arXiv research

We classify incompressible, boundary-incompressible, nonorientable surfaces in punctured-torus bundles over $S^1$. We use the ideas of Floyd, Hatcher, and Thurston. The main tool is to put our surface in the "Morse position" with respect to the projection of the bundle into the basis S^1.

Culler and Shalen, and later Yoshida, give ways to construct incompressible surfaces in 3-manifolds from ideal points of the character and deformation varieties, respectively. We work in the case of hyperbolic punctured torus bundles, for which the incompressible surfaces were classified by Floyd and Hatcher. We conver…

We study the effect of the mapping class group of a reducible 3-manifold $M$ on each incompressible surface that is invariant under a self-homeomorphism of $M$. As an application of this study we answer a question of F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures: A reducible 3-manifold admits an Anosov torus if an…

The paper proves finiteness of cosmetic fillings on a specific type of 3-manifold.

We show that for certain hyperbolic 3-manifolds, all boundary slopes are slopes of immersed incompressible surfaces, covered by incompressible embeddings in some finite cover. The manifolds include hyperbolic punctured torus bundles and hyperbolic two-bridge knots.

Study satellite knots and their quandles related to incompressible tori.

Let $M$ be an irreducible, compact, connected, orientable 3-manifold whose boundary is a torus. We show that if $M$ is hyperbolic, then it admits at most six finite/cyclic fillings of maximal distance 5. Further, the distance of a finite/cyclic filling to a cyclic filling is at most 2. If $M$ has a non-boundary-paralle…

The paper proves conditions for positive scalar curvature metrics on manifolds with incompressible hypersurfaces.

Let M be a hyperbolic manifold of finite volume which fibers over the circle with fiber a once punctured torus, and let S be an arbitrary incompressible surface in M. We determine the characteristic JSJ-subpair of M-S and show, in particular, that the guts of (M,S) is empty.

Extends Frohman and Rannard's result to Seifert fiber spaces with singular surfaces.

We show that on any hyperbolic knot in $S^3$ there is at most one non-integral Dehn surgery which yields a manifold containing an incompressible torus.

Study handles in sutured manifolds and knots, finding varied handle numbers and unique surfaces.

Study shows torsion in knot module for specific Montesinos knots.

Let M be an orientable and irreducible 3-manifold whose boundary is an incompressible torus. Suppose that M does not contain any closed nonperipheral embedded incompressible surfaces. We will show in this paper that the immersed surfaces in M with the 4-plane property can realize only finitely many boundary slopes. Mor…

We investigate certain $4$-dimensional analogues of the classical $3$-dimensional Dehn's lemma, giving examples where such analogues do or do not hold, in the smooth and topological categories. In particular, we show that an essential $2$-sphere $S$ in the boundary of a simply connected $4$-manifold $W$ such that $S$ i…

Let M be $S^3$, $S^1\times S^2$, or a lens space L(p,q), and let k be a (1,1)-knot in M, i.e., a knot which is of 1-bridge with respect to a Heegaard torus. We show that if there is a closed meridionally incompressible surface in the complement of k, then the surface and the knot can be put in a special position, namel…

Proves positive mass theorems for specific ALF and ALG manifolds.

The study determines $\mathbb{Z}_2$-Thurston norms in Sol manifolds and embeds non-orientable surfaces.

We give a complete description of exceptional surgeries on pretzel knots of type $(-2, p, p)$ with $p \ge 5$. It is known that such a knot admits a unique toroidal surgery yielding a toroidal manifold with a unique incompressible torus. By cutting along the torus, we obtain two connected components, one of which is a t…

This is a "software upgrade" to a paper originally published in 1976, with cleaner statements and improved proofs. The main result is that, in a Haken 3-manifold, the space of all incompressible surfaces in a single isotopy class is contractible, except when the surface is the fiber of a surface bundle structure, in wh…

The exceptional Dehn filling conjecture of the second author concerning the relationship between exceptional slopes $α, β$ on the boundary of a hyperbolic knot manifold $M$ has been verified in all cases other than small Seifert filling slopes. In this paper we verify it when $α$ is a small Seifert filling slope and $β…

Let M be a compact, connected, orientable, irreducible 3-manifold and T' an incompressible torus boundary component of M such that the pair (M,T') is not cabled. By a result of C. Gordon, if S and T are incompressible punctured tori in M with boundary on T' and boundary slopes at distance d, then d is at most 8, and th…

Let M be a closed, orientable, irreducible, non-simply connected 3-manifold. We prove that if M admits a sequence of Riemannian metrics whose sectional curvature is locally controlled and whose thick part becomes asymptotically hyperbolic and has a sufficiently small volume, then M is Seifert fibred or contains an inco…

Upper bound on 3-manifold volumes from surface homeomorphisms.

Proves SU(2) representations for certain 3-spheres with embedded tori.

Suppose $K$ is a hyperbolic knot in a solid torus $V$ intersecting a meridian disk $D$ twice. We will show that if $K$ is not the Whitehead knot and the frontier of a regular neighborhood of $K \cup D$ is incompressible in the knot exterior, then $K$ admits at most one exceptional surgery, which must be toroidal. Embed…

Let $T$ be a separating incompressible torus in a 3-manifold $M$. Assuming that a genus $g$ Heegaard splitting $V \cup_S W$ can be positioned nicely with respect to $T$ (e.g. $V \cup_S W$ is strongly irreducible), we obtain an upper bound on the number of stabilizations required for $V \cup_S W$ to become isotopic to a…

The study finds counterexamples to a conjecture about incompressible planar surfaces in hyperbolic link exteriors.

The paper introduces Slope Conjecture which relates the degree of the Jones polynomial of a knot and its parallels with the slopes of incompressible surfaces in the knot complement. More precisely, we introduce two knot invariants, the Jones slopes (a finite set of rational numbers) and the Jones period (a natural numb…

The main results of the paper is that we give a characteristics for an annulus sum and a once-punctured torus sum of two handlebodies to be a handlebody as follows: 1. The annulus sum $H=H_1\cup_A H_2$ of two handlebodies $H_1$ and $H_2$ is a handlebody if and only if the core curve of $A$ is a longitude for either $H_…

For a hyperbolic knot in the 3-sphere, at most finitely many Dehn surgeries yield non-hyperbolic 3-manifolds. As a typical case of such an exceptional surgery, a toroidal surgery is one that yields a closed 3-manifold containing an incompressible torus. The slope corresponding to a toroidal surgery, called a toroidal s…

Unique incompressible surface in fibered 3-manifolds.

We show that a knot in $S^3$ with an infinite number of distinct incompressible Seifert surfaces contains a closed incompressible surface in its complement.

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 lengt…

We consider a numerical approach for the incompressible surface Navier-Stokes equation. The approach is based on the covariant form and uses discrete exterior calculus (DEC) in space and a semi-implicit discretization in time. The discretization is described in detail and related to finite difference schemes on stagger…

Let M be a compact, connected, orientable, irreducible 3-manifold and T an incompressible torus boundary component of M such that the pair (M,T) is not cabled. In the paper "Toroidal and Klein bottle boundary slopes" [arXiv:math/0601034] by the author it was established that for any K-incompressible tori F,F' in (M,T) …

Let $M$ be a 3-manifold with torus boundary components $T_1$ and $T_2$. Let $φ\colon T_1 \to T_2$ be a homeomorphism, $M_φ$ the manifold obtained from $M$ by gluing $T_1$ to $T_2$ via the map $φ$, and $T$ the image of $T_1$ in $M_φ$. We show that if $φ$ is "sufficiently complicated" then any incompressible or strongly …

Develops slope detection for 3-manifolds with torus boundaries.

Study the exponential map on surfaces using fluid dynamics.

Standardizes surfaces in 3D handlebodies using Morse theory.

In this article, we show how to embed the so-called CH2 equations into the geodesic flow of the Hdiv metric in 2D, which, itself, can be embedded in the incompressible Euler equation of a non compact Riemannian manifold. The method consists in embedding the incompressible Euler equation with a potential term coming fro…

For a submanifold M in a Euclidean space, the tangential component x^T of the position vector field x of M is the most natural vector field tangent to the Euclidean submanifold, called the canonical vector field of M. In this article, first we prove that the canonical vector field of every Euclidean submanifold is alwa…

Incompressible fluid dynamics on special manifolds.

Cooper-Manning and Louder gave examples of maps of surface groups to PSL(2,C) which are not injective, but are incompressible (i.e. no simple loop is in the kernel). We construct more examples with very simple certificates for their incompressibility arising from the theory of stable commutator length.

We show that any closed incompressible surface in the complement of a positive knot is algebraically non-split from the knot, positive knots cannot bound non-free incompressible Seifert surfaces and that the splitability and the primeness of positive knots and links can be seen from their positive diagrams.

We show that any noncompact oriented surface is homeomorphic to the leaf of a minimal foliation of a closed $3$-manifold. These foliations are (or are covered by) suspensions of continuous minimal actions of surface groups on the circle. Moreover, the above result is also true for any prescription of a countable family…

Let M be a compact 3-manifold whose interior admits a complete hyperbolic structure. We let Lambda(M) be the supremum of the bottom eigenvalue of the Laplacian of N, where N varies over all hyperbolic 3-manifolds homeomorphic to the interior of M. Similarly, we let D(M) be the infimum of the Hausdorff dimensions of lim…

We introduce a new technique for finding lower bounds on the Heegaard genus of a 3-manifold obtained by gluing a pair of 3-manifolds together along an incompressible torus or annulus. We deduce a number of inequalities, including one which implies that $t(K_1# K_2)\geq \max {t(K_1),t(K_2)}$, where $t(-)$ denotes tunnel…