arXiv subjects
Publications and source records attributed to Makoto Ozawa.
For a compact connected 3-submanifold with connected boundary in the 3-sphere, we relate the existence of a Seifert surface system for a surface with a Dehn surgery along a null-homologous link. As its corollary, we obtain a refinement of the Fox's re-embedding theorem.
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 characterize composite tunnel number one genus two handlebody-knots.
We give a locally minimal, but not globally minimal bridge position of a knot, that is, an unstabilized, nonminimal bridge position of a knot. It implies that a bridge position cannot always be simplified so that the bridge number monotonically decreases to the minimal.
It is a well-known procedure for constructing a torus knot or link that first we prepare an unknotted torus and meridian disks in the complementary solid tori of it, and second smooth the intersections of the boundary of meridian disks uniformly. Then we obtain a torus knot or link on the unknotted torus and its Seifert surface made of meridian disks. In the present paper, we generalize this procedure by a closed fake surface and show that the resultant two surfaces obtained by smoothing triple points uniformly are essential. We also show that a knot obtained by this procedure satisfies the Neuwirth conjecture, and the distance of two boundary slopes for the knot is equal to the number of triple points of the closed fake surface.
We show that a handlebody-knot whose exterior is boundary-irreducible has a unique maximal unnested set of knotted handle decomposing spheres up to isotopies and annulus-moves. As an application, we show that the handlebody-knots $6_{14}$ and $6_{15}$ are not equivalent. We also show that some genus two handlebody-knots with a knotted handle decomposing sphere can be determined by their exteriors. As an application, we show that the exteriors of $6_{14}$ and $6_{15}$ are not homeomorphic.
Neuwirth asked if any non-trivial knot in the 3-sphere can be embedded in a closed surface so that the complement of the surface is a connected essential surface for the knot complement. In this paper, we examine some variations on this question and prove it for all knots up to 11 crossings except for two examples. We also establish the conjecture for all Montesinos knots and for all generalized arborescently alternating knots. For knot exteriors containing closed incompressible surfaces satisfying a simple homological condition, we establish that the knots satisfy the Neuwirth conjecture. If there is a proper degree one map from knot $K$ to knot $K'$ and $K'$ satisfies the Neuwirth conjecture then we prove the same is true for knot $K$. Algorithms are given to decide if a knot satisfies the various versions of the Neuwirth conjecture and also the related conjectures about whether all non-trivial knots have essential surfaces at integer boundary slopes.
First, we extend Otal's result for the trivial knot to trivial spatial graphs, namely, we show that for any bridge tangle decomposing sphere $S^2$ for a trivial spatial graph $Γ$, there exists a 2-sphere $F$ such that $F$ contains $Γ$ and $F$ intersects $S^2$ in a single loop. Next, we introduce two invariants for spatial graphs. As a generalization of the bridge number for knots, we define the {\em bridge string number} $bs(Γ)$ of a spatial graph $Γ$ as the minimal number of $|Γ\cap S^2|$ for all bridge tangle decomposing sphere $S^2$. As a spatial version of the representativity for a graph embedded in a surface, we define the {\em representativity} of a non-trivial spatial graph $Γ$ as \[ r(Γ)=\max_{F\in\mathcal{F}} \min_{D\in\mathcal{D}_F} |\partial D\cap Γ|, \] where $\mathcal{F}$ is the set of all closed surfaces containing $Γ$ and $\mathcal{D}_F$ is the set of all compressing disks for $F$ in $S^3$. Then we show that for a non-trivial spatial graph $Γ$, \[ \displaystyle r(Γ)\le \frac{bs(Γ)}{2}. \] In particular, if $Γ$ is a knot, then $r(Γ)\le b(Γ)$, where $b(Γ)$ denotes the bridge number. This generalizes Schubert's result on torus knots.
We study a canonical spanning surface obtained from a knot or link diagram depending on a given Kauffman state, and give a sufficient condition for the surface to be essential. By using the essential surface, we can see the triviality and splittability of a knot or link from its diagrams. This has been done on the extended knot or link class which includes all of semiadequate, homogeneous, and most of algebraic knots and links. In the process of the proof of main theorem, Gabai's Murasugi sum theorem is extended to the case of nonorientable spanning surfaces.
We show that any non-minimal bridge decomposition of a torus knot is stabilized and that $n$-bridge decompositions of a torus knot are unique for any integer $n$. This implies that a knot in a bridge position is a torus knot if and only if there exists a torus containing the knot such that it intersects the bridge sphere in two essential loops.
In the present paper, we will show that a $(p,q,r)$-pretzel knot has the representativity 3 if and only if $(p,q,r)$ is either $\pm(-2,3,3)$ or $\pm(-2,3,5)$. We also show that a large algebraic knot has the representativity less than or equal to 3.
We introduce two numerical invariants, the waist and the trunk of knots. The waist of a closed incompressible surface in the complement of a knot is defined as the minimal intersection number of all compressing disks for the surface in the 3-sphere and the knot. Then the waist of a knot is defined as the maximal waist of all closed incompressible surfaces in the complement of the knot. On the other hand, the trunk of a knot is defined as the minimal number of the intersection of the most thick level 2-sphere and the knot over all Morse positions of the knot. In this paper, we obtain an inequality between the waist and the trunk of knots and show that the inequality is best possible. We also define the supertrunk of a knot and relate it to the hull number and the 3-width.
Let $F$ be an incompressible, meridionally incompressible and not boundary-parallel surface with boundary in the complement of an algebraic tangle $(B,T)$. Then $F$ separates the strings of $T$ in $B$ and the boundary slope of $F$ is uniquely determined by $(B,T)$ and hence we can define the slope of the algebraic tangle. In addition to the Conway's tangle sum, we define a natural product of two tangles. The slopes and binary operation on algebraic tangles lead an algebraic structure which is isomorphic to the rational numbers. We introduce a new knot and link class, algebraically alternating knots and links, roughly speaking which are constructed from alternating knots and links by replacing some crossings with algebraic tangles. We give a necessary and sufficient condition for a closed surface to be incompressible and meridionally incompressible in the complement of an algebraically alternating knot or link $K$, in particular we show that if $K$ is a knot, then the complement of $K$ does not contain such a surface.
We introduce a new numerical invariant of knots and links from the descending diagrams. It is considered to live between the unknotting number and the bridge number.
In this paper, we give a necessary condition for a diagram to represent the trivial knot.
In this paper, we characterize closed incompressible surfaces of genus two in the complements of 3-bridge knots and links. This characterization includes that of essential 2-string tangle decompositions for 3-bridge knots and links.
In this paper, we study on knots and closed incompressible surfaces in the 3-sphere via Morse functions. We show that both of knots and closed incompressible surfaces can be isotoped into a "related Morse position" simultaneously. As an application, we have following results. *Smallness of Montesinos tangles with length two and Kinoshita's theta curve *Classification of closed incompressible and meridionally incompressible surfaces in 2-bridge theta-curve and handcuff graph complements and the complements of links which admit Hopf tangle decompositions.
In this article, we consider alternating knots on a closed surface in the 3-sphere, and show that these are not parallel to any closed surface disjoint from the prescribed one.