Torus Boundary Component Research Articles

A meromorphic extension of the 3D index

Using the locally compact abelian group mathbb {T}times mathbb {Z}, we assign a meromorphic function to each ideal triangulation of a 3-manifold with torus boundary components. The function is invariant under all 2–3 Pachner moves, and thus is a topological invariant of the underlying manifold. If the ideal triangulation has a strict angle structure, our meromorphic function can be expanded into a Laurent power series whose coefficients are formal power series in q with integer coefficients that coincide with the 3D index of (Dimofte et al. in Adv Theor Math Phys 17(5):975–1076, 2013). Our meromorphic function can be computed explicitly from the matrix of the gluing equations of a triangulation, and we illustrate this with several examples.

Dimension of character varieties for $3$-manifolds

Let M be an orientable 3-manifold, compact with boundary and Γ its fundamental group. Consider a complex reductive algebraic group G. The character variety X(Γ, G) is the GIT quotient Hom(Γ, G)//G of the space of morphisms Γ → G by the natural action by conjugation of G. In the case G = SL(2, C) this space has been thoroughly studied. Following work of Thurston [Thu80], as presented by Culler-Shalen [CS83], we give a lower bound for the dimension of irreducible components of X(Γ, G) in terms of the Euler characteristic χ(M) of M , the number t of torus boundary components of M , the dimension d and the rank r of G. Indeed, under mild assumptions on an irreducible component X0 of X(Γ, G), we prove the inequality dim(X0) ≥ t · r − dχ(M).

Bordered Heegaard Floer homology and graph manifolds

We perform two explicit computations of bordered Heegaard Floer invariants. The first is the type D trimodule associated to the trivial S^1 bundle over the pair of pants P. The second is a bimodule that is necessary for self-gluing, when two torus boundary components of a bordered manifold are glued to each other. Using the results of these two computations, we describe an algorithm for computing HF-hat of any graph manifold.

The 3D-index and normal surfaces

Dimofte, Gaiotto and Gukov introduced a powerful invariant, the 3D-index, associated to a suitable ideal triangulation of a 3-manifold with torus boundary components. The 3D-index is a collection of formal power series in $q^{1/2}$ with integer coefficients. Our goal is to explain how the 3D-index is a generating series of normal surfaces associated to the ideal triangulation. This shows a connection of the 3D-index with classical normal surface theory, and fulfills a dream of constructing topological invariants of 3-manifolds using normal surfaces.

Heegaard structure respects complicated JSJ decompositions

Let M be a 3-manifold with torus boundary components \(T_{1}\) and \(T_2\). Let \(\phi :T_{1} \rightarrow T_{2}\) be a homeomorphism, \(M_\phi \) the manifold obtained from M by gluing \(T_{1}\) to \(T_{2}\) via the map \(\phi \), and T the image of \(T_{1}\) in \(M_\phi \). We show that if \(\phi \) is “sufficiently complicated” then any incompressible or strongly irreducible surface in \(M_\phi \) can be isotoped to be disjoint from T. It follows that every Heegaard splitting of a 3-manifold admitting a “sufficiently complicated” JSJ decomposition is an amalgamation of Heegaard splittings of the components of the JSJ decomposition.

Surfaces that become isotopic after Dehn filling

We show that after generic filling along a torus boundary component of a 3-manifold, no two closed, 2-sided, essential surfaces become isotopic, and no closed, 2-sided, essential surface becomes inessential. That is, the set of essential surfaces (considered up to isotopy) survives unchanged in all suitably generic Dehn fillings. Furthermore, for all but finitely many non-generic fillings, we show that two essential surfaces can only become isotopic in a constrained way.

Band-taut sutured manifolds

Attaching a 2-handle to a genus two or greater boundary component of a 3-manifold is a natural generalization of Dehn filling a torus boundary component. We prove that there is an interesting relationship between an essential surface in a sutured 3-manifold, the number of intersections between the boundary of the surface and one of the sutures, and the cocore of the 2-handle in the manifold after attaching a 2-handle along the suture. We use this result to show that tunnels for tunnel number one knots or links in any 3-manifold can be isotoped to lie on a branched surface corresponding to a certain taut sutured manifold hierarchy of the knot or link exterior. In a subsequent paper, we use the theorem to prove that band sums satisfy the cabling conjecture, and to give new proofs that unknotting number one knots are prime and that genus is superadditive under band sum. To prove the theorem, we introduce band taut sutured manifolds and prove the existence of band taut sutured manifold hierarchies.

The quantum content of the gluing equations

The gluing equations of a cusped hyperbolic 3‐manifold M are a system of polynomial equations in the shapes of an ideal triangulation T of M that describe the complete hyperbolic structure of M and its deformations. Given a Neumann‐Zagier datum (comprising the shapes together with the gluing equations in a particular canonical form) we define a formal power series with coefficients in the invariant trace field of M that should (a) agree with the asymptotic expansion of the Kashaev invariant to all orders, and (b) contain the nonabelian Reidemeister‐Ray‐Singer torsion of M as its first subleading “1‐loop” term. As a case study, we prove topological invariance of the 1‐loop part of the constructed series and extend it into a formal power series of rational functions on the PSL.2;C/ character variety of M . We provide a computer implementation of the first three terms of the series using the standard SnapPy toolbox and check numerically the agreement of our torsion with the Reidemeister‐Ray‐Singer for all 59924 hyperbolic knots with at most 14 crossings. Finally, we explain how the definition of our series follows from the quantization of 3‐dimensional hyperbolic geometry, using principles of topological quantum field theory. Our results have a straightforward extension to any 3‐manifold M with torus boundary components (not necessarily hyperbolic) that admits a regular ideal triangulation with respect to some PSL.2;C/ representation.

A generalisation of the deformation variety

Given an ideal triangulation of a connected 3-manifold with non-empty boundary consisting of a disjoint union of tori, a point of the deformation variety is an assignment of complex numbers to the dihedral angles of the tetrahedra subject to Thurston's gluing equations. From this, one can recover a representation of the fundamental group of the manifold into the isometries of 3-dimensional hyperbolic space. However, the deformation variety depends crucially on the triangulation: there may be entire components of the representation variety which can be obtained from the deformation variety with one triangulation but not another. We introduce a generalisation of the deformation variety, which again consists of assignments of complex variables to certain dihedral angles subject to polynomial equations, but together with some extra combinatorial data concerning degenerate tetrahedra. This "extended deformation variety" deals with many situations that the deformation variety cannot. In particular we show that for any ideal triangulation of a small orientable 3-manifold with a single torus boundary component, we can recover all of the irreducible non-dihedral representations from the associated extended deformation variety. More generally, we give an algorithm to produce a triangulation of a given orientable 3-manifold with torus boundary components for which the same result holds. As an application, we show that this extended deformation variety detects all factors of the PSL(2,C) A-polynomial associated to the components consisting of the representations it recovers.

Depth of pleated surfaces in toroidal cusps of hyperbolic 3–manifolds

Let $F$ be a closed essential surface in a hyperbolic 3-manifold $M$ with a toroidal cusp $N$. The depth of $F$ in $N$ is the maximal distance from points of $F$ in $N$ to the boundary of $N$. It will be shown that if $F$ is an essential pleated surface which is not coannular to the boundary torus of $N$ then the depth of $F$ in $N$ is bounded above by a constant depending only on the genus of $F$. The result is used to show that an immersed closed essential surface in $M$ which is not coannular to the torus boundary components of $M$ will remain essential in the Dehn filling manifold $M(\gamma)$ after excluding $C_g$ curves from each torus boundary component of $M$, where $C_g$ is a constant depending only on the genus $g$ of the surface.

Boundary Structure of Hyperbolic 3-Manifolds Admitting Annular and Toroidal Fillings at Large Distance

AbstractFor a hyperbolic 3-manifold M with a torus boundary component, all but finitely many Dehn fillings yield hyperbolic 3-manifolds. In this paper, we will focus on the situation where M has two exceptional Dehn fillings: an annular filling and a toroidal filling. For such a situation, Gordon gave an upper bound of 5 for the distance between such slopes. Furthermore, the distance 4 is realized only by two specific manifolds, and 5 is realized by a single manifold. These manifolds all have a union of two tori as their boundaries. Also, there is a manifold with three tori as its boundary which realizes the distance 3. We show that if the distance is 3 then the boundary of the manifold consists of at most three tori.

Toroidal Dehn fillings on hyperbolic 3-manifolds

We determine all hyperbolic 3-manifolds $M$ admitting two toroidal Dehn fillings at distance 4 or 5. We show that if $M$ is a hyperbolic 3-manifold with a torus boundary component $T_0$, and $r,s$ are two slopes on $T_0$ with $\Delta(r,s) = 4$ or 5 such that $M(r)$ and $M(s)$ both contain an essential torus, then $M$ is either one of 14 specific manifolds $M_i$, or obtained from $M_1, M_2, M_3$ or $M_{14}$ by attaching a solid torus to $\partial M_i - T_0$. All the manifolds $M_i$ are hyperbolic, and we show that only the first three can be embedded into $S^3$. As a consequence, this leads to a complete classification of all hyperbolic knots in $S^3$ admitting two toroidal surgeries with distance at least 4.

Exceptional Dehn fillings on hyperbolic 3-manifolds with at least two boundary components

We estimate the number of exceptional slopes for hyperbolic 3-manifolds with a torus boundary component and at least one other boundary component.

Toroidal and Klein bottle boundary slopes

Let M be a compact, connected, orientable, irreducible 3-manifold and T 0 an incompressible torus boundary component of M such that the pair ( M , T 0 ) is not cabled. By a result of C. Gordon, if ( S , ∂ S ) , ( T , ∂ T ) ⊂ ( M , T 0 ) are incompressible punctured tori with boundary slopes at distance Δ = Δ ( ∂ S , ∂ T ) , then Δ ⩽ 8 , and the cases where Δ = 6 , 7 , 8 are very few and classified. We give a simplified proof of this result (or rather, of its reduction process), using an improved estimate for the maximum possible number of mutually parallel negative edges in the graphs of intersection of S and T. We also extend Gordon's result by allowing either S or T to be an essential Klein bottle.

Constructions controlees de champs de Reeb et applications

On every compact, orientable, irreducible 3-manifold V which is toroidal or has torus boundary components we construct a contact 1-form whose Reeb vector field R does not have any contractible periodic orbits and is tangent to the boundary. Moreover, if bdry V is nonempty, then the Reeb vector field R is transverse to a taut foliation. By appealing to results of Hofer, Wysocki, and Zehnder, we show that, under certain conditions, the 3-manifold obtained by Dehn filling along bdry V is irreducible and different from the 3-sphere. Resum\'e On construit, sur toute variete V de dimension trois orientable, compacte, irreductible, bordee par des tores ou toroidale, une forme de contact dont le champ de Reeb R est sans orbite periodique contractible et tangent au bord. De plus, si le bord de V est non vide, le champ R est transversal a un feuilletage tendu. En utilisant des resultats de Hofer, Wysocki et Zehnder, on obtient sous certaines conditions que la variete obtenue par obturation de Dehn le long du bord de V est irreductible et differente de la sphere S^3.

On hyperbolic 3–manifolds realizing the maximal distance between toroidal Dehn fillings

For a hyperbolic 3-manifold M with a torus boundary component, all but finitely many Dehn fillings on the torus component yield hyperbolic 3-manifolds. In this paper, we will focus on the situation where M has two exceptional Dehn fillings, both of which yield toroidal manifolds. For such situation, Gordon gave an upper bound for the distance between two slopes of Dehn fillings. In particular, if M is large, then the distance is at most 5. We show that this upper bound can be improved by 1 for a broad class of large manifolds.

Immersed essential surfaces and Dehn surgery

It is known that an embedded essential surface F in a hyperbolic manifold M remains essential in Dehn filling spaces M( γ) for most slopes γ on a torus boundary component T of M. The main theorem of this paper is to generalize this result to immersed surfaces. More explicitly, if an immersed essential surface F has coannular slopes β 1,…, β n on T, then there is a constant K such that F remains essential in M( γ) when Δ( γ, β i )> K for all i. It will also be shown that all but finitely many Freedman tubings of a geometrically finite surface in M are π 1-injective.

P2-reducing and toroidal Dehn fillings

Let M be a compact, connected, orientable 3-manifold with a torus boundary component $\partial_{0}M$. A slope on $\partial_{0}M$ is the isotopy class of an unoriented essential simple loop. For a slope r, the manifold obtained from M by r-Dehn filling is M(r) = M$\cup$Vr, where Vr is a solid torus glued to M along $\partial_{0}M$ in such a way that r bounds a meridian disc in Vr. If r and s are two slopes on $\partial_{0}M$, then $\Delta$(r, s) denotes their minimal geometric intersection number.

Dehn fillings on a two torus boundary components 3-manifold

Dehn fillings on a two torus boundary components 3-manifold

Annular and boundary reducing Dehn fillings

Let M be a simple 3-manifold, i.e. one that contains no essential sphere, disk, annulus or torus, with a torus boundary component ∂ 0 M. One is interested in obtaining upper bounds for the distance (intersection number) Δ(α, β) between slopes α, β on ∂ 0 M such that Dehn filling M along α, β produces manifolds M( α), M( β) that are not simple. There are ten cases, according to whether M(α) (M(β)) contains an essential sphere, disk, annulus or torus. Here we show that if M( α) contains an essential annulus and M( β) contains an essential disk then Δ(α, β)⩽2 . This completes the determination of upper bounds for Δ(α, β) in all ten cases.