arXiv research

Generalizes fully augmented links to doubled 3-manifolds with geometric bounds.

The study of double coset growth in specific groups confirms a conjecture about generic 3-manifolds.

Complete minimal surfaces with any genus found in a specific 3-manifold.

The study finds generalized torsion elements in 3-manifolds from specific knot surgeries.

Paper proves existence of minimal doublings on surfaces with specific properties.

New invariants from quantum group theory for hyperbolic 3-manifolds.

Let M = H^3 / Γbe a hyperbolic 3-manifold of finite volume. We show that if H and K are abelian subgroups of Γand g is in Γ, then the double coset HgK is separable in Γ. As a consequence we prove that if M is a closed, orientable, Haken 3-manifold and the fundamental group of every hyperbolic piece of the torus decompo…

Gabai showed that the Whitehead manifold is the union of two submanifolds each of which is homeomorphic to $\mathbb R^3$ and whose intersection is again homeomorphic to $\mathbb R^3$. Using a family of generalizations of the Whitehead Link, we show that there are uncountably many contractible 3-manifolds with this doub…

The tetrus is a sort of big brother to the tripus, W.P. Thurston's example of a compact hyperbolic 3-manifold with totally geodesic boundary. We describe a sixfold cover of the double of the tetrus, itself a double, which fibers over the circle with fiber a closed surface of genus 19. We also record arithmeticity of th…

Invariant of 3-manifolds using Hopf algebra elements.

We study the double slice genus of a knot, a natural generalization of slice genus. We define a notion called band number, a natural generalization of band unknotting number, and prove it is an upper bound on double slice genus. Our bound is based on an analysis of broken surface diagrams and embedding properties of 3-…

Study on fibered knots in 3-manifolds, proving unrelated volume and genus.

The branched virtual fibering theorem by Sakuma states that every closed orientable $3$-manifold with a Heegaard surface of genus $g$ has a branched double cover which is a genus $g$ surface bundle over the circle. It is proved by Brooks that such a surface bundle can be chosen to be hyperbolic. We prove that the minim…

We show that the infinite cyclic cover of the exterior of the untwisted Whitehead double of a non-trivial knot does not embed in any compact 3-manifold, answering a question of Jiang, Ni, Wang and Zhou.

In this paper, we compute the subgroup distortion of all finitely generated subgroups of all finitely generated 3-manifold groups, and the subgroup distortion in this case can only be linear, quadratic, exponential and double exponential. It turns out that the subgroup distortion of a subgroup of a 3-manifold group is …

Given a 3-manifold that can be written as the double of a compression body, we compute the Chern-Simons critical values for arbitrary compact connected structure groups. We also show that the moduli space of flat connections is connected when there are no reducibles.

A Dehn sphere in a closed 3-manifold M is a 2-sphere immersed in M with only double curve and triple point singularities. The Dehn sphere S fills M if it defines a cell-decomposition of M. The inverse image in S^{2} of the double curves of S is the Johansson diagram of S and if S fills M it is possible to reconstruct M…

New examples show high twisting doesn't guarantee open book maximality.

We establish an $h$-principle for exact Lagrangian immersions with transverse self-intersections and the minimal, or near-minimal number of double points. One corollary of our result is that any orientable closed 3-manifold admits an exact Lagrangian immersion into standard symplectic 6-space $\R^6_\st$ with exactly on…

Study shows inequality in Floer homologies for 3-manifold covers.

Simplified combinatorial descriptions of branched spines for 3-manifolds using primary MP move and sliding moves.

We consider a union of two pants decompositions of the same orientable 2-dimensional surface of any genus g. Each pants decomposition corresponds to some handlebody bounded by this surface, so two pants decompositions correspond to a Heegaard splitting of a 3-manifold. We introduce a groupoid FT acting on double pants …

New real invariants for 3-manifolds and links.

We prove the existence of a spectral sequence for Lagrangian Floer homology which converges to the Floer homology of the image of a Lagrangian submanifold under multiple fibred Dehn twists. The $E_1$ term of the sequence is given by the hypercube of "resolutions" of the Dehn twists involved. The proof relies on the exa…

Study fiber-preserving, orientation-reversing involutions on Seifert fibered 3-manifolds.

Let $g \colon S \looparrowright N$ be a properly immersed $π_1$--injective surface in a non-geometric $3$--manifold $N$. We compute the distortion of $π_1(S)$ in $π_1(N)$ and show that how it is related to separability of $π_1(S)$ in $π_1(N)$. The only possibility of the distortion is linear, quadratic, exponential, an…

Invariants for 3-manifolds with embedded links using Hopf algebras.

A Margulis spacetime is a complete flat affine Lorentzian 3-manifold with free fundamental group. Associated to $M$ is a noncompact complete hyperbolic surface $Σ$. We study double extensions of $π_1 (M) \cong π_1 (Σ)$ when $Σ$ is homeomorphic to a projective plane minus two discs. We classify proper actions of this do…

A 1-bridge torus knot in a 3-manifold of genus $\le 1$ is a knot drawn on a Heegaard torus with one bridge. We give two types of normal forms to parameterize the family of 1-bridge torus knots that are similar to the Schubert's normal form and the Conway's normal form for 2-bridge knots. For a given Schubert's normal f…

A rational number $r$ is called a left orderable slope of a knot $K \subset S^3$ if the 3-manifold obtained from $S^3$ by $r$-surgery along $K$ has left orderable fundamental group. In this paper we consider the double twist knots $C(k,l)$ in the Conway notation. For any positive integers $m$ and $n$, we show that if $…

Study of lifting maps in branched covers of 3-manifolds, showing non-injectivity.

The paper extends knot theory to 4-manifolds, defining new genera and obstructions.

Motivated by a result of L.P. Roberts on rational blow-downs in Heegaard-Floer homology, we study such operations along 3-manifolds that arise as branched double covers of $S^{3}$ along several non-alternating, slice knots.

The study characterizes slopes for hyperbolic knots and Whitehead doubles.

New Langlands duality conjectures for 3-manifold skein modules.

We discuss some additivity properties of the simplicial volume for manifolds with boundary: we give proofs of additivity for glueing amenable boundary components and of superadditivity for glueing amenable submanifolds of the boundary, and we discuss doubling of 3-manifolds.

Anosov flows in hyperbolic 3-manifolds are quasigeodesic if not R-covered.

Researchers compute twisted Reidemeister torsion for hyperbolic 3-manifolds.

We study configurations of immersed curves in surfaces and surfaces in 3-manifolds. Among other results, we show that primitive curves have only finitely many configurations which minimize the number of double points. We give examples of minimal configurations not realized by geodesics in any hyperbolic metric.

Many key algorithms in 3-manifold topology involve the enumeration of normal surfaces, which is based upon the double description method for finding the vertices of a convex polytope. Typically we are only interested in a small subset of these vertices, thus opening the way for substantial optimization. Here we give an…

3-manifolds have covers with infinitely many ideal triangulations.

We prove hyperbolic 3-manifolds are geometrically inflexible: a unit quasiconformal deformation of a Kleinian group extends to an equivariant bi-Lipschitz diffeomorphism between quotients whose pointwise bi-Lipschitz constant decays exponentially in the distance form the boundary of the convex core for points in the th…

In low-dimensional topology, many important decision algorithms are based on normal surface enumeration, which is a form of vertex enumeration over a high-dimensional and highly degenerate polytope. Because this enumeration is subject to extra combinatorial constraints, the only practical algorithms to date have been v…

A 3-manifold is foliar if it supports a codimension-one co-oriented taut foliation. Suppose $M$ is an oriented 3-manifold with connected boundary a torus, and suppose $M$ contains a properly embedded, compact, oriented, surface $R$ with a single boundary component that is Thurston norm minimizing in $H_2(M, \partial M)…

Geometrization theorem, fibered case: Every three-manifold that fibers over the circle admits a geometric decomposition. Double limit theorem: for any sequence of quasi-Fuchsian groups whose controlling pair of conformal structures tends toward a pair of projectively measured laminations that bind the surface, there is…

New quantum invariant for framed 3-manifolds using ideal triangulations.

Constructs knots from 3-manifolds with specified geometric limits.

The twisted torsion of a 3-manifold is well-known to be zero whenever the corresponding twisted Alexander module is non-torsion. Under mild extra assumptions we introduce a new twisted torsion invariant which is always non-zero. We show how this torsion invariant relates to the twisted intersection form of a bounding 4…