invariant minimal genus Seifert surface. We also prove variants of Edmonds' theorem, which are useful in studying invariant Seifert surfaces for strongly invertible knots.

math.GT↗

arXiv2022

Homotopy motions of surfaces in $3$-manifolds

We introduce the concept of a homotopy motion of a subset in a manifold, and give a systematic study of homotopy motions of surfaces in closed orientable 3-manifolds. This notion arises from various natural problems in 3-manifold theory such as domination of manifold pairs, homotopical behavior of simple loops on a Heegaard surface, and monodromies of virtual branched covering surface bundles associated to a Heegaard splitting.

math.GT↗

arXiv2021

A survey of the impact of Thurston's work on Knot Theory

This is a survey of the impact of Thurston's work on knot theory, laying emphasis on the two characteristic features, rigidity and flexibility, of 3-dimensional hyperbolic structures. We also lay emphasis on the role of the classical invariants, the Alexander polynomial and the homology of finite branched/unbranched coverings.

math.GT↗

arXiv2020

Classification of parabolic generating pairs of Kleinian groups with two parabolic generators

We give an alternative proof to Agol's classification of parabolic generating pairs of non-free Kleinian groups generated by two parabolic transformations. As an application, we give a complete characterisation of epimorphims between $2$-bridge knot groups and a complete characterisation of degree one maps between the exteriors of hyperbolic $2$-bridge links.

math.GT↗

arXiv2020

Classification of non-free Kleinian groups generated by two parabolic transformations

We give a full proof to Agol's announcement on the classification of non-free Kleinian groups generated by two parabolic transformations.

math.GT↗

arXiv2018

Prime amphicheiral knots with free period 2

We construct prime amphicheiral knots that have free period 2. This settles an open question raised by the second named author, who proved that amphicheiral hyperbolic knots cannot admit free periods and that prime amphicheiral knots cannot admit free periods of order >2.

math.GT↗

arXiv2016

An application of non-positively curved cubings of alternating links

By using non-positively curved cubings of prime alternating link exteriors, we prove that certain ideal triangulations of their complements, derived from reduced alternating diagrams, are non-degenerate, in the sense that none of the edges is homotopic relative its endpoints to a peripheral arc. This guarantees that the hyperbolicity equations for those triangulations for hyperbolic alternating links have solutions corresponding to the complete hyperbolic structures. Since the ideal triangulations considered in this paper are often used in the study of the volume conjecture, this result has a potential application to the volume conjecture.

math.GT↗

arXiv2016

A family of two generator non-Hopfian groups

We construct $2$-generator non-Hopfian groups $G_m, m=3, 4, 5, \dots$, where each $G_m$ has a specific presentation $G_m=\langle a, b \, | \, u_{r_{m,0}}=u_{r_{m,1}}=u_{r_{m,2}}= \cdots =1 \rangle$ which satisfies small cancellation conditions $C(4)$ and $T(4)$. Here, $u_{r_{m,i}}$ is the single relator of the upper presentation of the $2$-bridge link group of slope $r_{m,i}$, where $r_{m,0}=[m+1,m,m]$ and $r_{m,i}=[m+1,m-1,(i-1)\langle m \rangle,m+1,m]$ in continued fraction expansion for every integer $i \ge 1$.

math.GR↗

arXiv2016

The action of the mapping class group on the space of geodesic rays of a punctured hyperbolic surface

Let $Σ$ be a complete finite-area orientable hyperbolic surface with one cusp, and let $\mathcal{R}$ be the space of complete geodesic rays in $Σ$ emanating from the puncture. Then there is a natural action of the mapping class group of $Σ$ on $\mathcal{R}$. We show that this action is "almost everywhere" wandering.

math.GT↗

arXiv2015

Parabolic generating pairs of genus-one 2-bridge knot groups

We show that any parabolic generating pair of a genus-one hyperbolic 2-bridge knot group is equivalent to the upper or lower meridian pair. As an application, we obtain a complete classification of the epimorphisms from 2-bridge knot groups to genus-one hyperbolic 2-bridge knot groups.

math.GR↗

arXiv2015

Simple loops on 2-bridge spheres in Heckoid orbifolds for the trivial knot

In this paper, we give a necessary and sufficient condition for an essential simple loop on a $2$-bridge sphere in an even Heckoid orbifold for the trivial knot to be null-homotopic, peripheral or torsion in the orbifold. We also give a necessary and sufficient condition for two essential simple loops on a $2$-bridge sphere in an even Heckoid orbifold for the trivial knot to be homotopic in the orbifold.

math.GR↗

arXiv2014

Homotopically equivalent simple loops on 2-bridge spheres in Heckoid orbifolds for 2-bridge links (I)

In this paper and its sequel, we give a necessary and sufficient condition for two essential simple loops on a 2-bridge sphere in an even Heckoid orbifold for a 2-bridge link to be homotopic in the orbifold. We also give a necessary and sufficient condition for an essential simple loop on a 2-bridge sphere in an even Heckoid orbifold for a 2-bridge link to be peripheral or torsion in the orbifold. This paper treats the case when the 2-bridge link is a $(2,p)$-torus link, and its sequel will treat the remaining cases.

math.GR↗

arXiv2014

Homotopically equivalent simple loops on 2-bridge spheres in Heckoid orbifolds for 2-bridge links (II)

In this paper and its prequel, we give a necessary and sufficient condition for two essential simple loops on a 2-bridge sphere in an even Heckoid orbifold for a 2-bridge link to be homotopic in the orbifold. We also give a necessary and sufficient condition for an essential simple loop on a 2-bridge sphere in an even Heckoid orbifold for a 2-bridge link to be peripheral or torsion in the orbifold. The prequel treated the case when the 2-bridge link is a $(2,p)$-torus link, and this paper treats the remaining cases.

math.GR↗

arXiv2013

Subgroups of mapping class groups related to Heegaard splittings and bridge decompositions

Let $M=H_1\cup_S H_2$ be a Heegaard splitting of a closed orientable 3-manifold $M$ (or a bridge decomposition of a link exterior). Consider the subgroup $\mathrm{MCG}^0(H_j)$ of the mapping class group of $H_j$ consisting of mapping classes represented by auto-homeomorphisms of $H_j$ homotopic to the identity, and let $G_j$ be the subgroup of the automorphism group of the curve complex $\mathcal{CC}(S)$ obtained as the image of $\mathrm{MCG}^0(H_j)$. Then the group $G= $ generated by $G_1$ and $G_2$ preserve the homotopy class in $M$ of simple loops on $S$. In this paper, we study the structure of the group $G$ and the problem to what extent the converse to this observation holds.

math.GT↗

arXiv2013

Homotopically equivalent simple loops on 2-bridge spheres in 2-bridge link complements (III)

This is the last of a series of papers which give a necessary and sufficient condition for two essential simple loops on a 2-bridge sphere in a 2-bridge link complement to be homotopic in the link complement. The first paper of the series treated the case of the 2-bridge torus links, and the second paper treated the case of 2-bridge links of slope $n/(2n+1)$ and $(n+1)/(3n+2)$, where $n \ge 2$ is an arbitrary integer. In this paper, we first treat the case of 2-bridge links of slope $n/(mn+1)$ and $(n+1)/((m+1)n+m)$, where $m \ge 3$ is an arbitrary integer, and then treat the remaining cases by induction.

math.GR↗