Degeneracy pressure of relic neutrinos and cosmic coincidence problem

We consider the universe as a huge $\nu_R$-sphere formed with degenerate relic neutrinos and suggest that its constant energy density play a role of an effective cosmological constant. We construct the sphere as a bubble of true vacuum in a field theory model with a spontaneously broken U(1) global symmetry, and we interpret the sphere-forming time as the transition time for recent acceleration of the universe. The coincidence problem may be regarded as naturally resolved in this model, because the relic neutrinos can make the $\nu_R$-sphere at the recent past time during the matter-dominated era.Comment: 6 pages, added reference

A type of the Lefschetz hyperplane section theorem on \Q-Fano 3-folds with Picard number one and 1/2(1,1,1)1/2(1,1,1)-singularities

We prove a type of the Lefschetz hyperplane section theorem on Q-Fano 3-folds with Picard number one and $1/2(1,1,1)$-singularities by using some degeneration method. As a byproduct, we obtain a new example of a Calabi-Yau 3-fold $X$ with Picard number one whose invariants are $$(H_X^3, c_2 (X) \cdot H_X, {e} (X)) = (8, 44, -88),$$ where $H_X$, $e(X)$ and $c_2(X)$ are an ample generator of \Pic(X), the topological Euler characteristic number and the second Chern class of $X$ respectively.Comment: 9 page

Reduction of bridge positions along a bridge disk

Suppose a knot in a $3$-manifold is in $n$-bridge position. We consider a reduction of the knot along a bridge disk $D$ and show that the result is an $(n-1)$-bridge position if and only if there is a bridge disk $E$ such that $(D, E)$ is a cancelling pair. We apply this to an unknot $K$, in $n$-bridge position with respect to a bridge sphere $S$ in the $3$-sphere, to consider the relationship between a bridge disk $D$ and a disk in the $3$-sphere that $K$ bounds. We show that if a reduction of $K$ along $D$ yields an $(n-1)$-bridge position, then $K$ bounds a disk that contains $D$ as a subdisk and intersects $S$ in $n$ arcs.Comment: 11 pages, 16 figure

Calabi-Yau construction by smoothing normal crossing varieties

We investigate a method of construction of Calabi--Yau manifolds, that is, by smoothing normal crossing varieties. We develop some theories for calculating the Picard groups of the Calabi--Yau manifolds obtained in this method. Some applications are included, such as construction of new examples of Calabi--Yau 3-folds with Picard number one with some interesting properties.Comment: A large revision was made. Basically the paper returned to its previous version (v.2). The Kaehlerness arguement in Propositon 9.1 of v.3 needs correctio

On the connectedness of subcomplexes of a disk complex

For a boundary-reducible $3$-manifold $M$ with $\partial M$ a genus $g$ surface, we show that if $M$ admits a genus $g+1$ Heegaard surface $S$, then the disk complex of $S$ is simply connected. Also we consider the connectedness of the complex of reducing spheres. We investigate the intersection of two reducing spheres for a genus three Heegaard splitting of $\mathrm{(torus)} \times I$.Comment: 11 pages, 4 figure

(Disk, Essential surface) pairs of Heegaard splittings that intersect in one point

We consider a Heegaard splitting M=H_1 \cup_S H_2 of a 3-manifold M having an essential disk D in H_1 and an essential surface F in H_2 with |D \cap F|=1. (We require that boundary of F is in S when H_2 is a compressionbody with non-empty "minus" boundary.) Let F be a genus g surface with n boundary components. From S, we obtain a genus g(S)+2g+n-2 Heegaard splitting M=H'_1 \cup_S' H'_2 by cutting H_2 along F and attaching F \times [0,1] to H_1. As an application, by using a theorem due to Casson and Gordon, we give examples of 3-manifolds having two Heegaard splittings of distinct genera where one of the two Heegaard splittings is a strongly irreducible non-minimal genus splitting and it is obtained from the other by the above construction.Comment: 9 pages, 5 figure

Twisted torus knots T(p,q,3,s) are tunnel number one

We show that twisted torus knots $T(p,q,3,s)$ are tunnel number one. A short spanning arc connecting two adjacent twisted strands is an unknotting tunnel.Comment: 3 pages, 1 figur

Calabi-Yau coverings over some singular varieties and new Calabi-Yau 3-folds with Picard number one

This note is a report on the observation that some singular varieties admit Calabi--Yau coverings. As an application, we construct 18 new Calabi--Yau 3-folds with Picard number one that have some interesting properties.Comment: Several typos were corrected and a condition in Theorem 3.2 was adde

Mirror pairs of Calabi-Yau threefolds from mirror pairs of quasi-Fano threefolds

We present a new construction of mirror pairs of Calabi-Yau manifolds by smoothing normal crossing varieties, consisting of two quasi-Fano manifolds. We introduce a notion of mirror pairs of quasi-Fano manifolds with anticanonical Calabi-Yau fibrations using recent conjectures about Landau-Ginzburg models. Utilizing this notion, we give pairs of normal crossing varieties and show that the pairs of smoothed Calabi-Yau manifolds satisfy the Hodge number relations of mirror symmetry. We consider quasi-Fano threefolds that are some blow-ups of Gorenstein toric Fano threefolds and build 6518 mirror pairs of Calabi-Yau threefolds, including 79 self-mirrors.Comment: This version will appear in J. Math. Pures App

On weak reducing disks and disk surgery

Let $K$ be an unknot in $8$-bridge position in the $3$-sphere. We give an example of a pair of weak reducing disks $D_1$ and $D_2$ for $K$ such that both disks obtained from $D_i$ ($i = 1, 2$) by a surgery along any outermost disk in $D_{3-i}$, cut off by an outermost arc of $D_i \cap D_{3-i}$ in $D_{3-i}$, are not weak reducing disks, i.e. the property of weak reducibility of compressing disks is not preserved by a disk surgery.Comment: 8 pages, 9 figure