arXiv research

We show that there exist knots K in S^3 with g(E(K))=2 and g(E(K#K#K))=6. Together with Theorem~1.5 of [1], this proves existence of counterexamples to Morimoto's Conjecture (Conjecture 1.5 of [2]). This is a special case of arxiv.org/abs/math.GT/0701765 [1] Tsuyoshi Kobayashi and Yo'av Rieck. On the growth rate of the…

Let $X$ be the exterior of connected sum of knots and $X_i$ the exteriors of the individual knots. In \cite{morimoto1} Morimoto conjectured (originally for $n=2$) that $g(X) < σ_{i=1}^n g(X_i)$ if and only if there exists a so-called \em primitive meridian \em in the exterior of the connected sum of a proper subset of …

We abstract Morimoto's construction of complex structures on product manifolds to pairs of certain generalized $F$-structures on manifolds that are not necessarily global products. As applications we characterize invariant generalized complex structures on product manifolds in which one factor is a Lie group and we gen…

Given integers g_i > 1 (i=1,...,n) we prove that there exist infinitely may knots K_i in S^3 so that g(E(K_i)) = g_i and the Heegaard genus of the exterior of the connected sum of K_1,...,K_n is the sum the Heegaard genera of K_1,...,K_n, that is: g(E(K_1#...#K_n)) = g(E(K_1)) +...+ g(E(K_n)). (Here, E() denotes the ex…

Study symplectification of rank 2 distributions and their connections.

Method finds MAEs on contactified para-Kähler manifolds.

In this paper, we show that any unknotting tunnel for a two bridge knot is isotopic to either one of known ones. This together with Morimoto-Sakuma's result gives the complete classification of unknotting tunnels for two bridge knots up to isotopies and homeomorphisms.

The only knots that are tunnel number one and genus one are those that are already known: 2-bridge knots obtained by plumbing together two unknotted annuli and the satellite examples classified by Eudave-Munoz and by Morimoto-Sakuma. This confirms a conjecture first made by Goda and Teragaito.

We show that, for any integer $n\ge 3$, there is a prime knot $k$ such that (1) $k$ is not meridionally primitive, and (2) for every $m$-bridge knot $k'$ with $m\leq n$, the tunnel numbers satisfy $t(k\# k')\le t(k)$. This gives counterexamples to a conjecture of Morimoto and Moriah on tunnel number under connected sum…

We consider normal almost contact structures on a Riemannian manifold and, through their associated sections of an ad-hoc twistor bundle, study their harmonicity, as sections or as maps. We rewrite these harmonicity equations in terms of the Riemann curvature tensor and find conditions relating the harmonicity of the a…

We consider manifolds endowed with a contact pair structure. To such a structure are naturally associated two almost complex structures. If they are both integrable, we call the structure a normal contact pair. We generalize the Morimoto's Theorem on product of almost contact manifolds to flat bundles. We construct som…

According to the work of Laitinen, Morimoto, Oliver and Pawałowski, a finite group $G$ has a smooth effective one fixed point action on some sphere if and only if $G$ is an Oliver group. For some finite Oliver groups $G$ of order up to $216$, and for $G=A_5\times C_n$ for $n=3,5,7$, we present a strategy of excluding o…

In this paper, we give an isotopy classification of 3-bridge spheres of 3-bridge arborescent links, which are not Montesinos links. To this end, we prove a certain refinement of a theorem of J.S. Birman and H.M. Hilden on the relation between bridge presentations of links and Heegaard splittings of 3-manifolds. In the …

We prove that if $K_1 \subset M_1,...,K_n \subset M_n$ are m-small knots in closed orientable 3-manifolds then the Heegaard genus of $E(#_{i=1}^n K_i)$ is strictly less than the sum of the Heegaard genera of the $E(K_i)$ ($i=1,...,n$) if and only if there exists a proper subset $I$ of $\{1,...,n\}$ so that $#_{i \in I}…

New families of twisted torus knots found with essential surfaces.

This paper studies the question of whether minimal genus Heegaard splittings of exterior spaces of knots which are connected sums are weakly reducible or not. Furthermore it is shown that the Heegaard splittings of the knots used by Morimoto to show that tunnel number can be sub-additive are all strongly irreducible. T…

Given a knot $K$ in a closed orientable manifold $M$ we define the growth rate of the tunnel number of $K$ to be $gr_t(K) = \limsup_{n \to \infty} \frac{t(nK) - n t(K)}{n-1}$. As our main result we prove that the Heegaard genus of $M$ is strictly less than the Heegaard genus of the knot exterior if and only if the grow…

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…

The braid axis of a closed 3-braid lifts to a genus one fibered knot in the double cover of S^3 branched over the closed braid. Every (null homologous) genus one fibered knot in a 3-manifold may be obtained in this way. Using this perspective we answer a question of Morimoto about the number of genus one fibered knots …

Little is known on the classification of Heegaard splittings for hyperbolic 3-manifolds. Although Kobayashi gave a complete classification of Heegaard splittings for the exteriors of 2-bridge knots, our knowledge of other classes is extremely limited. In particular, there are very few hyperbolic manifolds that are know…

Study harmonicity of normal almost contact structures on Riemannian manifolds.

It is a consequence of theorems of Gordon-Reid [Tangle decompositions of tunnel number one knots and links, J. Knot Theory and its Ramifications, 4 (1995) 389-409] and Thompson [Thin position and bridge number for knots in the 3-sphere, Topology, 36 (1997) 505-507] that a tunnel number one knot, if put in thin position…

We construct some families of complex structures on compact manifolds by means of normal almost contact structures (nacs) so that each complex manifold in the family has a non-singular holomorphic flow. These families include as particular cases the Hopf and Calabi-Eckmann manifolds and the complex structures on the pr…

Contact path geometries are curved geometric structures on a contact manifold comprising smooth families of paths modeled on the family of all isotropic lines in the projectivization of a symplectic vector space. Locally such a structure is equivalent to the graphs in the space of independent and depedent variables of …

Connected sum and trivalent vertex sum are natural operations on genus 2 spatial graphs and, as with knots, tunnel number behaves in interesting ways under these operations. We prove sharp Scharlemann-Schultens type bounds for the tunnel number of a composite genus 2 spatial graph. For the tunnel number of a composite …

A conjecture about rational curves' formal principle and convergence proved for Goursat type families.

An explicit classification of simply connected compact homogeneous CR manifolds G/L of codimension one, with non-degenerate Levi form, is given. There are three classes of such manifolds: a) the standard CR homogeneous manifolds which are homogeneous S^1-bundles over a flag manifold F, with CR structure induced by an i…

A filtered manifold is a smooth manifold $M$ together with a filtration of the tangent bundle by smooth subbundles which is compatible with the Lie bracket of vector fields in a certain sense. The Lie bracket of vector fields then induces a bilinear operation on the associated graded of each tangent space of $M$ making…

Unified approach to geometric structure equivalence problem.