arXiv subjects

William Jaco

Publications and source records attributed to William Jaco.

14 recordsLinked to original sources

arXiv2022

Complexity of 3-manifolds obtained by Dehn filling

Let $M$ be a compact 3--manifold with boundary a single torus. We present upper and lower complexity bounds for closed 3--manifolds obtained as even Dehn fillings of $M.$ As an application, we characterise some infinite families of even Dehn fillings of $M$ for which our method determines the complexity of its members up to an additive constant. The constant only depends on the size of a chosen triangulation of $M$, and the isotopy class of its boundary. We then show that, given a triangulation $\mathcal T$ of $M$ with $2$--triangle torus boundary, there exist infinite families of even Dehn fillings of $M$ for which we can determine the complexity of the filled manifolds with a gap between upper and lower bound of at most $13 |\mathcal T| + 7.$ This result is bootstrapped to obtain the gap as a function of the size of an ideal triangulation of the interior of $M$, or the number of crossings of a knot diagram. We also show how to compute the gap for explicit families of fillings of knot complements in the three-sphere. The practicability of our approach is demonstrated by determining the complexity up to a gap of at most 10 for several infinite families of even fillings of the figure eight knot, the pretzel knot $P(-2,3,7)$, and the trefoil.

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arXiv2021

Slope norm and an algorithm to compute the crosscap number

We give three algorithms to determine the crosscap number of a knot in the 3-sphere using $0$-efficient triangulations and normal surface theory. Our algorithms are shown to be correct for a larger class of complements of knots in closed 3-manifolds. The crosscap number is closely related to the minimum over all spanning slopes of a more general invariant, the slope norm. For any irreducible 3-manifold $M$ with incompressible boundary a torus, we give an algorithm that, for every slope on the boundary that represents the trivial class in $H_1(M; \mathbb{Z}_2)$, determines the maximal Euler characteristic of any properly embedded surface having a boundary curve of this slope. We complement our theoretical work with an implementation of our algorithms, and compute the crosscap number of knots for which previous methods would have been inconclusive. In particular, we determine 196 previously unknown crosscap numbers in the census of all knots with up to 12 crossings.

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arXiv2020

Efficient triangulations and boundary slopes

For a compact, irreducible, $\partial$-irreducible, an-annular bounded 3-manifold $M\ne\mathbb{B}^3$, then any triangulation $\mathcal{T}$ of $M$ can be modified to an ideal triangulation $\mathcal{T}^*$ of $\stackrel{\circ}{M}$. We use the inverse relationship of crushing a triangulation along a normal surface and that of inflating an ideal triangulation to introduce and study boundary-efficient triangulations and end-efficient ideal triangulations. We prove that the topological conditions necessary for a compact 3-manifold $M$ admitting an annular-efficient triangulation are sufficient to modify any triangulation of $M$ to a boundary-efficient triangulation which is also annular-efficient. From the proof we have for any ideal triangulation $T^*$ and any inflation $\mathcal{T}_Λ$, there is a bijective correspondence between the closed normal surfaces in $\mathcal{T}^*$ and the closed normal surfaces in $\mathcal{T}_Λ$ with corresponding normal surfaces being homeomorphic. It follows that for an ideal triangulation $\mathcal{T}^*$ that is $0$-efficient, $1$-efficient, or end-efficient, then any inflation $\mathcal{T}_Λ$ of $\mathcal{T}^*$ is $0$-efficient, $1$-efficient, or $\partial$-efficient, respectively. There are algorithms to decide if a given triangulation or ideal triangulation of a $3$-manifold is one of these efficient triangulations. Finally, it is shown that for an annular-efficient triangulation, there are only a finite number of boundary slopes for normal surfaces of a bounded Euler characteristic; hence, in a compact, orientable, irreducible, $\partial$-irreducible, and an-annular $3$-manifold, there are only finitely many boundary slopes for incompressible and $\partial$-incompressible surfaces of a bounded Euler characteristic.

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arXiv2019

On minimal ideal triangulations of cusped hyperbolic 3-manifolds

Previous work of the authors studies minimal triangulations of closed 3-manifolds using a characterisation of low degree edges, embedded layered solid torus subcomplexes and 1-dimensional $\mathbb{Z}_2$-cohomology. The underlying blueprint is now used in the study of minimal ideal triangulations. As an application, it is shown that the monodromy ideal triangulations of the hyperbolic once-punctured torus bundles are minimal.

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arXiv2017

Z2-Thurston Norm and Complexity of 3-Manifolds, II

In this sequel to earlier papers by three of the authors, we obtain a new bound on the complexity of a closed 3--manifold, as well as a characterisation of manifolds realising our complexity bounds. As an application, we obtain the first infinite families of minimal triangulations of Seifert fibred spaces modelled on Thurston's geometry $\widetilde{\text{SL}_2(\mathbb{R})}.$

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arXiv2014

Bounds for the genus of a normal surface

This paper gives sharp linear bounds on the genus of a normal surface in a triangulated compact, orientable 3--manifold in terms of the quadrilaterals in its cell decomposition---different bounds arise from varying hypotheses on the surface or triangulation. Two applications of these bounds are given. First, the minimal triangulations of the product of a closed surface and the closed interval are determined. Second, an alternative approach to the realisation problem using normal surface theory is shown to be less powerful than its dual method using subcomplexes of polytopes.

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arXiv2011

Annular-Efficient Triangulations of 3-manifolds

A triangulation of a compact 3-manifold is annular-efficient if it is 0-efficient and the only normal, incompressible annuli are thin edge-linking. If a compact 3-manifold has an annular-efficient triangulation, then it is irreducible, boundary-irreducible, and an-annular. Conversely, it is shown that for a compact, irreducible, boundary-irreducible, and an-annular 3-manifold, any triangulation can be modified to an annular-efficient triangulation. It follows that for a manifold satisfying this hypothesis, there are only a finite number of boundary slopes for incompressible and boundary-incompressible surfaces of a bounded Euler characteristic.

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arXiv2009

Z2-Thurston Norm and Complexity of 3-Manifolds

A new lower bound on the complexity of a 3-manifold is given using the Z2-Thurston norm. This bound is shown to be sharp, and the minimal triangulations realising it are characterised using normal surfaces consisting entirely of quadrilateral discs.

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arXiv2009

Coverings and Minimal Triangulations of 3-Manifolds

This paper uses results on the classification of minimal triangulations of 3-manifolds to produce additional results, using covering spaces. Using previous work on minimal triangulations of lens spaces, it is shown that the lens space $L(4k, 2k-1)$ and the generalised quaternionic space $S^3/Q_{4k}$ have complexity $k,$ where $k\ge 2.$ Moreover, it is shown that their minimal triangulations are unique.

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arXiv2008

Minimal triangulations for an infinite family of lens spaces

The notion of a layered triangulation of a lens space was defined by Jaco and Rubinstein in earlier work, and, unless the lens space is L(3,1), a layered triangulation with the minimal number of tetrahedra was shown to be unique and termed its "minimal layered triangulation." This paper proves that for each integer n>1, the minimal layered triangulation of the lens space L(2n,1) is its unique minimal triangulation. More generally, the minimal triangulations (and hence the complexity) are determined for an infinite family of lens spaces containing the lens spaces L(2n,1).

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arXiv2006

Finding planar surfaces in knot- and link-manifolds

It is shown that given any link-manifold, there is an algorithm to decide if the manifold contains an embedded, essential planar surface; if it does, the algorithm will construct one. If a slope on the boundary of the link-manifold is given, there is an algorithm to determine if the slope bounds an embedded punctured-disk; if a meridian slope is given, then it can be determined if a longitude bounds an embedded punctured-disk. The methods use normal surface theory but do not follow the classical approach. Properties of minimal vertex triangulations, layered-triangulations, 0--efficient triangulations and especially triangulated Dehn fillings are central to our methods. We also use an average length estimate for boundary curves of embedded normal surfaces; a version of the average length estimate with boundary conditions also is derived. An algorithm is given to construct precisely those slopes on the boundary of a given link-manifold that bound an embedded punctured-disk in the link-manifold.

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arXiv2006

Layered-triangulations of 3-manifolds

A family of one-vertex triangulations of 3-manifolds, layered-triangulations, is defined. Layered-triangulations are first described for handlebodies and then extended to all 3-manifolds via Heegaard splittings. A complete and detailed analysis of layered-triangulations is given in the cases of the solid torus and lens spaces, including the classification of all normal and almost normal surfaces in these triangulations. Minimal layered-triangulations of lens spaces provide a common setting for new proofs of the classification of lens spaces admitting an embedded non orientable surface and the classification of embedded non orientable surfaces in each such lens space, as well as a new proof of the uniqueness of Heegaard splittings of lens spaces. Canonical triangulations of Dehn fillings, triangulated Dehn fillings, are constructed and applied to the study of Heegaard splittings and efficient triangulations of Dehn fillings. A new presentation of 3-manifolds as being obtained from special layered-triangulations of handlebodies with one-vertex, 2-symmetric triangulations on their boundaries, called triangulated Heegaard splittings, is defined and explored. The 1-skeleton (L-graph) of the complex determined as the quotient of the flip-complex by considering those homemorphisms of the genus g surface that extend to a homeomorphism of the genus g handlebody is used to organize much of the work. Numerous questions remain open, particularly in relation to the L-graph, 2-symmetric triangulations of a closed orientable surface, minimal layered-triangulations of genus-g-handlebodies, and the relationship of layered-triangulations to foliations.

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arXiv2002

0-efficient triangulations of 3-manifolds

0-efficient triangulations of 3-manifolds are defined and studied. It is shown that any triangulation of a closed, orientable, irreducible 3-manifold M can be modified to a 0-efficient triangulation or M can be shown to be one of the manifolds S^3, RP^3 or L(3,1). Similarly, any triangulation of a compact, orientable, irreducible, boundary-irreducible 3-manifold can be modified to a 0-efficient triangulation. The notion of a 0-efficient ideal triangulation is defined. It is shown if M is a compact, orientable, irreducible, boundary-irreducible 3-manifold having no essential annuli and distinct from the 3-cell, then the interior of M admits an ideal triangulation; furthermore, it is shown that any ideal triangulation of such a 3-manifold can be modified to a 0-efficient ideal triangulation. A 0-efficient triangulation of a closed manifold has only one vertex or the manifold is S^3 and the triangulation has precisely two vertices. 0-efficient triangulations of 3-manifolds with boundary, and distinct from the 3-cell, have all their vertices in the boundary and then just one vertex in each boundary component. As tools, we introduce the concepts of barrier surface and shrinking, as well as the notion of crushing a triangulation along a normal surface. A number of applications are given, including an algorithm to construct an irreducible decomposition of a closed, orientable 3-manifold, an algorithm to construct a maximal collection of pairwise disjoint, normal 2-spheres in a closed 3-manifold, an alternate algorithm for the 3-sphere recognition problem, results on edges of low valence in minimal triangulations of 3-manifolds, and a construction of irreducible knots in closed 3-manifolds.

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arXiv1998

Decision problems in the space of Dehn fillings

In this paper, we use normal surface theory to study Dehn filling on a knot-manifold. First, it is shown that there is a finite computable set of slopes on the boundary of a knot-manifold that bound normal and almost normal surfaces in a one-vertex triangulation of that knot-manifold. This is combined with existence theorems for normal and almost normal surfaces to construct algorithms to determine precisely which manifolds obtained by Dehn filling: 1) are reducible, 2) contain two--sided incompressible surfaces, 3) are Haken, 4) fiber over the circle, 5) are the 3--sphere, and 6) are a lens space. Each of these algorithms is a finite computation. Moreover, in the case of essential surfaces, we show that the topology of the filled manifolds is strongly reflected in the triangulation of the knot-manifold. If a filled manifold contains an essential surface then the knot-manifold contains an essential vertex solution that caps off to an essential surface of the same type in the filled manifold. (Vertex solutions are the premier class of normal surface and are computable.)

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