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Discovery Projects - Grant ID: DP130103694

Title
Discovery Projects - Grant ID: DP130103694
Funding
ARC | Discovery Projects
Contract (GA) number
DP130103694
Start Date
2013/01/01
End Date
2015/12/31
Open Access mandate
no
Organizations
-
More information
http://purl.org/au-research/grants/arc/DP130103694

 

  • Thin position for incompressible surfaces in 3-manifolds

    Ichihara, Kazuhiro; Ozawa, Makoto; Rubinstein, J. Hyam (2015)
    Projects: ARC | Discovery Projects - Grant ID: DP130103694 (DP130103694)
    In this paper, we give an algorithm to build all compact orientable atoroidal Haken 3-manifolds with tori boundary or closed orientable Haken 3-manifolds, so that in both cases, there are embedded closed orientable separating incompressible surfaces which are not tori. Next, such incompressible surfaces are related to Heegaard splittings. For simplicity, we focus on the case of separating incompressible surfaces, since non-separating ones have been extensively studied. After putting the surfa...

    A birationality result for character varieties

    Klaff, Ben; Tillmann, Stephan (2013)
    Projects: ARC | Discovery Projects - Grant ID: DP130103694 (DP130103694)
    Let M be an orientable, cusped hyperbolic 3-manifold of finite volume. We show that the restriction map from a Dehn surgery component in the PSL(2,C)-character variety of M to the character variety of the boundary of M is a birational isomorphism onto its image. This generalises a result by Nathan Dunfield. A key step in our proof is the exactness of Craig Hodgson's volume differential on the eigenvalue variety.

    Even triangulations of n-dimensional pseudo-manifolds

    Rubinstein, J. Hyam; Tillmann, Stephan (2014)
    Projects: ARC | Discovery Projects - Grant ID: DP130103694 (DP130103694)
    This paper introduces even triangulations of n-dimensional pseudo-manifolds and links their combinatorics to the topology of the pseudo-manifolds. This is done via normal hypersurface theory and the study of certain symmetric representation. In dimension 3, necessary and sufficient conditions for the existence of even triangulations having one or two vertices are given. For Haken n-manifolds, an interesting connection between very short hierarchies and even triangulations is observed.

    Triangulations of 3-manifolds with essential edges

    Hodgson, Craig D.; Rubinstein, J. Hyam; Segerman, Henry; Tillmann, Stephan (2014)
    Projects: ARC | Discovery Projects - Grant ID: DP130103694 (DP130103694)
    We define essential and strongly essential triangulations of 3-manifolds, and give four constructions using different tools (Heegaard splittings, hierarchies of Haken 3-manifolds, Epstein-Penner decompositions, and cut loci of Riemannian manifolds) to obtain triangulations with these properties under various hypotheses on the topology or geometry of the manifold. We also show that a semi-angle structure is a sufficient condition for a triangulation of a 3-manifold to be essential, and a stric...

    Asymmetric hyperbolic L-spaces, Heegaard genus, and Dehn filling

    Dunfield, Nathan M.; Hoffman, Neil R.; Licata, Joan E. (2014)
    Projects: ARC | Discovery Projects - Grant ID: DP130103694 (DP130103694)
    An L-space is a rational homology 3-sphere with minimal Heegaard Floer homology. We give the first examples of hyperbolic L-spaces with no symmetries. In particular, unlike all previously known L-spaces, these manifolds are not double branched covers of links in S^3. We prove the existence of infinitely many such examples (in several distinct families) using a mix of hyperbolic geometry, Floer theory, and verified computer calculations. Of independent interest is our technique for using inter...

    Multisections of piecewise linear manifolds

    Rubinstein, J. Hyam; Tillmann, Stephan (2016)
    Projects: ARC | Discovery Projects - Grant ID: DP130103694 (DP130103694)
    Recently Gay and Kirby described a new decomposition of smooth closed $4$-manifolds called a trisection. This paper generalises Heegaard splittings of $3$-manifolds and trisections of $4$-manifolds to all dimensions, using triangulations as a key tool. In particular, we prove that every closed piecewise linear $n$-manifold has a multisection, i.e. can be divided into $k+1$ $n$-dimensional $1$-handlebodies, where $n=2k+1$ or $n=2k$, such that intersections of the handlebodies have spines of sm...

    The 3D-index and normal surfaces

    Garoufalidis, Stavros; Hodgson, Craig; Hoffman, Neil; Rubinstein, Hyam (2016)
    Projects: ARC | Discovery Projects - Grant ID: DP130103694 (DP130103694)
    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 to...

    Decomposing Heegaard splittings along separating incompressible surfaces in 3-manifolds

    Ichihara, Kazuhiro; Ozawa, Makoto; Rubinstein, J. Hyam (2018)
    Projects: ARC | Discovery Projects - Grant ID: DP130103694 (DP130103694)
    In this paper, by putting a separating incompressible surface in a 3-manifold into Morse position relative to the height function associated to a strongly irreducible Heegaard splitting, we show that an incompressible subsurface of the Heegaard splitting can be found, by decomposing the 3-manifold along the separating surface. Further if the Heegaard surface is of Hempel distance at least 4, then there is a pair of such subsurfaces on both sides of the given separating surface. This gives a p...

    Verified computations for hyperbolic 3-manifolds

    Hoffman, Neil; Ichihara, Kazuhiro; Kashiwagi, Masahide; Masai, Hidetoshi; Oishi, Shin'ichi; Takayasu, Akitoshi (2013)
    Projects: ARC | Discovery Projects - Grant ID: DP130103694 (DP130103694)
    For a given cusped 3-manifold $M$ admitting an ideal triangulation, we describe a method to rigorously prove that either $M$ or a filling of $M$ admits a complete hyperbolic structure via verified computer calculations. Central to our method are an implementation of interval arithmetic and Krawczyk's Test. These techniques represent an improvement over existing algorithms as they are faster, while accounting for error accumulation in a more direct and user friendly way.

    Bounds for the genus of a normal surface

    Jaco, William; Johnson, Jesse; Spreer, Jonathan; Tillmann, Stephan (2014)
    Projects: ARC | Discovery Projects - Grant ID: DP130103694 (DP130103694)
    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 theor...
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