arXiv research

Takamura established a theory on splitting families of degenerations of complex curves. He introduced a powerful method for constructing a splitting family, called a barking family, in which there appear not only a singular fiber over the origin but also singular fibers over other points, called subordinate fibers. In …

Constructs a family to handle unstable fibers on complex surfaces.

Describes 3-manifolds by families of singularly fibered surfaces.

We construct two infinite families of knots each of which admits a Seifert fibered surgery with none of these surgeries coming from Dean's primitive/Seifert-fibered construction. This disproves a conjecture that all Seifert fibered surgeries arise from Dean's primitive/Seifert-fibered construction. The (-3,3,5)-pretzel…

New bases found for Kauffman bracket skein module of fibered torus.

Study finds infinite non-fibered twisted torus knots.

Manifolds with fibered hyperbolic cusp metrics include hyperbolic manifolds with cusps and locally symmetric spaces of Q-rank one. We extend Vaillant's treatment of Dirac-type operators associated to these metrics by weaking the hypotheses on the boundary families through the use of Fredholm perturbations as in the fam…

In this paper, we derive the family switching formula of -n two-sphere fiber bundle embedded in a smooth four-manifold fiber bundle. In the smooth category, it is a partial generalization of Fintushel-Stern's argument for four-manifolds. We also derive an algebraic analogue of the family switching formula, allowing the…

We find an infinite family of Seifert fibered surgeries on strongly invertible knots which do not have primitive/Seifert positions. Each member of the family is obtained from a trefoil knot after alternate twists along a pair of seiferters for a Seifert fibered surgery on a trefoil knot.

The paper explores fibered and quasi-positive links, introducing new families and invariants.

In this paper we give a Chern-Weil-type construction of characteristic classes of fiber bundles, based on homotopy theory of C-infinity algebras. Our idea is to replace a family of closed manifolds to a family of C-infinity morphisms with family of metrics.

Associated to every state surface for a knot or link is a state graph, which embeds as a spine of the state surface. A state graph can be decomposed along cut-vertices into graphs with induced planar embeddings. Associated with each such planar graph is a checkerboard surface, and each state surface is a fiber if and o…

Researchers determine the Thurston unit ball for a family of $n$-chained links and find conditions for fibered faces.

Essential dimension of a family of complex manifolds is the dimension of the image of its base in the Kuranishi space of the fiber. We prove that any family of hyperkähler manifolds over a compact simply connected base has essential dimension not greater than $1$. A similar result about families of complex tori is also…

Proved boundary Dehn twist is exotic for Milnor fibers with specific conditions.

We consider the family of constant curvature fiber metrics for a Lefschetz fibration with regular fibers of genus greater than one. A result of Obitsu and Wolpert is refined by showing that on an appropriate resolution of the total space, constructed by iterated blow-up, this family is log-smooth, i.e. polyhomogeneous …

Extends width estimates to family case using index theory.

Given a polarized family of varieties over the unit disc, smooth except over the origin and with smooth fibers Calabi-Yau, we show that the origin lies at finite Weil-Petersson distance if and only if after a finite base change the family is birational to one with central fiber a Calabi-Yau variety with at worst canoni…

The waist inequality states that for a continuous map from S^n to R^q, not all fibers can have small (n-q)-dimensional volume. We construct maps for which most fibers have small (n-q)-dimensional volume and all fibers have bounded (n-q)-dimensional volume.

Thurston's fibered face theory allows us to partition the set of pseudo-Anosov mapping classes on different compact oriented surfaces into subclasses with related dynamical behavior. This is done via a correspondence between the rational points on fibered faces in the first cohomology of a hyperbolic 3-manifold and the…

The study computes invariants of satellite knots using bordered Floer homology.

We prove that the indices of fibered-cusp and $d$-Dirac operators on a spin manifold with fibered boundary coincide if the associated family of Dirac operators on the fibers of the boundary is invertible. This answers a question raised by Piazza. Under this invertibility assumption, our method yields an index formula f…

New exotic 4-manifolds found with fiber bundles.

Unique incompressible surface in fibered 3-manifolds.

The paper classifies tight contact structures on Seifert fiber spaces.

We construct an infinite family of knots in rational homology spheres with irreducible, non-fibered complements, for which every non-longitudinal filling is an L-space.

We call a pair (K, m) of a knot K in the 3-sphere S^3 and an integer m a Seifert fibered surgery if m-surgery on K yields a Seifert fiber space. For most known Seifert fibered surgeries (K, m), K can be embedded in a genus 2 Heegaard surface of S^3 in a primitive/Seifert position, the concept introduced by Dean as a na…

We study the relationship between fibered ribbon 1-knots and fibered ribbon 2-knots by studying fibered slice disks with handlebody fibers. We give a characterization of fibered homotopy-ribbon disks and give analogues of the Stallings twist for fibered disks and 2-knots. As an application, we produce infinite families…

Study singular fibers in genus 2 algebraic fibrations and their monodromy factorizations.

Let $M$ be a hyperkaehler manifold, and $η$ a closed, positive (1,1)-form which is degenerate everywhere on $M$. We associate to $η$ a family of complex structures on $M$, called a degenerate twistor family, and parametrized by a complex line. When $η$ is a pullback of a Kaehler form under a Lagrangian fibration $L$, a…

Classifies fibered ribbon pretzels, except for a few cases.

Infinite family of hyperbolic 3-manifolds with large volumes.

The paper proves a precise SYZ conjecture for toric Calabi-Yau manifolds with singular fibers.

Study Turaev-Viro invariants of Seifert fibered 3-manifolds, proving volume conjecture.

In this paper, we deduce some rigidity results in warped product spaces under normal variations of CMC hypersurfaces. In particular, we prove the existence of one-parameter families locally rigid on the spatial fiber of Anti-de Sitter Schwarzschild spacetime and one-parameter families with bifurcation points on the spa…

The paper studies deformations of Lagrangian fibrations on symplectic manifolds.

Study characteristic classes for manifold bundles, focusing on fiber families.

New bounds found for minimal surfaces in hyperbolic 3-manifolds.

Study volume growth in Milnor fibers using real Lagrangians.

The study constructs and analyzes new symplectic 4-manifolds from hyperelliptic Lefschetz fibrations.

Using contact surgery we define families of contact structures on certain Seifert fibered three-manifolds. We prove that all these contact structures are tight using contact Ozsath-Szabo invariants. We use these examples to show that, given a natural number n, there exists a Seifert fibered three-manifold carrying at l…

This paper extends previous work on genus two fibrations by studying and resolving singular fibers.

Given some type of fibration on a 4-manifold $X$ with a torus regular fiber $T$, we may produce a new 4-manifold $X_T$ by performing torus surgery on $T$. There is a natural way to extend the fibration to $X_T$, but a multiple fiber (non-generic) singularity is introduced. We construct explicit generic fibrations (with…

Study Kauffman bracket skein modules of Seifert fibered spaces.

The paper finds infinite families of non-left-orderable L-spaces from tangles.

We consider a gauge invariant one parameter family of families of fiberwise twisted Dirac type operators on a fiberation with the typical fiber an even dimensional compact manifold with boundary, i.e., a family $\{D_u\}, u\in [0,1]$ with $D_1=gD_0g^{-1}$ for a suitable unitary automorphism $g$ of the twisted bundle. Su…

In this article we construct a family of knot surgery $4$-manifolds admitting arbitrarily many nonisomorphic Lefschetz fibration structures with the same genus fiber. We obtain such families by performing knot surgery on an elliptic surface $E(2)$ using connected sums of fibered knots obtained by Stallings twist from a…

The twisted torus knots lie on the standard genus 2 Heegaard surface for $S^3$, as do the primitive/primitive and primitive/Seifert knots. It is known that primitive/primitive knots are fibered, and that not all primitive/Seifert knots are fibered. Since there is a wealth of primitive/Seifert knots that are twisted tor…